Pub Date : 2025-07-01Epub Date: 2025-04-09DOI: 10.1016/j.aam.2025.102890
Irem Portakal , Daniel Windisch
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 -games.
{"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":"2025-07-01","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}
Pub Date : 2025-07-01Epub Date: 2025-04-08DOI: 10.1016/j.aam.2025.102899
Colin Defant, Mitchell Lee
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 troupes. Using a generalization of troupes that we call weighted troupes, 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 troupe transform. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes.
{"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":"2025-07-01","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}
Pub Date : 2025-07-01Epub Date: 2025-04-30DOI: 10.1016/j.aam.2025.102902
Tim Huber , James McLaughlin , Dongxi Ye
<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
{"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":"2025-07-01","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}
Pub Date : 2025-07-01Epub Date: 2025-04-14DOI: 10.1016/j.aam.2025.102888
Honglin Zhu
Given a grid polygon P in a grid of equilateral triangles, Defant and Jiradilok considered a billiards system where beams of light bounce around inside P. We study the relationship between the perimeter of P and the number of different trajectories that the billiards system has. Resolving a conjecture of Defant and Jiradilok, we prove the sharp inequality and characterize the equality cases.
{"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":"2025-07-01","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}
Pub Date : 2025-07-01Epub Date: 2025-04-10DOI: 10.1016/j.aam.2025.102889
Sen-Peng Eu , Louis Kao , Juei-Yin Lin
Cai and Readdy proposed a new framework for studying the q-analogue of a combinatorial structure S. Specifically, the aim is to identify two statistics over S and a proper subset of S such that represents the q--expansion over , 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 q-Stirling numbers of both kinds, as well as colored q-Lah numbers. We also briefly discuss q-Stirling and q-Lah numbers of type D.
蔡和瑞迪提出了一个研究组合结构 S 的 q-analogue f(q) 的新框架。具体来说，其目的是找出 S 上的两个统计量和 S 的一个适当子集 S′，从而使 f(q) 代表 S′上的 q-(1+q)- 展开，并探索这种展开的正集和拓扑解释。Cai 和 Readdy 在此框架内提供了两种经典斯特林数的全面剖面图。在这项工作中，我们将蔡和雷迪的结果扩展到两种彩色 q-Stirling 数以及彩色 q-Lah 数。我们还简要讨论了 D 型的 q-Stirling 和 q-Lah 数。
{"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":"2025-07-01","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}
Pub Date : 2025-07-01Epub Date: 2025-04-22DOI: 10.1016/j.aam.2025.102900
Dein Wong , Qi Zhou , Xinlei Wang
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) [19] 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.
1.
Based on matrix analysis, two conditions on what kinds of vertices have the same PageRank score are obtained.
2.
Four techniques for constructing Co-PR graphs are established.
3.
A non-regular connected graph of order n, with as PR scores of most of its vertices, is constructed, which provides a negative answer to Modjtaba's conjecture above.
{"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":"2025-07-01","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}
Pub Date : 2025-06-01Epub Date: 2025-03-18DOI: 10.1016/j.aam.2025.102881
Meng Liu , Ye Wang
Let be the graph consisting of three internally disjoint paths of length four sharing common endpoints. It is shown as by computing polynomial resultants.
{"title":"Polynomial resultants and Ramsey numbers of a theta graph","authors":"Meng Liu , Ye Wang","doi":"10.1016/j.aam.2025.102881","DOIUrl":"10.1016/j.aam.2025.102881","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow></msub></math></span> be the graph consisting of three internally disjoint paths of length four sharing common endpoints. It is shown <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>Θ</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span> as <span><math><mi>k</mi><mo>→</mo><mo>∞</mo></math></span> by computing polynomial resultants.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102881"},"PeriodicalIF":1.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143642896","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}
Pub Date : 2025-06-01Epub Date: 2025-03-17DOI: 10.1016/j.aam.2025.102880
Franquiz Caraballo Alba
The linear orbit of a degree d hypersurface in is its orbit under the natural action of , in the projective space of dimension parameterizing such hypersurfaces. This action restricted to a specific hypersurface X extends to a rational map from the projectivization of the space of matrices to . The class of the graph of this map is the predegree polynomial of its corresponding hypersurface. The objective of this paper is threefold. First, we formally define the predegree polynomial of a hypersurface in , introduced in the case of plane curves by Aluffi and Faber, and prove some results in the general case. A key result in the general setting is that a partial resolution of said rational map can contain enough information to compute the predegree polynomial of a hypersurface. Second, we compute the leading term of the predegree polynomial of a smooth quadric in over an algebraically closed field with characteristic 0, and compute the other coefficients in the specific case . In analogy to Aluffi and Faber's work, the tool for computing this invariant is producing a (partial) resolution of the previously mentioned rational map which contains enough information to obtain the invariant. Third, we provide a complete resolution of the rational map in the case , which in principle could be used to compute more refined invariants.
