Pub Date : 2025-04-04DOI: 10.1016/j.aam.2025.102887
Farid Aliniaeifard, Stephanie van Willigenburg
We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the x-basis in the Hopf algebra of symmetric functions in noncommuting variables, NCSym, which arises in the theory of Grothendieck bialgebras. We achieve this by applying the theory of Hopf monoids and the Fock functor. We also determine combinatorial expansions of this basis in terms of the monomial and power sum symmetric functions in NCSym, and by taking the commutative image of the x-basis we discover a new multiplicative basis for the algebra of symmetric functions.
我们回答了Bergeron， Hohlweg, Rosas， and Zabrocki从2006年提出的一个问题，给出了非交换变量NCSym对称函数的Hopf代数中x基的余积的组合描述，该问题出现在Grothendieck双代数理论中。我们利用Hopf半群理论和Fock函子实现了这一点。我们还用NCSym中的单项式和幂和对称函数确定了该基的组合展开式，并通过取x基的交换像发现了对称函数代数的一种新的乘法基。
{"title":"The extra basis in noncommuting variables","authors":"Farid Aliniaeifard, Stephanie van Willigenburg","doi":"10.1016/j.aam.2025.102887","DOIUrl":"10.1016/j.aam.2025.102887","url":null,"abstract":"<div><div>We answer a question of Bergeron, Hohlweg, Rosas, and Zabrocki from 2006 to give a combinatorial description for the coproduct of the <em>x</em>-basis in the Hopf algebra of symmetric functions in noncommuting variables, NCSym, which arises in the theory of Grothendieck bialgebras. We achieve this by applying the theory of Hopf monoids and the Fock functor. We also determine combinatorial expansions of this basis in terms of the monomial and power sum symmetric functions in NCSym, and by taking the commutative image of the <em>x</em>-basis we discover a new multiplicative basis for the algebra of symmetric functions.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"168 ","pages":"Article 102887"},"PeriodicalIF":1.0,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767926","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}
Pub 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":"2025-04-01","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}
Pub Date : 2025-03-28DOI: 10.1016/j.aam.2025.102883
Persi Diaconis , Nathan Tung
Let be a group of permutations of kn objects which permutes things independently in disjoint blocks of size k and then permutes the blocks. We investigate the probabilistic and enumerative aspects of random elements of . This includes novel limit theorems for cycles of various lengths, number of cycles, and inversions. The limits include compound Poisson distributions with interesting dependence structure.
{"title":"Poisson approximation for large permutation groups","authors":"Persi Diaconis , Nathan Tung","doi":"10.1016/j.aam.2025.102883","DOIUrl":"10.1016/j.aam.2025.102883","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> be a group of permutations of <em>kn</em> objects which permutes things independently in disjoint blocks of size <em>k</em> and then permutes the blocks. We investigate the probabilistic and enumerative aspects of random elements of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. This includes novel limit theorems for cycles of various lengths, number of cycles, and inversions. The limits include compound Poisson distributions with interesting dependence structure.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102883"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143714822","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 : 2025-03-26DOI: 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":"2025-03-26","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 : 2025-03-20DOI: 10.1016/j.aam.2025.102882
Kaimei Huang, Sherry H.F. Yan
<div><div>As natural generalizations of the descent number (<span><math><mi>des</mi></math></span>) and the major index (<span><math><mi>maj</mi></math></span>), Rawlings introduced the notions of the <em>r</em>-descent number (<span><math><mi>r</mi><mrow><mi>des</mi></mrow></math></span>) and the <em>r</em>-major index (<span><math><mi>r</mi><mrow><mi>maj</mi></mrow></math></span>) for a given positive integer <em>r</em>. A pair <span><math><mo>(</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of permutation statistics is said to be <em>r</em>-Euler-Mahonian if <span><math><mo>(</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> are equidistributed over the set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of all permutations 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>. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> is <span><math><mo>(</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-Euler-Mahonian for all positive integers <em>g</em> and <em>ℓ</em>, where <span><math><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level excedance number and <span><math><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> where <span><math><mi>r</mi><mo>=</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></math></span>. Setting <span><math><mi>g</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span>, our result recovers the equidistribution of <span><math><mo>(</mo><mrow><mi>des</mi></mrow><mo>,</mo><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>den</mi></mrow><mo>)</mo></math></span>, which was first conjectured by Denert and proved by Foata
作为下降数（des）和主索引（maj）的自然推广，Rawlings引入了给定正整数r的r-下降数（rdes）和r-主索引（rmaj）的概念。如果（st1,st2）和（rdes,rmaj）在{1,2，…，n}的所有排列的集合Sn上是均匀分布的，那么一对（st1,st2）排列统计量就是r- euler - mahonian。本文的主要目的是证实Liu最近提出的一个猜想，即对于所有正整数g和r， （gexc r,gden r）是(g+ r−1)-Euler-Mahonian，其中gexc r表示g-gap r -level的超越数，gden r表示g-gap r -level的Denert's统计量。这是通过客观证明（gexc r,gden r）和（rdes,rmaj）的均匀分布来实现的，其中r=g+ r−1。设g= r =1，我们的结果恢复了（des,maj）和（exc,den）的均匀分布，这是由Denert首先推测并由Foata和Zeilberger证明的。我们的第二个主要结果与（geexc r,gdeng+ r）的类似结果有关，该结果表明（geexc r,gdeng+ r）对于所有正整数g和r都是(g+ r−1)-欧拉-马霍尼量。
{"title":"Further results on r-Euler-Mahonian statistics","authors":"Kaimei Huang, Sherry H.F. Yan","doi":"10.1016/j.aam.2025.102882","DOIUrl":"10.1016/j.aam.2025.102882","url":null,"abstract":"<div><div>As natural generalizations of the descent number (<span><math><mi>des</mi></math></span>) and the major index (<span><math><mi>maj</mi></math></span>), Rawlings introduced the notions of the <em>r</em>-descent number (<span><math><mi>r</mi><mrow><mi>des</mi></mrow></math></span>) and the <em>r</em>-major index (<span><math><mi>r</mi><mrow><mi>maj</mi></mrow></math></span>) for a given positive integer <em>r</em>. A pair <span><math><mo>(</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of permutation statistics is said to be <em>r</em>-Euler-Mahonian if <span><math><mo>(</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> are equidistributed over the set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of all permutations 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>. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> is <span><math><mo>(</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-Euler-Mahonian for all positive integers <em>g</em> and <em>ℓ</em>, where <span><math><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level excedance number and <span><math><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> where <span><math><mi>r</mi><mo>=</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></math></span>. Setting <span><math><mi>g</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span>, our result recovers the equidistribution of <span><math><mo>(</mo><mrow><mi>des</mi></mrow><mo>,</mo><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>den</mi></mrow><mo>)</mo></math></span>, which was first conjectured by Denert and proved by Foata ","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102882"},"PeriodicalIF":1.0,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143686514","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 : 2025-03-19DOI: 10.1016/j.aam.2025.102885
Jagdeep Singh
For a matroid of rank r and a non-negative integer k, an element is called k-loose if every circuit containing it has size greater than . Zaslavsky and the author characterized all binary matroids with a 1-loose element. In this paper, we establish a sharp linear bound on the size of a binary matroid, in terms of its rank, that contains a k-loose element. A matroid is called k-paving if all its elements are k-loose. Rajpal showed that for a prime power q, the rank of a -matroid that is k-paving is bounded. We provide a bound on the rank of -matroids that are cosimple and have two k-loose elements. Consequently, we strengthen the result of Rajpal by providing a bound on the rank of -matroids that are k-paving. Additionally, we provide a bound on the size of binary matroids that are k-paving.
