We deal with Temperley-Lieb algebras of type B, extending the result in [3, §6]. By completing the relations coming from a presentation of the Temperley-Lieb algebra, we find its Gröbner-Shirshov basis to obtain the corresponding set of standard monomials. The explicit multiplication table between the monomials follows naturally.
{"title":"Standard monomials for temperley-lieb algebras","authors":"Sungsoon Kim, Dong-il Lee","doi":"10.1145/3055282.3055296","DOIUrl":"https://doi.org/10.1145/3055282.3055296","url":null,"abstract":"We deal with Temperley-Lieb algebras of type B, extending the result in [3, §6]. By completing the relations coming from a presentation of the Temperley-Lieb algebra, we find its Gröbner-Shirshov basis to obtain the corresponding set of standard monomials. The explicit multiplication table between the monomials follows naturally.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"59 1","pages":"179-181"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73976578","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}
It is with great sadness that we annonce you the death of our friend and colleague Marc Rybowicz. He passed away on November 11th after a long fight against cancer since June 2015.
{"title":"Death of Marc Rybowicz, aged 52","authors":"D. Duval, A. Poteaux","doi":"10.1145/3055282.3055300","DOIUrl":"https://doi.org/10.1145/3055282.3055300","url":null,"abstract":"It is with great sadness that we annonce you the death of our friend and colleague Marc Rybowicz. He passed away on November 11th after a long fight against cancer since June 2015.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"39 1","pages":"191"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90779639","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}
We employ two techniques to dramatically improve Maple's performance on the Fermat benchmarks for simplifying rational expressions. First, we factor expanded polynomials to ensure that gcds are identified and cancelled automatically. Second, we replace all expanded polynomials by new variables and normalize the result. To undo the substitutions, we use a C routine for sparse multivariate division by a set of polynomials. The resulting times for the first Fermat benchmark are a factor of 17x faster than Fermat and 39x faster than Magma.
{"title":"Fermat benchmarks for rational expressionals in maple","authors":"M. Monagan, Roman Pearce","doi":"10.1145/3055282.3055299","DOIUrl":"https://doi.org/10.1145/3055282.3055299","url":null,"abstract":"We employ two techniques to dramatically improve Maple's performance on the Fermat benchmarks for simplifying rational expressions. First, we factor expanded polynomials to ensure that gcds are identified and cancelled automatically. Second, we replace all expanded polynomials by new variables and normalize the result. To undo the substitutions, we use a C routine for sparse multivariate division by a set of polynomials. The resulting times for the first Fermat benchmark are a factor of 17x faster than Fermat and 39x faster than Magma.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"44 1","pages":"188-190"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84061780","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}
The exact probability, dependent on the matrix structure, is given that the block Wiedemann algorithm correctly computes the leading invariant factors of a matrix. A tight lower bound, structure independent, is derived.
{"title":"Probabilistic analysis of block wiedemann for leading invariant factors","authors":"Gavin Harrison, Jeremy R. Johnson, B. D. Saunders","doi":"10.1145/3055282.3055294","DOIUrl":"https://doi.org/10.1145/3055282.3055294","url":null,"abstract":"The exact probability, dependent on the matrix structure, is given that the block Wiedemann algorithm correctly computes the leading invariant factors of a matrix. A tight lower bound, structure independent, is derived.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"1 1","pages":"173-175"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90898473","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}
Polynomials eigenvalue problems with structured matrix polynomials arise in many applications. The standard way to solve polynomial eigenvalue problems is through the classical Frobenius companion linearizations, which may not retain the structure of the matrix polynomial. Particularly, the structure of the symmetric matrix polynomials can be lost, while from the computational point of view, it is advisable to construct a linearization which preserves the symmetry structure. Recently, new families of block-Kronecker pencils have been introduced in [5]. Applying block-Kronecker pencils, we present structure-preserving strong linearizations for symmetric matrix polynomials. When the matrix polynomial has an odd degree, these linearizations are strong regardless of whether the matrix polynomial is regular or singular. Additionally, we construct structure-preserving strong linearizations for regular symmetric matrix polynomials of even degree under some simple nonsingularity conditions.
