{"title":"Separating hash families with large universe","authors":"Xin Wei , Xiande Zhang , Gennian Ge","doi":"10.1016/j.jcta.2025.106075","DOIUrl":"10.1016/j.jcta.2025.106075","url":null,"abstract":"<div><div>Separating hash families are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory. Let <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> denote the maximum size of universe for a <em>t</em>-perfect hash family of length <em>N</em> over an alphabet of size <em>q</em>. In this paper, we show that <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><mo>−</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo><</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><mi>o</mi><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> for all <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span>, which answers an open problem about separating hash families raised by Blackburn et al. in 2008 for certain parameters. Previously, this result was known only for <span><math><mi>t</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span>. Our proof is obtained by establishing the existence of a large set of integers avoiding nontrivial solutions to a set of correlated linear equations.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106075"},"PeriodicalIF":0.9,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Vertex-transitive graphs with small motion and transitive permutation groups with small minimal degree","authors":"Antonio Montero , Primož Potočnik","doi":"10.1016/j.jcta.2025.106065","DOIUrl":"10.1016/j.jcta.2025.106065","url":null,"abstract":"<div><div>The motion of a graph is the minimum number of vertices that are moved by a non-trivial automorphism. Equivalently, it can be defined as the minimal degree of its automorphism group (as a permutation group on the vertices). In this paper, we develop some results on permutation groups (primitive and imprimitive) with small minimal degree. As a consequence of such results, we classify vertex-transitive graphs whose motion is 4 or a prime number.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106065"},"PeriodicalIF":0.9,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144106349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Studying the divisibility of power LCM matrices by power GCD matrices on gcd-closed sets","authors":"Jianrong Zhao , Chenxu Wang , Yu Fu","doi":"10.1016/j.jcta.2025.106063","DOIUrl":"10.1016/j.jcta.2025.106063","url":null,"abstract":"<div><div>Let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> be a gcd-closed set (i.e. <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>S</mi></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets <em>S</em> with <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>≥</mo><mn>4</mn></math></span> such that the GCD matrix (<em>S</em>) divides the LCM matrix <span><math><mo>[</mo><mi>S</mi><mo>]</mo></math></span> in the ring <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. For <span><math><mi>x</mi><mo>∈</mo><mi>S</mi></math></span>, let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>S</mi><mo>:</mo><mi>z</mi><mo><</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>|</mo><mi>x</mi><mtext> and </mtext><mo>(</mo><mi>z</mi><mo>|</mo><mi>y</mi><mo>|</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>)</mo><mo>⇒</mo><mi>y</mi><mo>∈</mo><mo>{</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>}</mo><mo>}</mo></math></span>. In 2009, Feng, Hong and Zhao answered this problem in the context where <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≤</mo><mn>2</mn></math></span>. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set <em>S</em> with <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>=</mo><mn>3</mn></math></span> such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span>. Meanwhile, they raised a conjecture on the necessary and sufficient condition such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span> holds for the remaining case <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≥</mo><mn>4</mn></math></span>. In this paper, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely.</d","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106063"},"PeriodicalIF":0.9,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complete 3-term arithmetic progression free sets of small size in vector spaces and other abelian groups","authors":"Bence Csajbók , Zoltán Lóránt Nagy","doi":"10.1016/j.jcta.2025.106061","DOIUrl":"10.1016/j.jcta.2025.106061","url":null,"abstract":"<div><div>A subset <em>S</em> of an abelian group <em>G</em> is called 3-AP free if it does not contain a three term arithmetic progression. Moreover, <em>S</em> is called complete 3-AP free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a 3-AP free set, which is necessarily complete. In this paper we are interested in the minimum size of complete 3-AP free sets. We define and study saturation w.r.t. 3-APs and present constructions of small complete 3-AP free sets and 3-AP saturating sets for several families of vector spaces and cyclic groups.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106061"},"PeriodicalIF":0.9,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aleksei L. Perezhogin , Vladimir N. Potapov , Anna A. Taranenko , Sergey Yu. Vladimirov
