{"title":"Dissimilar subalgebras of symmetry algebra of plasticity equations","authors":"Sergey I. Senashov , Alexander Yakhno","doi":"10.1016/j.jsc.2024.102358","DOIUrl":"10.1016/j.jsc.2024.102358","url":null,"abstract":"<div><p>In this paper we construct the optimal sets of dissimilar subalgebras up to dimension three for the Lie algebra of point symmetries of the system of three-dimensional stationary equations of perfect plasticity with the Huber–von Mises yield condition. The obtained results can be used to solve the problem of determining all invariant solutions of this system. It was necessary to design algorithms to facilitate some steps of the classification of subalgebras. The computational algebraic system SageMath was chosen to implement these algorithms. The most used functions and procedures are listed. The developed algorithms can be adapted to classify subalgebras of higher dimensions.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102358"},"PeriodicalIF":0.6,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141630273","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":"Determinant evaluations inspired by Di Francesco's determinant for twenty-vertex configurations","authors":"","doi":"10.1016/j.jsc.2024.102352","DOIUrl":"10.1016/j.jsc.2024.102352","url":null,"abstract":"<div><p>In his work on the twenty vertex model, <span><span>Di Francesco (2021)</span></span> found a determinant formula for the number of configurations in a specific such model, and he conjectured a closed form product formula for the evaluation of this determinant. We prove this conjecture here. Moreover, we actually generalize this determinant evaluation to a one-parameter family of determinant evaluations, and we present many more determinant evaluations of similar type — some proved, some left open as conjectures.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102352"},"PeriodicalIF":0.6,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574411","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":"Graceful bases in solution spaces of differential and difference equations","authors":"","doi":"10.1016/j.jsc.2024.102355","DOIUrl":"10.1016/j.jsc.2024.102355","url":null,"abstract":"<div><p>We construct fundamental systems of solutions to linear ordinary differential equations, linear difference equations, and systems of partial differential equations whose elements remain linearly independent for all values of algebraically independent symbolic parameters.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102355"},"PeriodicalIF":0.6,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574413","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 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 , 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 , 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 , 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, 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}
Amir Hashemi , Mahshid Mirhashemi , Werner M. Seiler
{"title":"Machine learning parameter systems, Noether normalisations and quasi-stable positions","authors":"Amir Hashemi , Mahshid Mirhashemi , 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}
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