{"title":"Constructing symmetric structure-preserving strong linearizations","authors":"H. Faßbender, J. Pérez, N. Shayanfar","doi":"10.1145/3055282.3055292","DOIUrl":"https://doi.org/10.1145/3055282.3055292","url":null,"abstract":"Polynomials eigenvalue problems with structured matrix polynomials arise in many applications. The standard way to solve polynomial eigenvalue problems is through the classical Frobenius companion linearizations, which may not retain the structure of the matrix polynomial. Particularly, the structure of the symmetric matrix polynomials can be lost, while from the computational point of view, it is advisable to construct a linearization which preserves the symmetry structure. Recently, new families of block-Kronecker pencils have been introduced in [5]. Applying block-Kronecker pencils, we present structure-preserving strong linearizations for symmetric matrix polynomials. When the matrix polynomial has an odd degree, these linearizations are strong regardless of whether the matrix polynomial is regular or singular. Additionally, we construct structure-preserving strong linearizations for regular symmetric matrix polynomials of even degree under some simple nonsingularity conditions.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"47 1","pages":"167-169"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79964313","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}
Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA.
{"title":"Abstracts of recent doctoral dissertations in computer algebra","authors":"Hui-li Huang, Y. Zhang","doi":"10.1145/3151131.3151135","DOIUrl":"https://doi.org/10.1145/3151131.3151135","url":null,"abstract":"Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"36 4 1","pages":"70-72"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85003138","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}
A nearring N is an algebra that does not satisfy the ring axioms, but nevertheless satisfies a one-sided distributivity law. SONATA is a GAP package designed as a research tool for nearrings. Users are able to define and work with finite nearrings in order to generate and test conjectures.
{"title":"Sonata: a GAP tool for nearring computations","authors":"E. Aichinger, Rika Yatchak","doi":"10.1145/3015306.3015310","DOIUrl":"https://doi.org/10.1145/3015306.3015310","url":null,"abstract":"A nearring N is an algebra that does not satisfy the ring axioms, but nevertheless satisfies a one-sided distributivity law. SONATA is a GAP package designed as a research tool for nearrings. Users are able to define and work with finite nearrings in order to generate and test conjectures.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"40 1","pages":"89-92"},"PeriodicalIF":0.0,"publicationDate":"2016-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91179737","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}
M. Abánades, F. Botana, Z. Kovács, T. Recio, C. Sólyom-Gecse
Much effort has been put into the implementation of automatic proving in interactive geometric environments (e.g. Java Geometry Expert, GeoGebra). The closely related concept of automatic discovery, remains however almost unexplored. This software presentation will demonstrate our results towards the incorporation of automatic discovery capabilities into GeoGebra, an educational software with tens of millions of users worldwide. As main result, we report on a new command, currently available in the official version, that allows the automatic discovery of loci of points in diagrams defined by implicit conditions. This represents an extension of a previous command, similar in nature, but restricted to loci defined by the standard mover-tracer construction. Our proposal successfully automates the `dummy locus dragging' in dynamic geometry. This makes the cycle conjecturing-checking-proving accessible for general users in elementary geometry.