{"title":"Characterization of polystochastic matrices of order 4 with zero permanent","authors":"Aleksei L. Perezhogin , Vladimir N. Potapov , Anna A. Taranenko , Sergey Yu. Vladimirov","doi":"10.1016/j.jcta.2025.106060","DOIUrl":"10.1016/j.jcta.2025.106060","url":null,"abstract":"<div><div>A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to 1. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if <em>d</em> is even, then the permanent of a <em>d</em>-dimensional polystochastic matrix of order 4 is positive, and for odd <em>d</em>, we give a complete characterization of <em>d</em>-dimensional polystochastic matrices with zero permanent.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106060"},"PeriodicalIF":0.9,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks","authors":"Sho Kubota , Kiyoto Yoshino","doi":"10.1016/j.jcta.2025.106064","DOIUrl":"10.1016/j.jcta.2025.106064","url":null,"abstract":"<div><div>We completely characterize circulant graphs with valency up to 4 that admit perfect state transfer. Those of valency 3 do not admit it. On the other hand, circulant graphs with valency 4 admit perfect state transfer only in two infinite families: one discovered by Zhan and another new family, while no others do. The main tools for deriving these results are symmetry of graphs and eigenvalues. We describe necessary conditions for perfect state transfer to occur based on symmetry of graphs, which mathematically refers to automorphisms of graphs. As for eigenvalues, if perfect state transfer occurs, then certain eigenvalues of the corresponding isotropic random walks must be the halves of algebraic integers. Taking this into account, we utilize known results on the rings of integers of cyclotomic fields.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106064"},"PeriodicalIF":0.9,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Proof of Frankl's conjecture on cross-intersecting families","authors":"Yongjiang Wu, Lihua Feng, Yongtao Li","doi":"10.1016/j.jcta.2025.106062","DOIUrl":"10.1016/j.jcta.2025.106062","url":null,"abstract":"<div><div>Two families <span><math><mi>F</mi></math></span> and <span><math><mi>G</mi></math></span> are called cross-intersecting if for every <span><math><mi>F</mi><mo>∈</mo><mi>F</mi></math></span> and <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span>, the intersection <span><math><mi>F</mi><mo>∩</mo><mi>G</mi></math></span> is non-empty. For any positive integers <em>n</em> and <em>k</em>, let <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span> denote the family of all <em>k</em>-element subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. Let <span><math><mi>t</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi></math></span> be non-negative integers with <span><math><mi>k</mi><mo>≥</mo><mi>s</mi><mo>+</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>+</mo><mi>t</mi></math></span>. In 2016, Frankl proved that if <span><math><mi>F</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span> and <span><math><mi>G</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> are cross-intersecting families, and <span><math><mi>F</mi></math></span> is <span><math><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-intersecting and <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≥</mo><mn>1</mn></math></span>, then <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn></math></span>. Furthermore, Frankl conjectured that under an additional condition <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>⊆</mo><mi>F</mi></math></span>, the following inequality holds:<span><span><span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munderover><mo>∑</mo><mr","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106062"},"PeriodicalIF":0.9,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Kernels for storage capacity and dual index coding","authors":"Ishay Haviv","doi":"10.1016/j.jcta.2025.106059","DOIUrl":"10.1016/j.jcta.2025.106059","url":null,"abstract":"<div><div>The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information.</div><div>In this paper, we initiate the study of the <figure><img></figure> and <figure><img></figure> problems from the perspective of parameterized complexity. We prove that the <figure><img></figure> problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the <figure><img></figure> problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes <span><span>[14]</span></span>, <span><span>[11]</span></span>. As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee <span><span>[13]</span></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106059"},"PeriodicalIF":0.9,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yuefeng Yang , Akihiro Munemasa , Kaishun Wang , Wenying Zhu
{"title":"Weakly distance-regular circulants, I","authors":"Yuefeng Yang , Akihiro Munemasa , Kaishun Wang , Wenying Zhu","doi":"10.1016/j.jcta.2025.106051","DOIUrl":"10.1016/j.jcta.2025.106051","url":null,"abstract":"<div><div>We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106051"},"PeriodicalIF":0.9,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dean Crnković , Maarten De Boeck , Francesco Pavese , Andrea Švob
{"title":"q-Analogs of divisible design graphs and Deza graphs","authors":"Dean Crnković , Maarten De Boeck , Francesco Pavese , Andrea Švob","doi":"10.1016/j.jcta.2025.106047","DOIUrl":"10.1016/j.jcta.2025.106047","url":null,"abstract":"<div><div>Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of <em>q</em>-analogs of divisible design graphs and show that all <em>q</em>-analogs of divisible design graphs come from spreads, and are actually <em>q</em>-analogs of strongly regular graphs.</div><div>Deza graphs were introduced by Erickson, Fernando, Haemers, Hardy and Hemmeter in 1999. In this paper, we introduce <em>q</em>-analogs of Deza graphs. Further, we determine possible parameters, give examples of <em>q</em>-analogs of Deza graphs and characterize all non-strongly regular <em>q</em>-analogs of Deza graphs with the smallest parameters.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106047"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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