Pub Date : 2015-01-01DOI: 10.1112/S1461157015000273
R. Guglielmetti
CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch). Supplementary materials are available with this article. Introduction LetH be the hyperbolic n-space, and let IsomH be the group of isometries ofH. For a given discrete hyperbolic Coxeter group Γ < IsomH and its associated fundamental polyhedron P ⊂ H, we are interested in geometrical and combinatorial properties of P . We want to know whether P is compact, has finite volume and, if the answer is yes, what its volume is. We also want to find the combinatorial structure of P , namely, the number of vertices, edges, 2-faces, and so on. Finally, it is interesting to find out whether Γ is arithmetic, that is, if Γ is commensurable to the reflection group of the automorphism group of a quadratic form of signature (n, 1). Most of these questions can be answered by studying finite and affine subgroups of Γ, but this involves a huge number of computations. This article presents the algorithms used in CoxIter, a computer program written in C++ designed to compute these invariants. The program is published under a free license (the GNU General Public License v3) and can be used freely in various projects. The source code and the documentation are available as supplementary material with the online version of this article and on the author’s website. The input of CoxIter is the graph of a hyperbolic Coxeter group (encoded in a simple way in a text file, see Appendix A) and a typical output can be the following. Reading file: ../graphs/14-vinb85.coxiter Number of vertices: 17 Dimension: 14 Vertices: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 Field generated by the entries of the Gram matrix: Q[sqrt(2)] File read Information Cocompact: no Finite covolume: yes Received 5 January 2015; revised 28 July 2015. 2010 Mathematics Subject Classification 5104 (primary), 52B05, 20F55 (secondary). Supported by the Schweizerischer Nationalfonds SNF no. 20002
{"title":"CoxIter – Computing invariants of hyperbolic Coxeter groups","authors":"R. Guglielmetti","doi":"10.1112/S1461157015000273","DOIUrl":"https://doi.org/10.1112/S1461157015000273","url":null,"abstract":"CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch). Supplementary materials are available with this article. Introduction LetH be the hyperbolic n-space, and let IsomH be the group of isometries ofH. For a given discrete hyperbolic Coxeter group Γ < IsomH and its associated fundamental polyhedron P ⊂ H, we are interested in geometrical and combinatorial properties of P . We want to know whether P is compact, has finite volume and, if the answer is yes, what its volume is. We also want to find the combinatorial structure of P , namely, the number of vertices, edges, 2-faces, and so on. Finally, it is interesting to find out whether Γ is arithmetic, that is, if Γ is commensurable to the reflection group of the automorphism group of a quadratic form of signature (n, 1). Most of these questions can be answered by studying finite and affine subgroups of Γ, but this involves a huge number of computations. This article presents the algorithms used in CoxIter, a computer program written in C++ designed to compute these invariants. The program is published under a free license (the GNU General Public License v3) and can be used freely in various projects. The source code and the documentation are available as supplementary material with the online version of this article and on the author’s website. The input of CoxIter is the graph of a hyperbolic Coxeter group (encoded in a simple way in a text file, see Appendix A) and a typical output can be the following. Reading file: ../graphs/14-vinb85.coxiter Number of vertices: 17 Dimension: 14 Vertices: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 Field generated by the entries of the Gram matrix: Q[sqrt(2)] File read Information Cocompact: no Finite covolume: yes Received 5 January 2015; revised 28 July 2015. 2010 Mathematics Subject Classification 5104 (primary), 52B05, 20F55 (secondary). Supported by the Schweizerischer Nationalfonds SNF no. 20002","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"754-773"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000273","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.1112/S1461157015000133
H. Duan, Xuan Zhao
Let G be a compact connected Lie group with a maximal torus T . In the context of Schubert calculus we present the integral cohomology H ∗ ( G/T ) by a minimal system of generators and relations.
{"title":"Schubert presentation of the cohomology ring of flag manifolds","authors":"H. Duan, Xuan Zhao","doi":"10.1112/S1461157015000133","DOIUrl":"https://doi.org/10.1112/S1461157015000133","url":null,"abstract":"Let G be a compact connected Lie group with a maximal torus T . In the context of Schubert calculus we present the integral cohomology H ∗ ( G/T ) by a minimal system of generators and relations.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"489-506"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000133","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.1112/S1461157015000248
R. Wilson
{"title":"Every in the Monster contains -elements","authors":"R. Wilson","doi":"10.1112/S1461157015000248","DOIUrl":"https://doi.org/10.1112/S1461157015000248","url":null,"abstract":"","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"667-674"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000248","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.1112/S1461157014000485
Z. Gu, Yanping Chen
Our main purpose in this paper is to propose the piecewise Legendre spectral-collocation method to solve Volterra integro-differential equations. We provide convergence analysis to show that the numerical errors in our method decay in $h^{m}N^{-m}$ -version rate. These results are better than the piecewise polynomial collocation method and the global Legendre spectral-collocation method. The provided numerical examples confirm these theoretical results.
