Advances in Applied Mathematics最新文献

{"title":"Refining a chain theorem from matroids to internally 4-connected graphs","authors":"Chanun Lewchalermvongs ,&nbsp;Guoli Ding","doi":"10.1016/j.aam.2024.102802","DOIUrl":"10.1016/j.aam.2024.102802","url":null,"abstract":"<div><div>Graph theory and matroid theory are interconnected with matroids providing a way to generalize and analyze the structural and independence properties within graphs. Chain theorems, vital tools in both matroid and graph theory, enable the analysis of matroid structures associated with graphs. In a significant contribution, Chun, Mayhew, and Oxley <span><span>[2]</span></span> established a chain theorem for internally 4-connected binary matroids, clarifying the operations involved. Our research builds upon this by specifying the matroid result to internally 4-connected graphs. The primary goal of our research is to refine this chain theorem for matroids into a chain theorem for internally 4-connected graphs, making it more accessible to individuals less acquainted with matroid theory.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593674","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 enumeration of series-parallel matroids","authors":"Nicholas Proudfoot ,&nbsp;Yuan Xu ,&nbsp;Benjamin Young","doi":"10.1016/j.aam.2024.102801","DOIUrl":"10.1016/j.aam.2024.102801","url":null,"abstract":"<div><div>By the work of Ferroni and Larson, Kazhdan–Lusztig polynomials and <em>Z</em>-polynomials of complete graphs have combinatorial interpretations in terms of quasi series-parallel matroids. We provide explicit formulas for the number of series-parallel matroids and the number of simple series-parallel matroids of a given rank and cardinality, extending results of Ferroni–Larson and Gao–Proudfoot–Yang–Zhang.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554364","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":"Identifiability of homoscedastic linear structural equation models using algebraic matroids","authors":"Mathias Drton,&nbsp;Benjamin Hollering,&nbsp;Jun Wu","doi":"10.1016/j.aam.2024.102794","DOIUrl":"10.1016/j.aam.2024.102794","url":null,"abstract":"<div><div>We consider structural equation models (SEMs), in which every variable is a function of a subset of the other variables and a stochastic error. Each such SEM is naturally associated with a directed graph describing the relationships between variables. When the errors are homoscedastic, recent work has proposed methods for inferring the graph from observational data under the assumption that the graph is acyclic (i.e., the SEM is recursive). In this work, we study the setting of homoscedastic errors but allow the graph to be cyclic (i.e., the SEM to be non-recursive). Using an algebraic approach that compares matroids derived from the parameterizations of the models, we derive sufficient conditions for when two simple directed graphs generate different distributions generically. Based on these conditions, we exhibit subclasses of graphs that allow for directed cycles, yet are generically identifiable. We also conjecture a strengthening of our graphical criterion which can be used to distinguish many more non-complete graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437729","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":"Minimal skew semistandard tableaux and the Hillman–Grassl correspondence","authors":"Alejandro H. Morales ,&nbsp;Greta Panova ,&nbsp;GaYee Park","doi":"10.1016/j.aam.2024.102792","DOIUrl":"10.1016/j.aam.2024.102792","url":null,"abstract":"<div><div>Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula <span><span>(NHLF)</span></span> as a positive sum over excited diagrams of products of hook-lengths. Subsequently, Morales, Pak, and Panova gave a <em>q</em>-analogue of this formula in terms of skew semistandard tableaux (SSYT). They also showed, partly algebraically, that the Hillman–Grassl bijection, restricted to skew semistandard tableaux, is behind their <em>q</em>-analogue. We study the problem of circumventing the algebraic part and proving the bijection completely combinatorially, which we do for the case of border strips. For general skew shapes, we define minimal semistandard Young tableaux, that are in correspondence with excited diagrams via a new description of the Hillman–Grassl bijection and have an analogue of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula <span><span>(OOF)</span></span> for counting standard tableaux of skew shape. Our construction immediately implies that the summands in the NHLF are less than the summands in the OOF and we characterize the shapes where both formulas have the same number of summands.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434090","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":"Proof of a conjecture about Parrondo's paradox for two-armed slot machines","authors":"Huaijin Liang ,&nbsp;Zengjing Chen","doi":"10.1016/j.aam.2024.102793","DOIUrl":"10.1016/j.aam.2024.102793","url":null,"abstract":"<div><div>The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter <em>J</em>. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm <em>A</em> and arm <em>B</em>, played individually, are asymptotically fair, but when alternated randomly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with <em>r A</em>s and <em>s B</em>s (e.g., <span><math><mi>A</mi><mi>B</mi><mi>A</mi><mi>B</mi><mi>B</mi></math></span> if <span><math><mi>r</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>s</mi><mo>=</mo><mn>3</mn></math></span>). They established the Parrondo effect if <span><math><mi>r</mi><mo>+</mo><mi>s</mi></math></span> divides <em>J</em>, and conjectured it in four other situations, including the case <span><math><mi>J</mi><mo>=</mo><mn>2</mn></math></span> with <span><math><mi>r</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>s</mi><mo>≥</mo><mn>1</mn></math></span>. We prove the conjecture in the latter case.