{"title":"Linear orbits of smooth quadric surfaces","authors":"Franquiz Caraballo Alba","doi":"10.1016/j.aam.2025.102880","DOIUrl":"10.1016/j.aam.2025.102880","url":null,"abstract":"<div><div>The <em>linear orbit</em> of a degree <em>d</em> hypersurface in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is its orbit under the natural action of <span><math><mi>P</mi><mi>GL</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>, in the projective space of dimension <span><math><mi>N</mi><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>d</mi></mrow></mtd></mtr><mtr><mtd><mi>d</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mn>1</mn></math></span> parameterizing such hypersurfaces. This action restricted to a specific hypersurface <em>X</em> extends to a rational map from the projectivization of the space of matrices to <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. The class of the graph of this map is the <em>predegree polynomial</em> of its corresponding hypersurface. The objective of this paper is threefold. First, we formally define the predegree polynomial of a hypersurface in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, introduced in the case of plane curves by Aluffi and Faber, and prove some results in the general case. A key result in the general setting is that a partial resolution of said rational map can contain enough information to compute the predegree polynomial of a hypersurface. Second, we compute the leading term of the predegree polynomial of a smooth quadric in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over an algebraically closed field with characteristic 0, and compute the other coefficients in the specific case <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>. In analogy to Aluffi and Faber's work, the tool for computing this invariant is producing a (partial) resolution of the previously mentioned rational map which contains enough information to obtain the invariant. Third, we provide a complete resolution of the rational map in the case <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>, which in principle could be used to compute more refined invariants.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102880"},"PeriodicalIF":1.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636655","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}
Pub Date : 2025-06-01Epub Date: 2025-03-26DOI: 10.1016/j.aam.2025.102884
Xiangyu Ding, Lisa Hui Sun
In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal number theorem and showed the nonnegativity related to partition functions. Meanwhile, Andrews and Merca, Guo and Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been confirmed analytically and also combinatorially. In 2022, Merca proposed a stronger version for this conjecture. In this paper, by applying Agarwal, Andrews and Bressoud's identity derived from the Bailey lattice, we obtain a truncated version for the Jacobi triple product series with odd basis, which reduces to the Andrews–Gordon identity as a special instance. As consequences, we obtain new truncated forms for Euler's pentagonal number theorem, Gauss' theta series on triangular numbers and square numbers, which lead to inequalities for certain partition functions. Moreover, by considering a truncated theta series involving ℓ-regular partitions, we confirm a conjecture proposed by Ballantine and Merca about 6-regular partitions and show that Merca's stronger conjecture on truncated Jacobi triple product series holds when for .
{"title":"Truncated theta series from the Bailey lattice","authors":"Xiangyu Ding, Lisa Hui Sun","doi":"10.1016/j.aam.2025.102884","DOIUrl":"10.1016/j.aam.2025.102884","url":null,"abstract":"<div><div>In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal number theorem and showed the nonnegativity related to partition functions. Meanwhile, Andrews and Merca, Guo and Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been confirmed analytically and also combinatorially. In 2022, Merca proposed a stronger version for this conjecture. In this paper, by applying Agarwal, Andrews and Bressoud's identity derived from the Bailey lattice, we obtain a truncated version for the Jacobi triple product series with odd basis, which reduces to the Andrews–Gordon identity as a special instance. As consequences, we obtain new truncated forms for Euler's pentagonal number theorem, Gauss' theta series on triangular numbers and square numbers, which lead to inequalities for certain partition functions. Moreover, by considering a truncated theta series involving <em>ℓ</em>-regular partitions, we confirm a conjecture proposed by Ballantine and Merca about 6-regular partitions and show that Merca's stronger conjecture on truncated Jacobi triple product series holds when <span><math><mi>R</mi><mo>=</mo><mn>3</mn><mi>S</mi></math></span> for <span><math><mi>S</mi><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102884"},"PeriodicalIF":1.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704752","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}
Pub Date : 2025-06-01Epub Date: 2025-04-01DOI: 10.1016/j.aam.2025.102886
David G.L. Wang , James Z.F. Zhou
We develop a composition method to unearth positive -expansions of chromatic symmetric functions , where the subscript I stands for compositions rather than integer partitions. Using this method, we derive positive and neat -expansions for the chromatic symmetric functions of tadpoles, barbells and generalized bulls, and establish the e-positivity of hats. We also obtain a compact ribbon Schur analog for the chromatic symmetric function of cycles.
{"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":"2025-06-01","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}
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