{"title":"Automatic sequences and parity of partition functions","authors":"Shi-Chao Chen","doi":"10.1016/j.aam.2025.102869","DOIUrl":"10.1016/j.aam.2025.102869","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> be an integer, <em>ℓ</em> a prime and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> the finite field with <em>ℓ</em> elements. A sequence <span><math><msub><mrow><mo>(</mo><mi>a</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> is called <em>k</em>-automatic if there exists a deterministic finite automaton with output that reads the canonical base-<em>k</em> representation of <em>n</em> and the outputs <span><math><mi>a</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. We apply the properties of automatic sequences to prove the transcendence of a formal power series over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> related to infinite products. As applications, the parity results of various partition functions are obtained, including the root partition function and the prime parts partition function. We also establish the transcendence of the power series associated with holomorphic modular forms with integer coefficients.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102869"},"PeriodicalIF":1.0,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480729","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":"Stable multivariate Narayana polynomials and labeled plane trees","authors":"Harold R.L. Yang , Philip B. Zhang","doi":"10.1016/j.aam.2025.102867","DOIUrl":"10.1016/j.aam.2025.102867","url":null,"abstract":"<div><div>In this paper, we introduce stable multivariate generalizations of Narayana polynomials of types <em>A</em> and <em>B</em>. We give an insertion algorithm for labeled plane trees and introduce the notion of improper edges. Our polynomials are multivariate generating polynomials of labeled plane trees and can be generated by a grammatical labeling based on a context-free grammar. Our proof of real stability uses a characterization of stable-preserving linear operators due to Borcea and Brändén. In particular, we get an alternative multivariate stable refinement of the second-order Eulerian polynomials, which is different from the one given by Haglund and Visontai.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102867"},"PeriodicalIF":1.0,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480728","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":"Coefficients of the Tutte polynomial and minimal edge cuts of a graph","authors":"Haiyan Chen, Mingxu Guo","doi":"10.1016/j.aam.2025.102868","DOIUrl":"10.1016/j.aam.2025.102868","url":null,"abstract":"<div><div>Let <em>G</em> be a <span><math><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-edge connected graph with order <em>n</em> and size <em>m</em>. From a general result on the coefficients of polymatroid Tutte polynomial, Guan et al. (2023) <span><span>[16]</span></span> derived that<span><span><span><math><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>g</mi><mo>−</mo><mi>i</mi></mrow></msup><mo>]</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is the Tutte polynomial of <em>G</em> and <span><math><mi>g</mi><mo>=</mo><mi>m</mi><mo>−</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span>. Recall that the coefficients of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>)</mo></math></span> have many combinatorial explanations, including spanning trees, parking functions, superstable configurations (or recurrent configurations) of the Abelian Sandpile Model (ASM), and so on. Here we find that the above result has a simple and direct proof in terms of the superstable configurations of ASM. Motivated by this, in this paper, by constructing mappings between different sets, we first establish a relationship between non-superstable configurations and minimal edge cuts of <em>G</em>, then we generalize the above result from <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span> to <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo><</mo><mfrac><mrow><mn>3</mn><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. In precise,<span><span><span><math><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>g</mi><mo>−</mo><mi>i</mi></mrow></msup><mo>]</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>i</mi></mrow></munderover><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mo>−</mo><mi>j</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>|</mo><msub><mrow><mi>EC</mi></mrow><mrow><mi>j</mi></mrow>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102868"},"PeriodicalIF":1.0,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453998","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":"Inconsequential results on the Merino-Welsh conjecture for Tutte polynomials","authors":"Joseph P.S. Kung","doi":"10.1016/j.aam.2025.102866","DOIUrl":"10.1016/j.aam.2025.102866","url":null,"abstract":"<div><div>We present two sufficient conditions on a rank-<em>r</em> coloop-free matroid <em>M</em> for the additive Merino-Welsh inequality to hold. The first is that the density of <em>M</em> is sufficiently large. The second is that all the cocircuits of <em>M</em> have at least <span><math><mi>r</mi><mo>+</mo><mn>1</mn></math></span> elements. These results are “inconsequential” in the sense that although they show that a version of the conjecture holds for many matroids, they are far from covering all the possible cases.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102866"},"PeriodicalIF":1.0,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437101","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 excluded minors for GF(5)-representable matroids on ten elements","authors":"Nick Brettell","doi":"10.1016/j.aam.2025.102864","DOIUrl":"10.1016/j.aam.2025.102864","url":null,"abstract":"<div><div>Mayhew and Royle (2008) showed that there are 564 excluded minors for the class of <span><math><mrow><mi>GF</mi></mrow><mo>(</mo><mn>5</mn><mo>)</mo></math></span>-representable matroids having at most 9 elements. We enumerate the excluded minors for <span><math><mrow><mi>GF</mi></mrow><mo>(</mo><mn>5</mn><mo>)</mo></math></span>-representable matroids having 10 elements: there are precisely 2128 such excluded minors. In the process we find, for each <span><math><mi>i</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></math></span>, the excluded minors for the class of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-representable matroids having at most 10 elements, and the excluded minors for the class of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-representable matroids having at most 13 elements.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102864"},"PeriodicalIF":1.0,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437100","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":"Explicit identities on zeta values over imaginary quadratic fields","authors":"Soumyarup Banerjee , Rahul Kumar","doi":"10.1016/j.aam.2025.102865","DOIUrl":"10.1016/j.aam.2025.102865","url":null,"abstract":"<div><div>In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field are quite well known in literature. We here exhibit the identities for both even and odd values of the Dedekind zeta function over an imaginary quadratic field which are analogous to Ramanujan's identities for even and odd zeta values over <span><math><mi>Q</mi></math></span>. Moreover, any complex zeta values over imaginary quadratic field may also be evaluated from our identities.