{"title":"Dissections of lacunary eta quotients and identically vanishing coefficients","authors":"Tim Huber , James McLaughlin , Dongxi Ye","doi":"10.1016/j.aam.2025.102902","DOIUrl":"10.1016/j.aam.2025.102902","url":null,"abstract":"<div><div>For any function <span><math><mi>A</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> define<span><span><span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></msub><mo>:</mo><mo>=</mo><mo>{</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>:</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>}</mo><mo>.</mo></math></span></span></span> Now suppose <span><math><mi>C</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><mi>D</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> are two functions whose <em>m</em>-dissections are given by<span><span><span><math><mi>C</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>q</mi><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>,</mo></math></span></span></span><span><span><span><math><mi>D</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>q</mi><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mo>.</mo></math></span></span></span> If it is the case that <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>⟺</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span>, <span><math><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span>, then we say that <span><math><mi>C</mi><mo>(</mo><mi","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102902"},"PeriodicalIF":1.0,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143888196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Influence of the automorphism group of a graph on its PageRank scores of vertices","authors":"Dein Wong , Qi Zhou , Xinlei Wang","doi":"10.1016/j.aam.2025.102900","DOIUrl":"10.1016/j.aam.2025.102900","url":null,"abstract":"<div><div>Google's success derives in large part from its PageRank algorithm, which assign a score to every web page according to its importance. Recently, G. Modjtaba et al. (2021) <span><span>[19]</span></span> proved that similar vertices in a graph have the same PageRank score and they proposed a conjecture, suspecting that two graphs are completely non-Co-PR if they are non-Co-PR graphs. The investigation of this paper mainly concerns the influence of the automorphism group of a graph on its PageRank scores of vertices. The main results of this article are as follows.<ul><li><span>1.</span><span><div>Based on matrix analysis, two conditions on what kinds of vertices have the same PageRank score are obtained.</div></span></li><li><span>2.</span><span><div>Four techniques for constructing Co-PR graphs are established.</div></span></li><li><span>3.</span><span><div>A non-regular connected graph of order <em>n</em>, with <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span> as PR scores of most of its vertices, is constructed, which provides a negative answer to Modjtaba's conjecture above.</div></span></li></ul></div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102900"},"PeriodicalIF":1.0,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143855383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Counting flows of b-compatible graphs","authors":"Houshan Fu , Xiangyu Ren , Suijie Wang","doi":"10.1016/j.aam.2025.102901","DOIUrl":"10.1016/j.aam.2025.102901","url":null,"abstract":"<div><div>Kochol introduced the assigning polynomial <span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></math></span> to count nowhere-zero <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-flows of a graph <em>G</em>, where <em>A</em> is a finite Abelian group and <em>α</em> is a <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning from a family <span><math><mi>Λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of certain nonempty vertex subsets of <em>G</em> to <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>. We introduce the concepts of <em>b</em>-compatible graph and <em>b</em>-compatible broken bond to give an explicit formula for the assigning polynomials and to examine their coefficients. More specifically, for a function <span><math><mi>b</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span>, let <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> be a <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning of <em>G</em> such that for each <span><math><mi>X</mi><mo>∈</mo><mi>Λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> if and only if <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>X</mi></mrow></msub><mi>b</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. We show that for any <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-assigning <em>α</em> of <em>G</em>, if there exists a function <span><math><mi>b</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span> such that <em>G</em> is <em>b</em>-compatible and <span><math><mi>α</mi><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>G</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span>, then the assigning polynomial <span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></math></span> has the <em>b</em>-compatible spanning subgraph expansion<span><span><span><math><mi>F</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mi>S</mi><mo>⊆</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mi>G</mi><mo>−</mo><mi>S</mi><mrow><mtext> is</mtext><mspace></mspace><mtext>b</mtext><mtext>-compatible</mtext></mrow></mtd></mtr></mtable></mrow></munder><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></msup><msup><mrow><mi>k</mi></mrow><mrow><mi>m</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></msup><mo>,</mo></math></span></span></span> and is the following form<span><span><span><math><mi>F</mi>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102901"},"PeriodicalIF":1.0,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The maximum number of cycles in a triangular-grid billiards system with a given perimeter","authors":"Honglin Zhu","doi":"10.1016/j.aam.2025.102888","DOIUrl":"10.1016/j.aam.2025.102888","url":null,"abstract":"<div><div>Given a grid polygon <em>P</em> in a grid of equilateral triangles, Defant and Jiradilok considered a billiards system where beams of light bounce around inside <em>P</em>. We study the relationship between the perimeter <span><math><mi>perim</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> of <em>P</em> and the number of different trajectories <span><math><mi>cyc</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> that the billiards system has. Resolving a conjecture of Defant and Jiradilok, we prove the sharp inequality <span><math><mi>cyc</mi><mo>(</mo><mi>P</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mi>perim</mi><mo>(</mo><mi>P</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>4</mn></math></span> and characterize the equality cases.