Journal of Symbolic Computation最新文献

{"title":"On the log-concavity of the n-th root of sequences","authors":"Ernest X.W. Xia , Zuo-Ru Zhang","doi":"10.1016/j.jsc.2024.102349","DOIUrl":"10.1016/j.jsc.2024.102349","url":null,"abstract":"<div><p>In recent years, the log-concavity of the <em>n</em>-th root of a sequence <span><math><msub><mrow><mo>{</mo><mroot><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></mroot><mo>}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math></span> has been received a lot of attention. Recently, Sun posed the following conjecture in his new book: the sequences <span><math><msub><mrow><mo>{</mo><mroot><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></mroot><mo>}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mo>{</mo><mroot><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></mroot><mo>}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math></span> are log-concave, where<span><span><span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac></math></span></span></span> and<span><span><span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo>(</mo><mn>3</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>.</mo></math></span></span></span> In this paper, two methods, semi-automatic and analytic methods, are used to confirm Sun's conjecture. The semi-automatic method relies on a criterion on the log-concavity of <span><math><msub><mrow><mo>{</mo><mroot><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></mroot><mo>}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math></span> given b","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102349"},"PeriodicalIF":0.6,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the dimension of the solution space of linear difference equations over the ring of infinite sequences","authors":"Sergei Abramov ,&nbsp;Gleb Pogudin","doi":"10.1016/j.jsc.2024.102350","DOIUrl":"10.1016/j.jsc.2024.102350","url":null,"abstract":"<div><p>For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior information on the dimension is available.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102350"},"PeriodicalIF":0.6,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Orthogonal-symplectic matrices and their parametric representation","authors":"Alexander Batkhin ,&nbsp;Alexander Petrov","doi":"10.1016/j.jsc.2024.102353","DOIUrl":"10.1016/j.jsc.2024.102353","url":null,"abstract":"<div><p>The method of computing the parametric representation of an even orthogonal symplectic matrix is considered. The dimension of the family of such matrices is calculated. The general structure of matrices of small even dimensions up to 8 is discussed in detail. Theorem on the structure of a skew symmetric matrix generating a generic orthogonal symplectic matrix is proven. The problem of constructing an orthogonal symplectic matrix of dimension 4 by a given vector is solved. The application of this transformation to the study of families of periodic solutions to an autonomous Hamiltonian system with two degrees of freedom is discussed.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102353"},"PeriodicalIF":0.6,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"MacMahon's partition analysis XV: Parity","authors":"George E. Andrews ,&nbsp;Peter Paule","doi":"10.1016/j.jsc.2024.102351","DOIUrl":"10.1016/j.jsc.2024.102351","url":null,"abstract":"<div><p>We apply the methods of partition analysis to partitions in which the parity of parts plays a role. We begin with an in-depth treatment of the generating function for the partitions from the first Göllnitz-Gordon identity. We then deduce a Schmidt-type theorem related to the false theta functions. We also consider: (1) position parity, (2) partitions with distinct even parts, (3) partitions with distinct odd parts. One of the corollaries of these last considerations is a new interpretation of Hei-Chi Chan's cubic partitions. A second part of our article is devoted to the algorithmic derivation of identities and arithmetic congruences related to the generating functions considered in part one, including cubic partitions. To this end, Smoot's implementation of Radu's Ramanujan-Kolberg algorithm is used. Finally, we give a short description which explains how to use the Omega package to derive special instances of the results of part one.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102351"},"PeriodicalIF":0.6,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000555/pdfft?md5=f2007103044731924521360849de8523&pid=1-s2.0-S0747717124000555-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Arithmetic of D-algebraic functions","authors":"Bertrand Teguia Tabuguia","doi":"10.1016/j.jsc.2024.102348","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102348","url":null,"abstract":"<div><p>We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations are algebraic (ordinary or partial) differential equations (ADEs). The general purpose is to find ADEs whose solutions contain specified rational expressions of solutions to given ADEs. For univariate D-algebraic functions, we show how to derive an ADE of smallest possible order. In the multivariate case, we introduce a general algorithm for these computations and derive conclusions on the order bound of the resulting algebraic PDE. Using our accompanying Maple software, we discuss applications in physics, statistics, and symbolic integration.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102348"},"PeriodicalIF":0.6,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S074771712400052X/pdfft?md5=e432083aacaf22074cdb4e298b830ed8&pid=1-s2.0-S074771712400052X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141483108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Enumerating seating arrangements that obey social distancing","authors":"George Spahn,&nbsp;Doron Zeilberger","doi":"10.1016/j.jsc.2024.102344","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102344","url":null,"abstract":"<div><p>We illustrate the power of symbolic computation and experimental mathematics by investigating maximal seating arrangements, either on a line, or in a rectangular auditorium with a fixed number of columns but an arbitrary number of rows, that obey any prescribed set of ‘social distancing’ restrictions. In addition to enumeration, we study the statistical distribution of the density, and give simulation algorithms for generating them.