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102865"},"PeriodicalIF":1.0,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445391","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":"On (joint) equidistributions of mesh patterns 123 and 132 with symmetric shadings","authors":"Shuzhen Lv , Sergey Kitaev","doi":"10.1016/j.aam.2025.102856","DOIUrl":"10.1016/j.aam.2025.102856","url":null,"abstract":"<div><div>A notable problem within permutation patterns that has attracted considerable attention in literature since 1973 is the search for a bijective proof demonstrating that 123-avoiding and 132-avoiding permutations are equinumerous, both counted by the Catalan numbers. Despite this equivalence, the distributions of occurrences of the patterns 123 and 132 are distinct. When considering 123 and 132 as mesh patterns and selectively shading boxes, similar scenarios arise, even when avoidance is defined by the Bell numbers or other sequences, rather than the Catalan numbers.</div><div>However, computer experiments suggest that mesh patterns 123 and 132 may indeed be jointly equidistributed. Furthermore, by considering symmetric shadings relative to the diagonal, a maximum of 93 equidistributed pairs can potentially exist. This paper establishes 75 joint equidistributions, leaving the justification of the remaining cases as open problems. As a by-product, we also prove 36 relevant non-symmetric joint equidistributions. All our proofs are bijective and involve swapping occurrences of the patterns in question, thereby demonstrating their joint equidistribution. Our findings are a continuation of the systematic study of distributions of short-length mesh patterns initiated by Kitaev and Zhang in 2019.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102856"},"PeriodicalIF":1.0,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394991","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":"Generalized Morse theory of distance functions to surfaces for persistent homology","authors":"Anna Song , Ka Man Yim , Anthea Monod","doi":"10.1016/j.aam.2025.102857","DOIUrl":"10.1016/j.aam.2025.102857","url":null,"abstract":"<div><div>This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology to characterize geometric and topological patterns in shapes. The most well-known and widely applied tool from this approach is persistent homology, which tracks the evolution of topological features in a dynamic manner as a barcode. Morse theory is a framework from differential topology that studies critical points of functions on manifolds; it has been used to characterize the birth and death of persistent homology features. However, a significant limitation to Morse theory is that it cannot be readily applied to distance functions because distance functions lack smoothness, which is required in Morse theory. Our contribution to addressing this issue is two fold. First, we generalize Morse theory to Euclidean distance functions of bounded sets with smooth boundaries. We focus in particular on distance fields for shape representation and we study the persistent homology of shape textures using a sublevel set filtration induced by the signed distance function. We use transversality theory to prove that for generic embeddings of a smooth compact surface in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, signed distance functions admit finitely many non-degenerate critical points. This gives rise to our second contribution, which is that shapes and textures can both now be quantified and rigorously characterized in the language of persistent homology: signed distance persistence modules of generic shapes admit a finite barcode decomposition whose birth and death points can be classified and described geometrically. We use this approach to quantify shape textures on both simulated data and real vascular data from biology.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102857"},"PeriodicalIF":1.0,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387109","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}
Janet J.W. Dong , Lora R. Du , Kathy Q. Ji , Dax T.X. Zhang
{"title":"New refinements of Narayana polynomials and Motzkin polynomials","authors":"Janet J.W. Dong , Lora R. Du , Kathy Q. Ji , Dax T.X. Zhang","doi":"10.1016/j.aam.2025.102855","DOIUrl":"10.1016/j.aam.2025.102855","url":null,"abstract":"<div><div>Chen, Deutsch and Elizalde introduced a refinement of the Narayana polynomials by distinguishing between old (leftmost child) and young leaves of plane trees. They also provided a refinement of Coker's formula by constructing a bijection. In fact, Coker's formula establishes a connection between the Narayana polynomials and the Motzkin polynomials, which implies the <em>γ</em>-positivity of the Narayana polynomials. In this paper, we introduce the polynomial <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, which further refines the Narayana polynomials by considering leaves of plane trees that have no siblings. We obtain the generating function for <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. To achieve further refinement of Coker's formula based on the polynomial <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, we consider a refinement <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>;</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of the Motzkin polynomials by classifying the old leaves of a tip-augmented plane tree into three categories and the young leaves into two categories. The generating function for <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102855"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387108","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":"Computing triple and double Hurwitz numbers involving a branch point with a two-part profile","authors":"Zi-Wei Bai, Ricky X.F. Chen","doi":"10.1016/j.aam.2025.102854","DOIUrl":"10.1016/j.aam.2025.102854","url":null,"abstract":"<div><div>The study of Hurwitz numbers intersects with many research areas including representation theory, algebraic geometry and mathematical physics. Though many beautiful general properties have been discovered, obtaining explicit elementary expressions computing these numbers is hard and pertains to a primary goal of the topic. In fact, known explicit formulas are mainly for Hurwitz numbers involving at most two nonsimple branch points (i.e., double Hurwitz numbers). Even for double Hurwitz numbers, only the case where one of the branch points is fully ramified (i.e., one-part double Hurwitz numbers) has been completely and explicitly determined. In this paper, we contribute explicit elementary formulas computing Hurwitz numbers with completed <em>r</em>-cycles involving up to three nonsimple branch points where one of them has a two-part profile, enriching several lines of researches. In particular, we discuss the piecewise polynomiality and the genus-zero case in detail.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"165 ","pages":"Article 102854"},"PeriodicalIF":1.0,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155600","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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