{"title":"Some remarks on permanental dominance conjecture","authors":"Kijti Rodtes","doi":"10.1016/j.aam.2024.102758","DOIUrl":"10.1016/j.aam.2024.102758","url":null,"abstract":"<div><p>In this paper we provide an identity between the determinant and other generalized matrix functions, and give a criterion for positive semi-definite matrices to satisfy the permanental dominance conjecture. As a consequence, infinitely many classes of positive semi-definite matrices satisfying the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"160 ","pages":"Article 102758"},"PeriodicalIF":1.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142076510","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}
Zhicong Lin , Jing Liu , Suijie Wang , Wenston J.T. Zang
{"title":"More bijective combinatorics of weakly increasing trees","authors":"Zhicong Lin , Jing Liu , Suijie Wang , Wenston J.T. Zang","doi":"10.1016/j.aam.2024.102755","DOIUrl":"10.1016/j.aam.2024.102755","url":null,"abstract":"<div><p>As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin–Ma–Ma–Zhou (2021). Various intriguing connections and bijections for weakly increasing trees have already been found and the purpose of this paper is to present yet more bijective combinatorics on this unified object. Two of our main contributions are</p><ul><li><span>•</span><span><p>extension of an equidistribution result on plane trees due to Eu–Seo–Shin (2017), regarding levels and degrees of nodes, to weakly increasing trees;</p></span></li><li><span>•</span><span><p>a new interpretation of the multiset Schett polynomials in terms of odd left/right chains on weakly increasing binary trees.</p></span></li></ul> Interesting consequences are discussed, including new tree interpretations for the Jacobi elliptic functions and Euler numbers. Relevant enumerative results are also presented, involving recurrence relations, exponential generating functions and context-free grammars.</div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"160 ","pages":"Article 102755"},"PeriodicalIF":1.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142011722","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":"Stratified simple homotopy type: Theory and computation","authors":"Markus Banagl , Tim Mäder , Filip Sadlo","doi":"10.1016/j.aam.2024.102753","DOIUrl":"10.1016/j.aam.2024.102753","url":null,"abstract":"<div><p>Generalizing the idea of elementary simplicial collapses and expansions in classical simple homotopy theory to a stratified setting, we find local combinatorial transformations on stratified simplicial complexes that leave the global stratified homotopy type invariant. In particular, we obtain the notions of stratified formal deformations generalizing J. H. C. Whitehead's formal deformations. We implement the algorithmic execution of such transformations and the computation of intersection homology to illustrate the behavior of stratified simple homotopy equivalences on Vietoris-Rips type complexes associated to point sets sampled near given, possibly singular, spaces.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"160 ","pages":"Article 102753"},"PeriodicalIF":1.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S019688582400085X/pdfft?md5=38b8418d116b6eea867db781229ca852&pid=1-s2.0-S019688582400085X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141954155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the s-Gaussian measure in Rn","authors":"Youjiang Lin , Sudan Xing","doi":"10.1016/j.aam.2024.102744","DOIUrl":"10.1016/j.aam.2024.102744","url":null,"abstract":"<div><p>We construct the <em>s</em>-Gauss probability space by introducing the <em>s</em>-Gaussian density function in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for <span><math><mi>s</mi><mo>≥</mo><mn>0</mn></math></span>, a generalization of the classic Gaussian density function. Based on the <em>s</em>-Gaussian density function, we propose the <span><math><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span>-Ehrhard symmetrization which is an extension of the traditional Ehrhard symmetrization for sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. In particular, we establish the <em>s</em>-Gaussian isoperimetric inequality with respect to <em>s</em>-Gaussian measure in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Furthermore, we propose and prove the <em>s</em>-Ehrhard-Borell inequalities for <span><math><mi>s</mi><mo>></mo><mn>0</mn></math></span> when one of the two sets is a Borel set whilst the other being a convex set as well as the case when two sets are convex in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> with different methods.