{"title":"Components of domino tilings under flips in quadriculated tori","authors":"Qianqian Liu , Yaxian Zhang , Heping Zhang","doi":"10.1016/j.disc.2025.114396","DOIUrl":"10.1016/j.disc.2025.114396","url":null,"abstract":"<div><div>In a region <em>R</em> consisting of unit squares, a (domino) tiling is a collection of dominoes (the union of two adjacent squares) which pave fully the region. The flip graph of <em>R</em> is defined on the set of all tilings of <em>R</em> where two tilings are adjacent if we change one from the other by a flip (a <span><math><msup><mrow><mn>90</mn></mrow><mrow><mo>∘</mo></mrow></msup></math></span> rotation of a pair of side-by-side dominoes). If <em>R</em> is simply-connected, then its flip graph is connected. By using homology and cohomology, Saldanha, Tomei, Casarin and Romualdo obtained a criterion to decide if two tilings are in the same component of flip graph of quadriculated surface. By a graph-theoretic method, we obtain that the flip graph of a non-bipartite quadriculated torus consists of two isomorphic components. As an application, we obtain that the forcing numbers of all perfect matchings of each non-bipartite quadriculated torus form an integer-interval. For a bipartite quadriculated torus, the components of the flip graph is more complicated, and we use homology to obtain a general lower bound for the number of components of its flip graph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114396"},"PeriodicalIF":0.7,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143352435","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":"Flag-transitive point-primitive quasi-symmetric 2-designs with block intersection numbers 0 and y ≤ 10","authors":"Jianbing Lu , Yu Zhuang","doi":"10.1016/j.disc.2025.114398","DOIUrl":"10.1016/j.disc.2025.114398","url":null,"abstract":"<div><div>In this paper, we show that for a non-trivial quasi-symmetric 2-design <span><math><mi>D</mi></math></span> with two block intersection numbers <span><math><mi>x</mi><mo>=</mo><mn>0</mn></math></span> and <span><math><mn>2</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>10</mn></math></span>, if <span><math><mi>G</mi><mo>≤</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>D</mi><mo>)</mo></math></span> is flag-transitive and point-primitive, then <em>G</em> is either of affine type or almost simple type. Moreover, we prove that the socle of <em>G</em> cannot be an alternating group. If the socle of <em>G</em> is a sporadic group, then <span><math><mi>D</mi></math></span> and <em>G</em> must be one of the following: <span><math><mi>D</mi></math></span> is a 2-<span><math><mo>(</mo><mn>12</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></span> design with block intersection numbers <span><math><mn>0</mn><mo>,</mo><mn>3</mn></math></span> and <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>11</mn></mrow></msub></math></span>, or <span><math><mi>D</mi></math></span> is a 2-<span><math><mo>(</mo><mn>22</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>5</mn><mo>)</mo></math></span> design with block intersection numbers <span><math><mn>0</mn><mo>,</mo><mn>2</mn></math></span> and <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>22</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>:</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114398"},"PeriodicalIF":0.7,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221730","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":"An algorithm for packing hypertrees","authors":"Mourad Baïou , Francisco Barahona","doi":"10.1016/j.disc.2025.114397","DOIUrl":"10.1016/j.disc.2025.114397","url":null,"abstract":"<div><div>We present a combinatorial algorithm for determining a maximum packing of hypertrees in a capacitated hypergraph. This is an algorithmic proof of a theorem by Frank et al. <span><span>[7]</span></span>. This allows the extension of several algorithms developed for graphs to hypergraphs, for the <em>k</em>-cut problem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114397"},"PeriodicalIF":0.7,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143289902","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 the largest independent sets in the Kneser graph on chambers of PG(4,q)","authors":"Philipp Heering","doi":"10.1016/j.disc.2024.114392","DOIUrl":"10.1016/j.disc.2024.114392","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> be the graph whose vertices are the chambers of the finite projective 4-space <span><math><mi>PG</mi><mo>(</mo><mn>4</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, with two vertices being adjacent if the corresponding chambers are in general position. For <span><math><mi>q</mi><mo>≥</mo><mn>749</mn></math></span> we show that <span><math><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>2</mn><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> is the independence number of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and the geometric structure of the largest independent sets is described.