{"title":"k-loose elements and k-paving matroids","authors":"Jagdeep Singh","doi":"10.1016/j.aam.2025.102885","DOIUrl":"10.1016/j.aam.2025.102885","url":null,"abstract":"<div><div>For a matroid of rank <em>r</em> and a non-negative integer <em>k</em>, an element is called <em>k</em>-loose if every circuit containing it has size greater than <span><math><mi>r</mi><mo>−</mo><mi>k</mi></math></span>. Zaslavsky and the author characterized all binary matroids with a 1-loose element. In this paper, we establish a sharp linear bound on the size of a binary matroid, in terms of its rank, that contains a <em>k</em>-loose element. A matroid is called <em>k</em>-paving if all its elements are <em>k</em>-loose. Rajpal showed that for a prime power <em>q</em>, the rank of a <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-matroid that is <em>k</em>-paving is bounded. We provide a bound on the rank of <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-matroids that are cosimple and have two <em>k</em>-loose elements. Consequently, we strengthen the result of Rajpal by providing a bound on the rank of <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-matroids that are <em>k</em>-paving. Additionally, we provide a bound on the size of binary matroids that are <em>k</em>-paving.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102885"},"PeriodicalIF":1.0,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644876","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 : 2025-03-18DOI: 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":"2025-03-18","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 : 2025-03-17DOI: 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":"2025-03-17","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 : 2025-03-17DOI: 10.1016/j.aam.2025.102879
Erhan Güler , Nurettin Cenk Turgay
In this paper, we investigate rotational hypersurfaces family in n-dimensional Euclidean space . Our focus is on studying the Gauss map of this family with respect to the operator , which acts on functions defined on the hypersurfaces. The operator can be viewed as a modified Laplacian and is known by various names, including the Cheng–Yau operator in certain cases. Specifically, we focus on the scenario where and . By applying the operator to the Gauss map , we establish a classification theorem. This theorem establishes a connection between the matrix , and the Gauss map through the equation .
{"title":"Rotational hypersurfaces family satisfying Ln−3G=AG in the n-dimensional Euclidean space","authors":"Erhan Güler , Nurettin Cenk Turgay","doi":"10.1016/j.aam.2025.102879","DOIUrl":"10.1016/j.aam.2025.102879","url":null,"abstract":"<div><div>In this paper, we investigate rotational hypersurfaces family in <em>n</em>-dimensional Euclidean space <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Our focus is on studying the Gauss map <span><math><mi>G</mi></math></span> of this family with respect to the operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, which acts on functions defined on the hypersurfaces. The operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> can be viewed as a modified Laplacian and is known by various names, including the Cheng–Yau operator in certain cases. Specifically, we focus on the scenario where <span><math><mi>k</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>3</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>. By applying the operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></math></span> to the Gauss map <span><math><mi>G</mi></math></span>, we establish a classification theorem. This theorem establishes a connection between the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix <span><math><mi>A</mi></math></span>, and the Gauss map <span><math><mi>G</mi></math></span> through the equation <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub><mi>G</mi><mo>=</mo><mi>A</mi><mi>G</mi></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102879"},"PeriodicalIF":1.0,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636651","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 : 2025-02-27DOI: 10.1016/j.aam.2025.102870
Antonio Rieser, Jonathan Treviño-Marroquín
We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial ‘coarse-like’ structures to compact metric spaces, something which is impossible in coarse geometry. We first study homotopy in this context, and we then construct homology groups which are invariant under semi-coarse homotopy equivalence. We further show that any undirected graph induces a semi-coarse structure on its set of vertices , and that the respective semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of undirected graphs.
{"title":"Semi-coarse spaces, homotopy and homology","authors":"Antonio Rieser, Jonathan Treviño-Marroquín","doi":"10.1016/j.aam.2025.102870","DOIUrl":"10.1016/j.aam.2025.102870","url":null,"abstract":"<div><div>We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial ‘coarse-like’ structures to compact metric spaces, something which is impossible in coarse geometry. We first study homotopy in this context, and we then construct homology groups which are invariant under semi-coarse homotopy equivalence. We further show that any undirected graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> induces a semi-coarse structure on its set of vertices <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, and that the respective semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of undirected graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102870"},"PeriodicalIF":1.0,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510565","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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