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"160 ","pages":"Article 102744"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141961406","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":"Two measures of efficiency for the secretary problem with multiple items at each rank","authors":"Ross G. Pinsky","doi":"10.1016/j.aam.2024.102751","DOIUrl":"10.1016/j.aam.2024.102751","url":null,"abstract":"<div><p>For <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>∈</mo><mi>N</mi></math></span>, consider the following adaptation of the classical secretary problem. There are <em>k</em> items at each of <em>n</em> linearly ordered ranks. The <em>kn</em> items are revealed, one item at a time, in a uniformly random order, to an observer whose objective is to select an item of highest rank. At each stage the observer only knows the relative ranks of the items that have arrived thus far, and must either select the current item, in which case the process terminates, or reject it and continue to the next item. For <span><math><mi>M</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>k</mi><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span>, let <span><math><mi>S</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>;</mo><mi>M</mi><mo>)</mo></math></span> denote the strategy whereby one allows the first <em>M</em> items to pass, and then selects the first later arriving item whose rank is either equal to or greater than the highest rank of the first <em>M</em> items (if such an item exists). Let <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>S</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>;</mo><mi>M</mi><mo>)</mo></mrow></msub></math></span> denote the event that one selects an item of highest rank using strategy <span><math><mi>S</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>;</mo><mi>M</mi><mo>)</mo></math></span> and let <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>S</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>;</mo><mi>M</mi><mo>)</mo></mrow></msub><mo>)</mo></math></span> denote the corresponding probability. We obtain a formula for <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>S</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>;</mo><mi>M</mi><mo>)</mo></mrow></msub><mo>)</mo></math></span>, and for <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo></mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>S</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>;</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></msub><mo>)</mo></math></span>, when <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∼</mo><mi>c</mi><mi>k</mi><mi>n</mi></math></span>, with <span><math><mi>c</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. In the classical secretary problem (<span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>), the asymptotic probability of success using an optimal strategy is <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>e</mi></mrow></mfrac><mo>≈</mo><mn>0.368</mn></math></span>. For <span><math><mi>k</mi><mo>=</","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"160 ","pages":"Article 102751"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141961407","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":"General coefficient-vanishing results associated with theta series","authors":"Shane Chern , Dazhao Tang","doi":"10.1016/j.aam.2024.102742","DOIUrl":"10.1016/j.aam.2024.102742","url":null,"abstract":"<div><p>There are a number of sporadic coefficient-vanishing results associated with theta series, which suggest certain underlying patterns. By expanding theta powers as linear combinations of products of theta functions, we present two strategies that will provide a unified treatment. Our approaches rely on studying the behavior of products of two theta series under the action of the huffing operator. For this purpose, some explicit criteria are given. We may use the presented methods to not only verify experimentally discovered coefficient-vanishing results, but also to produce a series of general phenomena.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102742"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141773995","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":"Upper bounds of dual flagged Weyl characters","authors":"Simon C.Y. Peng , Zhuowei Lin , Sophie C.C. Sun","doi":"10.1016/j.aam.2024.102752","DOIUrl":"10.1016/j.aam.2024.102752","url":null,"abstract":"<div><p>For a subset <em>D</em> of boxes in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> square grid, let <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> denote the dual character of the flagged Weyl module associated to <em>D</em>. It is known that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> specifies to a Schubert polynomial (resp., a key polynomial) in the case when <em>D</em> is the Rothe diagram of a permutation (resp., the skyline diagram of a composition). One can naturally define a lower and an upper bound of <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. Mészáros, St. Dizier and Tanjaya conjectured that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> attains the upper bound if and only if <em>D</em> avoids a certain single subdiagram. We provide a proof of this conjecture.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"160 ","pages":"Article 102752"},"PeriodicalIF":1.