Heather A. Harrington , Mike Stillman , Alan Veliz-Cuba
{"title":"Algebraic network reconstruction of discrete dynamical systems","authors":"Heather A. Harrington , Mike Stillman , Alan Veliz-Cuba","doi":"10.1016/j.aam.2024.102760","DOIUrl":"10.1016/j.aam.2024.102760","url":null,"abstract":"<div><p>We present a computational algebra solution to reverse engineering the network structure of discrete dynamical systems from data. We use pseudomonomial ideals to determine dependencies between variables that encode constraints on the possible wiring diagrams underlying the process generating the discrete-time, continuous-space data. Our work assumes that each variable is either monotone increasing or decreasing. We prove that with enough data, even in the presence of small noise, our method can reconstruct the correct unique wiring diagram.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136744","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}
JiSun Huh , Sangwook Kim , Seunghyun Seo , Heesung Shin
{"title":"Bijections on pattern avoiding inversion sequences and related objects","authors":"JiSun Huh , Sangwook Kim , Seunghyun Seo , Heesung Shin","doi":"10.1016/j.aam.2024.102771","DOIUrl":"10.1016/j.aam.2024.102771","url":null,"abstract":"<div><p>The number of inversion sequences avoiding two patterns 101 and 102 is known to be the same as the number of permutations avoiding three patterns 2341, 2431, and 3241. This sequence also counts the number of Schröder paths without triple descents, restricted bicolored Dyck paths, <span><math><mo>(</mo><mn>101</mn><mo>,</mo><mn>021</mn><mo>)</mo></math></span>-avoiding inversion sequences, and weighted ordered trees. We provide bijections to integrate them together by introducing <em>F</em>-paths. Moreover, we define three kinds of statistics for each of the objects and count the number of each object with respect to these statistics. We also discuss direct sums of each object.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136743","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":"Extended Schur functions and bases related by involutions","authors":"Spencer Daugherty","doi":"10.1016/j.aam.2024.102770","DOIUrl":"10.1016/j.aam.2024.102770","url":null,"abstract":"<div><p>We introduce two new bases of <span><math><mi>Q</mi><mi>S</mi><mi>y</mi><mi>m</mi></math></span>, the reverse extended Schur functions and the row-strict reverse extended Schur functions, as well as their duals in <em>NSym</em>, the reverse shin functions and row-strict reverse shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions <em>ρ</em> and <em>ω</em>, which generalize the classical involution <em>ω</em> on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of <em>NSym</em> and <span><math><mi>Q</mi><mi>S</mi><mi>y</mi><mi>m</mi></math></span> respectively. We then use the involutions <em>ρ</em> and <em>ω</em> to translate these and other known results to our reverse and row-strict reverse bases.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142129395","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":"Off-diagonally symmetric domino tilings of the Aztec diamond of odd order","authors":"Yi-Lin Lee","doi":"10.1016/j.aam.2024.102759","DOIUrl":"10.1016/j.aam.2024.102759","url":null,"abstract":"<div><p>We study the enumeration of off-diagonally symmetric domino tilings of odd-order Aztec diamonds in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetry property which states that the numbers of off-diagonally symmetric domino tilings of the Aztec diamond of order <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></math></span> are equal when the boundary defect is at the <em>k</em>th position and the <span><math><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo></math></span>th position on the boundary, respectively. This symmetry property proves a special case of a recent conjecture by Behrend, Fischer, and Koutschan.</p><p>In the second direction, a Pfaffian formula is obtained for the number of “nearly” off-diagonally symmetric domino tilings of odd-order Aztec diamonds, where the entries of the Pfaffian satisfy a simple recurrence relation. The numbers of domino tilings mentioned in the above two directions do not seem to have a simple product formula, but we show that these numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers. The proof of these results involves the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths. Finally, we propose conjectures concerning the log-concavity and asymptotic behavior of the number of off-diagonally symmetric domino tilings of odd-order Aztec diamonds.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142087152","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":"Stereologically adapted Crofton formulae for tensor valuations","authors":"Emil Dare","doi":"10.1016/j.aam.2024.102754","DOIUrl":"10.1016/j.aam.2024.102754","url":null,"abstract":"<div><p>The classical Crofton formula explains how intrinsic volumes of a convex body <em>K</em> in <em>n</em>-dimensional Euclidean space can be obtained from integrating a measurement function at sections of <em>K</em> with invariantly moved affine flats. We generalize this idea by constructing stereologically adapted Crofton formulae for translation invariant Minkowski tensors, expressing a prescribed tensor valuation as an invariant integral of a measurement function of section profiles with flats. The measurement functions are weighed sums of powers of the metric tensor times Minkowski valuations. The weights are determined explicitly from known Crofton formulae using Zeilberger's algorithm. The main result is an exhaustive set of measurement functions where the invariant integration is over flats.</p><p>With the main result at hand, a Blaschke-Petkantschin formula allows us to establish new measurement functions valid when the invariant integration over flats is replaced by an invariant integration over subspaces containing a fixed subspace of lower dimension. Likewise, a stereologically adapted Crofton formula valid in the scheme of vertical sections is constructed. Only some special cases of this result have been stated explicitly before, with even the three-dimensional case yielding a new stereological formula. Here, we obtain new vertical section formulae for the surface tensors of even rank.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824000861/pdfft?md5=7cfe89c5b6afba8f72d1292bc78aca12&pid=1-s2.0-S0196885824000861-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142083640","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":"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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