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142423378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A topological approach to mapping space signatures","authors":"Chad Giusti ,&nbsp;Darrick Lee ,&nbsp;Vidit Nanda ,&nbsp;Harald Oberhauser","doi":"10.1016/j.aam.2024.102787","DOIUrl":"10.1016/j.aam.2024.102787","url":null,"abstract":"<div><div>A common approach for describing classes of functions and probability measures on a topological space <span><math><mi>X</mi></math></span> is to construct a suitable map Φ from <span><math><mi>X</mi></math></span> into a vector space, where linear methods can be applied to address both problems. The case where <span><math><mi>X</mi></math></span> is a space of paths <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where <span><math><mi>X</mi></math></span> is a space of maps <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for any <span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span>, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>. The key ingredient to our approach is topological; in particular, our starting point is a generalization of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326570","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":"Permanent identities, combinatorial sequences, and permutation statistics","authors":"Shishuo Fu ,&nbsp;Zhicong Lin ,&nbsp;Zhi-Wei Sun","doi":"10.1016/j.aam.2024.102789","DOIUrl":"10.1016/j.aam.2024.102789","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we confirm six conjectures on the exact values of some permanents, relating them to the Genocchi numbers of the first and second kinds as well as the Euler numbers. For example, we prove that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;per&lt;/mi&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;⌊&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;⌋&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are the Bernoulli numbers. We also show that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;per&lt;/mi&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;sgn&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mtext&gt;if &lt;/mtext&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mtext&gt;if &lt;/mtext&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;sgn&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the sign function, and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are the Euler (zigzag) numbers.&lt;/div&gt;&lt;div&gt;In the course of linking the evaluation of these permanents to the aforementioned combinatorial sequences, the classical permutation statistic – the exceda","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319000","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":"Continued fractions for q-deformed real numbers, {−1,0,1}-Hankel determinants, and Somos-Gale-Robinson sequences","authors":"Valentin Ovsienko ,&nbsp;Emmanuel Pedon","doi":"10.1016/j.aam.2024.102788","DOIUrl":"10.1016/j.aam.2024.102788","url":null,"abstract":"<div><div><em>q</em>-deformed real numbers are power series with integer coefficients. We study Stieltjes and Jacobi type continued fraction expansions of <em>q</em>-deformed real numbers and find many new examples of such continued fractions. We also investigate the corresponding sequences of Hankel determinants and find an infinite family of power series for which several of the first sequences of Hankel determinants consist of <span><math><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn></math></span> and 1 only. These Hankel sequences satisfy Somos and Gale-Robinson recurrences.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001209/pdfft?md5=8ffb0f6262c5c3186d8020047fccd544&pid=1-s2.0-S0196885824001209-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315071","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":"Summing the “exactly one 42” and similar subsums of the harmonic series","authors":"Jean-François Burnol","doi":"10.1016/j.aam.2024.102791","DOIUrl":"10.1016/j.aam.2024.102791","url":null,"abstract":"<div><p>For <span><math><mi>b</mi><mo>&gt;</mo><mn>1</mn></math></span> and <em>αβ</em> a string of two digits in base <em>b</em>, let <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> be the subsum of the harmonic series with only those integers having exactly one occurrence of <em>αβ</em>. We obtain a theoretical representation of such <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> series which, say for <span><math><mi>b</mi><mo>=</mo><mn>10</mn></math></span>, allows computing them all to thousands of digits. This is based on certain specific measures on the unit interval and the use of their Stieltjes transforms at negative integers. Integral identities of a combinatorial nature both explain the relation to the <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> sums and lead to recurrence formulas for the measure moments allowing in the end the straightforward numerical implementation.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001234/pdfft?md5=2a1220cc0cdb8447beb302719d095400&pid=1-s2.0-S0196885824001234-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271498","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":"Betti numbers and torsions in homology groups of double coverings","authors":"Suguru Ishibashi ,&nbsp;Sakumi Sugawara ,&nbsp;Masahiko Yoshinaga","doi":"10.1016/j.aam.2024.102790","DOIUrl":"10.1016/j.aam.2024.102790","url":null,"abstract":"<div><p>Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial 2-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001222/pdfft?md5=69da8c583775517da2bb2711b6c0326e&pid=1-s2.0-S0196885824001222-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271499","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}