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141961408","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":"Global positioning: The uniqueness question and a new solution method","authors":"Mireille Boutin , Gregor Kemper","doi":"10.1016/j.aam.2024.102741","DOIUrl":"10.1016/j.aam.2024.102741","url":null,"abstract":"<div><p>We provide a new algebraic solution procedure for the global positioning problem in <em>n</em> dimensions using <em>m</em> satellites. We also give a geometric characterization of the situations in which the problem does not have a unique solution. This characterization shows that such cases can happen in any dimension and with any number of satellites, leading to counterexamples to some open conjectures. We fill a gap in the literature by giving a proof for the long-held belief that when <span><math><mi>m</mi><mo>≥</mo><mi>n</mi><mo>+</mo><mn>2</mn></math></span>, the solution is unique for almost all user positions. Even better, when <span><math><mi>m</mi><mo>≥</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></math></span>, almost all satellite configurations will guarantee a unique solution for <em>all</em> user positions. Our uniqueness results provide a basis for predicting the behavior of numerical solutions, as ill-conditioning is expected near the threshold between areas of nonuniqueness and uniqueness. Some of our results are obtained using tools from algebraic geometry.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"160 ","pages":"Article 102741"},"PeriodicalIF":1.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824000733/pdfft?md5=afdf7a184841d258a70711fe7d252f55&pid=1-s2.0-S0196885824000733-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141773994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A grammar of Dumont and a theorem of Diaconis-Evans-Graham","authors":"William Y.C. Chen , Amy M. Fu","doi":"10.1016/j.aam.2024.102743","DOIUrl":"10.1016/j.aam.2024.102743","url":null,"abstract":"<div><p>We came across an unexpected connection between a remarkable grammar of Dumont for the joint distribution of <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>fix</mi></mrow><mo>)</mo></math></span> over <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and a beautiful theorem of Diaconis-Evans-Graham on successions and fixed points of permutations. With the grammar in hand, we demonstrate the advantage of the grammatical calculus in deriving the generating functions, where the constant property plays a substantial role. On the grounds of left successions of a permutation, we present a grammatical treatment of the joint distribution investigated by Roselle. Moreover, we obtain a left succession analogue of the Diaconis-Evans-Graham theorem, exemplifying the idea of a grammar assisted bijection. The grammatical labelings give rise to an equidistribution of <span><math><mo>(</mo><mrow><mi>jump</mi></mrow><mo>,</mo><mrow><mi>des</mi></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>drop</mi></mrow><mo>)</mo></math></span> restricted to the set of left successions and the set of fixed points, where jump is defined to be the number of ascents minus the number of left successions.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"160 ","pages":"Article 102743"},"PeriodicalIF":1.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141773993","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":"Boltzmann distribution on “short” integer partitions with power parts: Limit laws and sampling","authors":"Jean C. Peyen, Leonid V. Bogachev, Paul P. Martin","doi":"10.1016/j.aam.2024.102739","DOIUrl":"10.1016/j.aam.2024.102739","url":null,"abstract":"<div><p>The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set <span><math><msup><mrow><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msup></math></span> of <em>strict</em> integer partitions (i.e., with unequal parts) into perfect <em>q</em>-th powers. A combinatorial link is provided via a suitable conditioning by fixing the partition <em>weight</em> (the sum of parts) and <em>length</em> (the number of parts), leading to uniform distribution on the corresponding subspaces of partitions. The Boltzmann measure is calibrated through the hyper-parameters <span><math><mo>〈</mo><mi>N</mi><mo>〉</mo></math></span> and <span><math><mo>〈</mo><mi>M</mi><mo>〉</mo></math></span> controlling the expected weight and length, respectively. We study “short” partitions, where the parameter <span><math><mo>〈</mo><mi>M</mi><mo>〉</mo></math></span> is either fixed or grows slower than for typical partitions in <span><math><msup><mrow><mover><mrow><mi>Λ</mi></mrow><mrow><mo>ˇ</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msup></math></span>. For this model, we obtain a variety of limit theorems including the asymptotics of the cumulative cardinality in the case of fixed <span><math><mo>〈</mo><mi>M</mi><mo>〉</mo></math></span> and a limit shape result in the case of slow growth of <span><math><mo>〈</mo><mi>M</mi><mo>〉</mo></math></span>. In both cases, we also characterize the joint distribution of the weight and length, as well as the growth of the smallest and largest parts. Using these results we construct suitable sampling algorithms and analyze their performance.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102739"},"PeriodicalIF":1.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S019688582400071X/pdfft?md5=c62597e3a64191348b9f6a6a0db0b908&pid=1-s2.0-S019688582400071X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141639163","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