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102888"},"PeriodicalIF":1.0,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Colored q-Stirling and q-Lah numbers: A new view continued","authors":"Sen-Peng Eu , Louis Kao , Juei-Yin Lin","doi":"10.1016/j.aam.2025.102889","DOIUrl":"10.1016/j.aam.2025.102889","url":null,"abstract":"<div><div>Cai and Readdy proposed a new framework for studying the <em>q</em>-analogue <span><math><mi>f</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> of a combinatorial structure <em>S</em>. Specifically, the aim is to identify two statistics over <em>S</em> and a proper subset <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <em>S</em> such that <span><math><mi>f</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> represents the <em>q</em>-<span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>)</mo></math></span>-expansion over <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, and to explore the poset and topological interpretations of this expansion. Cai and Readdy provided comprehensive profiles for classical Stirling numbers of both kinds within this framework. In this work, we extend Cai and Readdy's results to colored <em>q</em>-Stirling numbers of both kinds, as well as colored <em>q</em>-Lah numbers. We also briefly discuss <em>q</em>-Stirling and <em>q</em>-Lah numbers of type <em>D</em>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102889"},"PeriodicalIF":1.0,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dependency equilibria: Boundary cases and their real algebraic geometry","authors":"Irem Portakal , Daniel Windisch","doi":"10.1016/j.aam.2025.102890","DOIUrl":"10.1016/j.aam.2025.102890","url":null,"abstract":"<div><div>This paper is a significant step forward in understanding dependency equilibria within the framework of real algebraic geometry encompassing both pure and mixed equilibria. In alignment with Spohn's original definition of dependency equilibria, we propose two alternative definitions, allowing for an algebro-geometric comprehensive study of all dependency equilibria. We give a sufficient condition for the existence of a pure dependency equilibrium and show that every Nash equilibrium lies on the Spohn variety, the algebraic model for dependency equilibria. For generic games, the set of real points of the Spohn variety is Zariski dense. Furthermore, every Nash equilibrium in this case is a dependency equilibrium. Finally, we present a detailed analysis of the geometric structure of dependency equilibria for <span><math><mo>(</mo><mn>2</mn><mo>×</mo><mn>2</mn><mo>)</mo></math></span>-games.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102890"},"PeriodicalIF":1.0,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Boolean, free, and classical cumulants as tree enumerations","authors":"Colin Defant, Mitchell Lee","doi":"10.1016/j.aam.2025.102899","DOIUrl":"10.1016/j.aam.2025.102899","url":null,"abstract":"<div><div>Defant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called <em>troupes</em>. Using a generalization of troupes that we call <em>weighted troupes</em>, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his <em>troupe transform</em>. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102899"},"PeriodicalIF":1.0,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143792352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The extra basis in noncommuting variables","authors":"Farid Aliniaeifard, Stephanie van Willigenburg","doi":"10.1016/j.aam.2025.102887","DOIUrl":"10.1016/j.aam.2025.102887","url":null,"abstract":"<div><div>We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the <em>x</em>-basis in the Hopf algebra of symmetric functions in noncommuting variables, NCSym, which arises in the theory of Grothendieck bialgebras. We achieve this by applying the theory of Hopf monoids and the Fock functor. We also determine combinatorial expansions of this basis in terms of the monomial and power sum symmetric functions in NCSym, and by taking the commutative image of the <em>x</em>-basis we discover a new multiplicative basis for the algebra of symmetric functions.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102887"},"PeriodicalIF":1.0,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A composition method for neat formulas of chromatic symmetric functions","authors":"David G.L. Wang , James Z.F. Zhou","doi":"10.1016/j.aam.2025.102886","DOIUrl":"10.1016/j.aam.2025.102886","url":null,"abstract":"<div><div>We develop a composition method to unearth positive <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>I</mi></mrow></msub></math></span>-expansions of chromatic symmetric functions <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, where the subscript <em>I</em> stands for compositions rather than integer partitions. Using this method, we derive positive and neat <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>I</mi></mrow></msub></math></span>-expansions for the chromatic symmetric functions of tadpoles, barbells and generalized bulls, and establish the <em>e</em>-positivity of hats. We also obtain a compact ribbon Schur analog for the chromatic symmetric function of cycles.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102886"},"PeriodicalIF":1.0,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Poisson approximation for large permutation groups","authors":"Persi Diaconis , Nathan Tung","doi":"10.1016/j.aam.2025.102883","DOIUrl":"10.1016/j.aam.2025.102883","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> be a group of permutations of <em>kn</em> objects which permutes things independently in disjoint blocks of size <em>k</em> and then permutes the blocks. We investigate the probabilistic and enumerative aspects of random elements of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. This includes novel limit theorems for cycles of various lengths, number of cycles, and inversions. The limits include compound Poisson distributions with interesting dependence structure.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102883"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143714822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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