{"title":"The optimal upper bound on the MP-ratio for quaternary words","authors":"Kristina Ago, Bojan Bašić","doi":"10.1016/j.aam.2025.102984","DOIUrl":"10.1016/j.aam.2025.102984","url":null,"abstract":"<div><div>The so-called <em>MP-ratio</em> is a kind of measure of how “packed with palindromes” a given word is. The lower bound on the MP-ratio for the set of all <em>n</em>-ary words is (trivially) 1, while the best possible upper bound is an open problem in the general case. It is solved for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> (where the optimal upper bound is 4) and for <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span> (where the optimal upper bound is 6). Also, it is known that in the <em>n</em>-ary case the optimal bound is between 2<em>n</em> and the order of growth <span><math><mi>n</mi><msup><mrow><mn>2</mn></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>. In this article we solve this problem for quaternary words, for which we show that the best possible upper bound on the MP-ratio equals 8. We believe that this is the last case in which the result is 2<em>n</em>, that is, we believe that for <span><math><mi>n</mi><mo>⩾</mo><mn>5</mn></math></span> there are words whose MP-ratio is strictly larger than 2<em>n</em>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102984"},"PeriodicalIF":1.3,"publicationDate":"2025-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333347","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":"A study on T-equivalent graphs","authors":"Fengming Dong , Meiqiao Zhang","doi":"10.1016/j.aam.2025.102985","DOIUrl":"10.1016/j.aam.2025.102985","url":null,"abstract":"<div><div>In his article [<em>J. Comb. Theory Ser. B</em> <strong>16</strong> (1974), 168–174], Tutte called two graphs <em>T</em>-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs <em>G</em> and <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> are <em>T</em>-equivalent if <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is obtained from <em>G</em> by flipping a rotor (i.e., replacing it by its mirror) of order at most 5, where a rotor of order <em>k</em> in <em>G</em> is an induced subgraph <em>R</em> having an automorphism <em>ψ</em> with a vertex orbit <span><math><mo>{</mo><msup><mrow><mi>ψ</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>(</mo><mi>u</mi><mo>)</mo><mo>:</mo><mi>i</mi><mo>≥</mo><mn>0</mn><mo>}</mo></math></span> of size <em>k</em> such that every vertex of <em>R</em> is only adjacent to vertices in <em>R</em> unless it is in this vertex orbit. In this article, we show the above result due to Tutte can be extended to a rotor <em>R</em> of order <span><math><mi>k</mi><mo>≥</mo><mn>6</mn></math></span> if the subgraph of <em>G</em> induced by all those edges of <em>G</em> which are not in <em>R</em> satisfies certain conditions. Also, we provide a new method for generating infinitely many non-isomorphic <em>T</em>-equivalent pairs of graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102985"},"PeriodicalIF":1.3,"publicationDate":"2025-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363042","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 enumeration of double cosets and self-inverse double cosets","authors":"Ludovic Schwob","doi":"10.1016/j.aam.2025.102982","DOIUrl":"10.1016/j.aam.2025.102982","url":null,"abstract":"<div><div>Double cosets appear in many contexts in combinatorics, for example in the enumeration of certain objects up to symmetries. Double cosets in a quotient of the form <span><math><mi>H</mi><mo>﹨</mo><mi>G</mi><mo>/</mo><mi>H</mi></math></span> have an inverse, and can be their own inverse. In this paper we present various formulas enumerating double cosets, and in particular self-inverse double cosets. We study double cosets in classical groups, especially the symmetric groups and the general linear groups, explaining how to obtain the information on their conjugacy classes required to apply our formulas. We also consider double cosets of parabolic subgroups of Coxeter groups of type B.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102982"},"PeriodicalIF":1.3,"publicationDate":"2025-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145320793","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 symmetric strong circuit elimination property","authors":"Christine Cho , James Oxley , Suijie Wang","doi":"10.1016/j.aam.2025.102983","DOIUrl":"10.1016/j.aam.2025.102983","url":null,"abstract":"<div><div>If <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are circuits in a matroid <em>M</em> with <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <em>e</em> in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, then <em>M</em> has a circuit <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> such that <span><math><mi>e</mi><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>⊆</mo><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>e</mi></math></span>. This strong circuit elimination axiom is inherently asymmetric. A matroid <em>M</em> has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, there is a circuit <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> with <span><math><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo><mo>⊆</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>⊆</mo><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>e</mi></math></span>. We prove that a connected matroid has this property if and only if it has no two skew circuits. We also characterize such matroids in terms of forbidden series minors, and we give a new matroid axiom system that is built around a modification of SSCE.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102983"},"PeriodicalIF":1.3,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268469","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":"Enumerative proof of a curious congruence for Eulerian numbers","authors":"Xiangzi Meng , Hao Pan","doi":"10.1016/j.aam.2025.102977","DOIUrl":"10.1016/j.aam.2025.102977","url":null,"abstract":"<div><div>The Eulerian number <span><math><mo>〈</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>〉</mo></math></span> counts all permutations on <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span> having exactly <em>k</em> ascents. In this paper, we give an enumerative proof of the following congruence:<span><span><span><math><mrow><mo>〈</mo><mtable><mtr><mtd><mrow><mi>a</mi><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>b</mi><mi>p</mi><mo>+</mo><mi>l</mi></mrow></mtd></mtr></mtable><mo>〉</mo></mrow><mo>≡</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>b</mi></mrow></msup><msup><mrow><mo>(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>a</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable><mo>)</mo></mrow><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>p</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <em>p</em> is