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114392"},"PeriodicalIF":0.7,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143289905","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":"Even pairs in Berge graphs with no balanced skew-partitions","authors":"Tara Abrishami , Maria Chudnovsky , Yaqian Tang","doi":"10.1016/j.disc.2024.114388","DOIUrl":"10.1016/j.disc.2024.114388","url":null,"abstract":"<div><div>Let <em>G</em> be a Berge graph that has no odd prism and no antihole of length at least six as an induced subgraph. We show that every such graph <em>G</em> with no balanced skew-partition is either complete or has an even pair.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114388"},"PeriodicalIF":0.7,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221732","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":"Elementary derivations of the Rogers-Fine identity and other q-series identities","authors":"Heng Huat Chan , Song Heng Chan , Zhi-Guo Liu","doi":"10.1016/j.disc.2024.114387","DOIUrl":"10.1016/j.disc.2024.114387","url":null,"abstract":"<div><div>We begin the article with a proof of the Rogers-Fine identity. We then show that the Rogers-Fine identity implies the Rogers-Ramanujan identities as well as a new finite version of the quintuple identity. Motivated by the connections between these identities, we discover an identity which yields proofs of Rogers-Ramanujan-type identities associated with the Rogers-Ramanujan continued fraction, the Ramanujan-Göllnitz-Gordon continued fraction and Ramanujan's cubic continued fraction. We also discover a new generalization of the quintuple product identity which leads to a generalization of an identity due to R.J. Evans and a short proof of <em>q</em>-Chu-Vandermonde identity that does not require the knowledge of the <em>q</em>-binomial theorem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114387"},"PeriodicalIF":0.7,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143352436","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}
Gregory Baimetov, Ryan Bushling, Ansel Goh, Raymond Guo, Owen Jacobs, Sean Lee
{"title":"A decomposition theorem for balanced measures","authors":"Gregory Baimetov, Ryan Bushling, Ansel Goh, Raymond Guo, Owen Jacobs, Sean Lee","doi":"10.1016/j.disc.2024.114389","DOIUrl":"10.1016/j.disc.2024.114389","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> be a connected graph. A probability measure <em>μ</em> on <em>V</em> is called <em>balanced</em> if it has the following property: if <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo></math></span> denotes the “earth mover's” cost of transporting all the mass of <em>μ</em> from all over the graph to the vertex <em>v</em>, then <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span> attains its global maximum at each point in the support of <em>μ</em>. We prove a decomposition result that characterizes balanced measures as convex combinations of suitable “extremal” balanced measures that we call <em>basic</em>. An upper bound on the number of basic balanced measures on <em>G</em> follows, and an example shows that this estimate is essentially sharp.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114389"},"PeriodicalIF":0.7,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221697","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":"Spectral extremal graphs for fan graphs","authors":"Loujun Yu , Yongtao Li , Yuejian Peng","doi":"10.1016/j.disc.2024.114391","DOIUrl":"10.1016/j.disc.2024.114391","url":null,"abstract":"<div><div>A well-known result of Nosal states that a graph <em>G</em> with <em>m</em> edges and <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><msqrt><mrow><mi>m</mi></mrow></msqrt></math></span> contains a triangle. Nikiforov [Combin. Probab. Comput. 11 (2002)] extended this result to cliques by showing that if <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><msqrt><mrow><mn>2</mn><mi>m</mi><mo>(</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>r</mi><mo>)</mo></mrow></msqrt></math></span>, then <em>G</em> contains a copy of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> be the graph obtained from a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> by adding an edge to two vertices with distance two, and let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be the friendship graph consisting of <em>k</em> triangles that share a common vertex. Recently, Zhai, Lin and Shu [European J. Combin. 95 (2021)], Sun, Li and Wei [Discrete Math. 346 (2023)], and Li, Lu and Peng [Discrete Math. 346 (2023)] proved that if <span><math><mi>m</mi><mo>≥</mo><mn>8</mn></math></span> and <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>4</mn><mi>m</mi><mo>−</mo><mn>3</mn></mrow></msqrt><mo>)</mo></math></span>, then <em>G</em> contains a copy of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, respectively, unless <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∨</mo><mfrac><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. In this paper, we give a unified extension by showing that such a graph contains a copy of <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∨</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> is the join of a vertex and a path on four vertices. Our result extends the aforementioned results since <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> a","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114391"},"PeriodicalIF":0.7,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143289906","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}