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102344"},"PeriodicalIF":0.6,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000488/pdfft?md5=6a5aa36ac0cbbaa9177e66ebc47b5af0&pid=1-s2.0-S0747717124000488-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141483106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Machine learning parameter systems, Noether normalisations and quasi-stable positions","authors":"Amir Hashemi ,&nbsp;Mahshid Mirhashemi ,&nbsp;Werner M. Seiler","doi":"10.1016/j.jsc.2024.102345","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102345","url":null,"abstract":"<div><p>We discuss the use of machine learning models for finding “good coordinates” for polynomial ideals. Our main goal is to put ideals into quasi-stable position, as this generic position shares most properties of the generic initial ideal, but can be deterministically reached and verified. Furthermore, it entails a Noether normalisation and provides us with a system of parameters. Traditional approaches use either random choices which typically destroy all sparsity or rather simple human heuristics which are only moderately successful. Our experiments show that machine learning models provide us here with interesting alternatives that most of the time make nearly optimal choices.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102345"},"PeriodicalIF":0.6,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S074771712400049X/pdfft?md5=ecbd0dd906bdab6ee43183876959d620&pid=1-s2.0-S074771712400049X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141483107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solving second order homogeneous differential equations in terms of Heun's general function","authors":"Shayea Aldossari","doi":"10.1016/j.jsc.2024.102347","DOIUrl":"10.1016/j.jsc.2024.102347","url":null,"abstract":"<div><p>In this paper, we present an algorithm that checks if a second-order differential operator <span><math><mi>L</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>[</mo><mo>∂</mo><mo>]</mo></math></span> can be reduced to the general Heun's differential operator. The algorithm detects the parameters of the transformations in <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>[</mo><mo>∂</mo><mo>]</mo></math></span> that transfer the general Heun's differential operator to the operator <em>L</em> whose solutions are of the form<span><span><span>(1)</span><span><math><mrow><mi>exp</mi></mrow><mo>(</mo><mo>∫</mo><mi>r</mi><mspace></mspace><mi>d</mi><mi>x</mi><mo>)</mo><mo>⋅</mo><mrow><mi>HeunG</mi></mrow><mo>(</mo><mi>a</mi><mo>,</mo><mspace></mspace><mi>q</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>,</mo><mi>δ</mi><mo>;</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mo>{</mo><mi>α</mi><mo>,</mo><mspace></mspace><mi>β</mi><mo>,</mo><mspace></mspace><mi>δ</mi><mo>,</mo><mspace></mspace><mi>γ</mi><mo>}</mo><mo>∈</mo><mi>Q</mi><mo>∖</mo><mi>Z</mi></math></span>, the functions <span><math><mi>r</mi><mo>,</mo><mspace></mspace><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, and <span><math><mi>C</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> is a subfield of <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> of index 2 or 3 or <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mspace></mspace><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mi>b</mi></mrow><mrow><mi>c</mi><mspace></mspace><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mi>d</mi></mrow></mfrac></math></span> for some <em>n</em> in <span><math><mi>N</mi><mo>∖</mo><mo>{</mo><mn>1</mn><mo>}</mo></math></span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102347"},"PeriodicalIF":0.6,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141401722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The deviation on cranks of partitions","authors":"Julia Q.D. Du","doi":"10.1016/j.jsc.2024.102346","DOIUrl":"10.1016/j.jsc.2024.102346","url":null,"abstract":"<div><p>In this paper, we present an algorithm to compute the deviation of the cranks from the average by using the theory of modular forms and Jacobi forms. Then applying the Ramanujan-type algorithm developed by Chen, Du and Zhao to each term in the expression of the deviation, we can derive the corresponding dissection formulas. As applications, we revisit the deviation of the cranks modulo 5 and 7, which were given by Garvan, and Mortenson, and also obtain the deviation of the cranks modulo 9 and 14.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102346"},"PeriodicalIF":0.6,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141395493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A post-quantum key exchange protocol from the intersection of conics","authors":"Alberto Alzati ,&nbsp;Daniele Di Tullio,&nbsp;Manoj Gyawali,&nbsp;Alfonso Tortora","doi":"10.1016/j.jsc.2024.102343","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102343","url":null,"abstract":"<div><p>In this paper we present a key exchange protocol in which Alice and Bob have secret keys given by two conics embedded in a large ambient space by means of the Veronese embedding and public keys given by hyperplanes containing the embedded curves. Both of them construct some common invariants given by the intersection of two conics.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102343"},"PeriodicalIF":0.6,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000476/pdfft?md5=c1a759b0f1c037e879260d12c7a34bcb&pid=1-s2.0-S0747717124000476-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141483104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}