prime, <span><math><mn>0</mn><mo>≤</mo><mi>b</mi><mo><</mo><mi>a</mi></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>p</mi><mo>−</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102977"},"PeriodicalIF":1.3,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268467","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":"Extremal distance spectral radius of graphs with fixed size","authors":"Hongying Lin , Bo Zhou","doi":"10.1016/j.aam.2025.102980","DOIUrl":"10.1016/j.aam.2025.102980","url":null,"abstract":"<div><div>Let <em>m</em> be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum adjacency spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size <em>m</em>. After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We investigate the problem to determine the connected graphs with minimum distance spectral radius in the class of graphs with size <em>m</em>. Given <em>m</em>, there is exactly one positive integer <em>n</em> such that <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo><</mo><mi>m</mi><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We establish some structural properties of the extremal graphs for all <em>m</em> and solve the problem for <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>max</mi><mo></mo><mo>{</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>6</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>}</mo><mo>≤</mo><mi>m</mi><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We give a conjecture for the remaining case. To prove the main results, we also determine the complements of forests of fixed order with large and small distance spectral radius.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102980"},"PeriodicalIF":1.3,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268468","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":"Hybrid pipe dreams for key polynomials","authors":"Yihan Xiao , Rui Xiong , Haofeng Zhang","doi":"10.1016/j.aam.2025.102979","DOIUrl":"10.1016/j.aam.2025.102979","url":null,"abstract":"<div><div>We develop a family of new combinatorial models for key polynomials. It is similar to the hybrid pipe dream model for Schubert polynomials defined recently by Knutson and Udell.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102979"},"PeriodicalIF":1.3,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268470","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":"Formulas and conjectures for partitions with restrictions on interval of parts","authors":"George E. Andrews , Mohamed El Bachraoui","doi":"10.1016/j.aam.2025.102981","DOIUrl":"10.1016/j.aam.2025.102981","url":null,"abstract":"<div><div>We focus on certain integer partitions and their weighted analogues with conditions on the interval of their parts. The <em>q</em>-double series turn out to be more fruitful as generating functions for our sequences. We give explicit formulas for the number of such partitions, we derive identities involving integer partitions, and we prove that some of our weighted sequences are positive. Furthermore, we state two curious conjectures on the coefficients of two <em>q</em>-double series.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102981"},"PeriodicalIF":1.3,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268596","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":"A class of generalized chord Minkowski problems","authors":"Lu Zhang","doi":"10.1016/j.aam.2025.102978","DOIUrl":"10.1016/j.aam.2025.102978","url":null,"abstract":"<div><div>In this paper, we consider a class of generalized chord integrals in integral geometry, where the integrand is a generalized kernel that replaces the Riesz kernel. The generalized chord measure arises from the study of the generalized chord integral of convex bodies. We pose the Minkowski problem for the generalized chord measure and obtain the existence of solutions to the related Minkowski problem.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102978"},"PeriodicalIF":1.3,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268595","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":"d-Separated permutations and q-Stirling numbers of the first kind","authors":"Rosena R.X. Du, Yun Li","doi":"10.1016/j.aam.2025.102976","DOIUrl":"10.1016/j.aam.2025.102976","url":null,"abstract":"<div><div>Let <em>d</em> be a nonnegative integer, a <em>d</em>-separated permutation is a permutation in which every two left-to-right minima are at distance greater than <em>d</em>. More precisely, for <span><math><mi>π</mi><mo>=</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, suppose that <span><math><msub><mrow><mi>π</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>π</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>π</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub></math></span> are the left-to-right minima of <em>π</em> with <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mn>1</mn><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>n</mi></math></span>, then <em>π</em> is <em>d</em>-separated if <span><math><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>−</mo><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>></mo><mi>d</mi></math></span> for each <em>j</em>, <span><math><mn>1</mn><mo><</mo><mi>j</mi><mo>≤</mo><mi>k</mi></math></span>. In this paper we study different enumerative properties on <em>d</em>-separated permutations. We first give a recurrence formula of the numbers <span><math><msup><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> of <em>d</em>-separated permutations in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with exactly <em>k</em> left-to-right minima. Then we study the inversion and co-inversion polynomials of <em>d</em>-separated permutations, and give <em>q</em>-analogue <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-analogue <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> of <span><math><msup><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> for any <em>d</em>. Note that when <span><math><mi>d</mi><mo>=</mo><mn>0</mn></math></span>, 0-separated permutations are just all permutations in <span><math><msub><mrow><mi>S</mi></mrow","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"173 ","pages":"Article 102976"},"PeriodicalIF":1.3,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220933","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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