Kathie Cameron , Aristotelis Chaniotis , Celina M.H. de Figueiredo , Sophie Spirkl
{"title":"The sandwich problem for odd-hole-free and even-hole-free graphs","authors":"Kathie Cameron , Aristotelis Chaniotis , Celina M.H. de Figueiredo , Sophie Spirkl","doi":"10.1016/j.disc.2024.114383","DOIUrl":"10.1016/j.disc.2024.114383","url":null,"abstract":"<div><div>For a property <span><math><mi>P</mi></math></span> of graphs, the <span><math><mi>P</mi></math></span>-<span>Sandwich-Problem</span>, introduced by Golumbic and Shamir (1993), is the following: Given a pair of graphs <span><math><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> on the same vertex set <em>V</em>, does there exist a graph <em>G</em> such that <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>V</mi></math></span>, <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>⊆</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>⊆</mo><mi>E</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, and <em>G</em> satisfies <span><math><mi>P</mi></math></span>? A <em>hole</em> in a graph is an induced subgraph which is a cycle of length at least four. An odd (respectively even) hole is a hole of odd (respectively even) length. Given a class of graphs <span><math><mi>C</mi></math></span> and a graph <em>G</em> we say that <em>G</em> is <span><math><mi>C</mi></math></span><em>-free</em> if it contains no induced subgraph isomorphic to a member of <span><math><mi>C</mi></math></span>. In this paper we prove that if <span><math><mi>P</mi></math></span> is the property of being odd-hole-free or the property of being even-hole-free, then the <span><math><mi>P</mi></math></span>-<span>Sandwich-Problem</span> is <span><math><mtext>NP</mtext></math></span>-complete.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114383"},"PeriodicalIF":0.7,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143221696","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":"q-Parikh matrices and q-deformed binomial coefficients of words","authors":"Antoine Renard, Michel Rigo , Markus A. Whiteland","doi":"10.1016/j.disc.2024.114381","DOIUrl":"10.1016/j.disc.2024.114381","url":null,"abstract":"<div><div>We have introduced a <em>q</em>-deformation, i.e., a polynomial in <em>q</em> with natural coefficients, of the binomial coefficient of two finite words <em>u</em> and <em>v</em> counting the number of occurrences of <em>v</em> as a subword of <em>u</em>. In this paper, we examine the <em>q</em>-deformation of Parikh matrices as introduced by Eğecioğlu in 2004.</div><div>Many classical results concerning Parikh matrices generalize to this new framework: Our first important observation is that the elements of such a matrix are in fact <em>q</em>-deformations of binomial coefficients of words. We also study their inverses and we obtain new identities about <em>q</em>-binomials.</div><div>For a finite word <em>z</em> and for the sequence <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> of prefixes of an infinite word, we show that the polynomial sequence <span><math><msub><mrow><mo>(</mo><mtable><mtr><mtd><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> converges to a formal series. We present links with additive number theory and <em>k</em>-regular sequences. In the case of a periodic word <span><math><msup><mrow><mi>u</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>, we generalize a result of Salomaa: the sequence <span><math><msub><mrow><mo>(</mo><mtable><mtr><mtd><msup><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msup></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> satisfies a linear recurrence relation with polynomial coefficients. Related to the theory of integer partition, we describe the growth and the zero set of the coefficients of the series associated with <span><math><msup><mrow><mi>u</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>.</div><div>Finally, we show that the minors of a <em>q</em>-Parikh matrix are polynomials with natural coefficients and consider a generalization of Cauchy's inequality. We also compare <em>q</em>-Parikh matrices associated with an arbitrary word with those associated with a canonical word <span><math><mn>12</mn><mo>⋯</mo><mi>k</mi></math></span> made of pairwise distinct symbols.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 5","pages":"Article 114381"},"PeriodicalIF":0.7,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143289904","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