{"title":"Piecewise Legendre spectral-collocation method for Volterra integro-differential equations","authors":"Z. Gu, Yanping Chen","doi":"10.1112/S1461157014000485","DOIUrl":"https://doi.org/10.1112/S1461157014000485","url":null,"abstract":"Our main purpose in this paper is to propose the piecewise Legendre spectral-collocation method to solve Volterra integro-differential equations. We provide convergence analysis to show that the numerical errors in our method decay in $h^{m}N^{-m}$ -version rate. These results are better than the piecewise polynomial collocation method and the global Legendre spectral-collocation method. The provided numerical examples confirm these theoretical results.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"231-249"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000485","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.1112/S1461157014000412
Edmond W. H. Lee, Wen Ting Zhang
Two semigroups are distinct if they are neither isomorphic nor anti-isomorphic. Although there exist $15,973$ pairwise distinct semigroups of order six, only four are known to be non-finitely based. In the present article, the finite basis property of the other $15,969$ distinct semigroups of order six is verified. Since all semigroups of order five or less are finitely based, the four known non-finitely based semigroups of order six are the only examples of minimal order.
{"title":"Finite Basis Problem for Semigroups of Order Six","authors":"Edmond W. H. Lee, Wen Ting Zhang","doi":"10.1112/S1461157014000412","DOIUrl":"https://doi.org/10.1112/S1461157014000412","url":null,"abstract":"Two semigroups are distinct if they are neither isomorphic nor anti-isomorphic. Although there exist $15,973$ pairwise distinct semigroups of order six, only four are known to be non-finitely based. In the present article, the finite basis property of the other $15,969$ distinct semigroups of order six is verified. Since all semigroups of order five or less are finitely based, the four known non-finitely based semigroups of order six are the only examples of minimal order.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"1-129"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000412","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.1112/S1461157014000473
A. P. Pobegailo
Polynomials for blending parametric curves in Lie groups are defined. Properties of these polynomials are proved. Blending parametric curves in Lie groups with these polynomials is considered. Then application of the proposed technique to construction of spline curves on smooth manifolds is presented. As an example, construction of spherical spline curves using the proposed approach is depicted.
{"title":"Construction of spline curves on smooth manifolds by action of Lie groups","authors":"A. P. Pobegailo","doi":"10.1112/S1461157014000473","DOIUrl":"https://doi.org/10.1112/S1461157014000473","url":null,"abstract":"Polynomials for blending parametric curves in Lie groups are defined. Properties of these polynomials are proved. Blending parametric curves in Lie groups with these polynomials is considered. Then application of the proposed technique to construction of spline curves on smooth manifolds is presented. As an example, construction of spherical spline curves using the proposed approach is depicted.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"24 1","pages":"217-230"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000473","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.1112/S1461157015000200
G. Chatzarakis, Ö. Öcalan
Consider the first-order retarded differential equation $$begin{eqnarray}x^{prime }(t)+p(t)x({ittau}(t))=0,quad tgeqslant t_{0},end{eqnarray}$$ where $p(t)geqslant 0$ and ${ittau}(t)$ is a function of positive real numbers such that ${ittau}(t)leqslant t$ for $tgeqslant t_{0}$ , and $lim _{trightarrow infty }{ittau}(t)=infty$ . Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving $liminf$ , is established when the well-known oscillation condition $$begin{eqnarray}liminf _{trightarrow infty }int _{{ittau}(t)}^{t}p(s),ds>frac{1}{e}end{eqnarray}$$ is not satisfied. An example illustrating the result is also given.
{"title":"Oscillation of differential equations with non-monotone retarded arguments","authors":"G. Chatzarakis, Ö. Öcalan","doi":"10.1112/S1461157015000200","DOIUrl":"https://doi.org/10.1112/S1461157015000200","url":null,"abstract":"Consider the first-order retarded differential equation $$begin{eqnarray}x^{prime }(t)+p(t)x({ittau}(t))=0,quad tgeqslant t_{0},end{eqnarray}$$ where $p(t)geqslant 0$ and ${ittau}(t)$ is a function of positive real numbers such that ${ittau}(t)leqslant t$ for $tgeqslant t_{0}$ , and $lim _{trightarrow infty }{ittau}(t)=infty$ . Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving $liminf$ , is established when the well-known oscillation condition $$begin{eqnarray}liminf _{trightarrow infty }int _{{ittau}(t)}^{t}p(s),ds>frac{1}{e}end{eqnarray}$$ is not satisfied. An example illustrating the result is also given.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"660-666"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000200","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.1112/S1461157015000029
Jennifer S. Balakrishnan
The Coleman integral is a p-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Har- rison (11), we extend the Coleman integration algorithms in (3), (1) to even degree models of hyperelliptic curves. We illustrate our methods with numer- ical examples computed in Sage.
{"title":"COLEMAN INTEGRATION FOR EVEN DEGREE MODELS OF HYPERELLIPTIC CURVES","authors":"Jennifer S. Balakrishnan","doi":"10.1112/S1461157015000029","DOIUrl":"https://doi.org/10.1112/S1461157015000029","url":null,"abstract":"The Coleman integral is a p-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Har- rison (11), we extend the Coleman integration algorithms in (3), (1) to even degree models of hyperelliptic curves. We illustrate our methods with numer- ical examples computed in Sage.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"258-265"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000029","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.1112/S1461157015000078
T. Grant
Conservation laws provide important constraints on the solutions of partial differential equations (PDEs), therefore it is important to preserve them when discretizing such equations. In this paper, a new systematic method for discretizing a PDE, so as to preserve the local form of multiple conservation laws, is presented. The technique, which uses symbolic computation, is applied to the Korteweg–de Vries (KdV) equation to find novel explicit and implicit schemes that have finite difference analogues of its first and second conservation laws and its first and third conservation laws. The resulting schemes are numerically compared with a multisymplectic scheme.
{"title":"Bespoke finite difference schemes that preserve multiple conservation laws","authors":"T. Grant","doi":"10.1112/S1461157015000078","DOIUrl":"https://doi.org/10.1112/S1461157015000078","url":null,"abstract":"Conservation laws provide important constraints on the solutions of partial differential equations (PDEs), therefore it is important to preserve them when discretizing such equations. In this paper, a new systematic method for discretizing a PDE, so as to preserve the local form of multiple conservation laws, is presented. The technique, which uses symbolic computation, is applied to the Korteweg–de Vries (KdV) equation to find novel explicit and implicit schemes that have finite difference analogues of its first and second conservation laws and its first and third conservation laws. The resulting schemes are numerically compared with a multisymplectic scheme.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"372-403"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000078","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.1112/S1461157015000236
M. Nijmeijer
We present a parallel algorithm to calculate a numerical approximation of a single, isolated root ${italpha}$ of a function $f:mathbb{R}rightarrow mathbb{R}$ which is sufficiently regular at and around ${italpha}$ . The algorithm is derivative free and performs one function evaluation on each processor per iteration. It requires at least three processors and can be scaled up to any number of these. The order with which the generated sequence of approximants converges to ${italpha}$ is equal to $(n+sqrt{n^{2}+4})/2$ for $n+1$ processors with $ngeqslant 2$ . This assumes that particular combinations of the derivatives of $f$ do not vanish at ${italpha}$ .
{"title":"A parallel root-finding algorithm","authors":"M. Nijmeijer","doi":"10.1112/S1461157015000236","DOIUrl":"https://doi.org/10.1112/S1461157015000236","url":null,"abstract":"We present a parallel algorithm to calculate a numerical approximation of a single, isolated root ${italpha}$ of a function $f:mathbb{R}rightarrow mathbb{R}$ which is sufficiently regular at and around ${italpha}$ . The algorithm is derivative free and performs one function evaluation on each processor per iteration. It requires at least three processors and can be scaled up to any number of these. The order with which the generated sequence of approximants converges to ${italpha}$ is equal to $(n+sqrt{n^{2}+4})/2$ for $n+1$ processors with $ngeqslant 2$ . This assumes that particular combinations of the derivatives of $f$ do not vanish at ${italpha}$ .","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"713-729"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000236","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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