{"title":"Development of automatic reasoning tools in GeoGebra","authors":"M. Abánades, F. Botana, Z. Kovács, T. Recio, C. Sólyom-Gecse","doi":"10.1145/3015306.3015309","DOIUrl":"https://doi.org/10.1145/3015306.3015309","url":null,"abstract":"Much effort has been put into the implementation of automatic proving in interactive geometric environments (e.g. Java Geometry Expert, GeoGebra). The closely related concept of automatic discovery, remains however almost unexplored.\u0000 This software presentation will demonstrate our results towards the incorporation of automatic discovery capabilities into GeoGebra, an educational software with tens of millions of users worldwide. As main result, we report on a new command, currently available in the official version, that allows the automatic discovery of loci of points in diagrams defined by implicit conditions. This represents an extension of a previous command, similar in nature, but restricted to loci defined by the standard mover-tracer construction. Our proposal successfully automates the `dummy locus dragging' in dynamic geometry. This makes the cycle conjecturing-checking-proving accessible for general users in elementary geometry.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"1 1","pages":"85-88"},"PeriodicalIF":0.0,"publicationDate":"2016-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89885345","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}
We present the Mathematica package PlanarLinkages, which provides commands for constructing and visualizing planar linkages that draw a prescribed algebraic curve of genus 0, or more generally, that follow a prescribed rational motion. Since the implemented algorithms are heavily based on the concept of motion polynomials, the functionality of the package also includes basic arithmetic of motion polynomials and a factorization procedure.
{"title":"Motion polynomials and planar linkages","authors":"C. Koutschan","doi":"10.1145/3015306.3015315","DOIUrl":"https://doi.org/10.1145/3015306.3015315","url":null,"abstract":"We present the Mathematica package PlanarLinkages, which provides commands for constructing and visualizing planar linkages that draw a prescribed algebraic curve of genus 0, or more generally, that follow a prescribed rational motion. Since the implemented algorithms are heavily based on the concept of motion polynomials, the functionality of the package also includes basic arithmetic of motion polynomials and a factorization procedure.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"12 1","pages":"109-112"},"PeriodicalIF":0.0,"publicationDate":"2016-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74601799","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}
The problem of isolating the real roots of a univariate polynomial with integer coefficients is an important problem in computational mathematics. Given a polynomial with integer coefficients, [EQUATION], the objective is to isolate the real roots of f, that is to compute intervals with rational endpoints that contain one and only one root of f. SLV is an open source software package written in C that provides functions for isolating the real roots of univariate polynomials with integer coefficients. It also provides functionality to approximate the isolated roots up to an arbitrary precision. Currently, it is realizes a subdivision algorithm based on Descartes' rule of sign, with modifications to improve its performance. SLV assumes that the input is a square-free polynomial. It performs all the operations using exact arithmetic based on the library gmp and it exploits as much as possible computations with dyadic numbers, that is numbers of the form a/2k where a and k are integers. It builds upon a constant memory variant of Descartes' rule of sign [3]. To minimize the number of allocations in the memory we wrote a small wrapper to use the queue.h library of OpenBSD, in order to have access to a fast implementation of lists and queues. Even though queue.h is written in C, it is a library that follows the generic programming paradigm, and if it is programmed carefully, it could be used with various data sets.
{"title":"SLV: a software for real root isolation","authors":"Elias P. Tsigaridas","doi":"10.1145/3015306.3015317","DOIUrl":"https://doi.org/10.1145/3015306.3015317","url":null,"abstract":"The problem of isolating the real roots of a univariate polynomial with integer coefficients is an important problem in computational mathematics. Given a polynomial with integer coefficients, [EQUATION], the objective is to isolate the real roots of f, that is to compute intervals with rational endpoints that contain one and only one root of f. SLV is an open source software package written in C that provides functions for isolating the real roots of univariate polynomials with integer coefficients. It also provides functionality to approximate the isolated roots up to an arbitrary precision. Currently, it is realizes a subdivision algorithm based on Descartes' rule of sign, with modifications to improve its performance. SLV assumes that the input is a square-free polynomial. It performs all the operations using exact arithmetic based on the library gmp and it exploits as much as possible computations with dyadic numbers, that is numbers of the form a/2k where a and k are integers. It builds upon a constant memory variant of Descartes' rule of sign [3]. To minimize the number of allocations in the memory we wrote a small wrapper to use the queue.h library of OpenBSD, in order to have access to a fast implementation of lists and queues. Even though queue.h is written in C, it is a library that follows the generic programming paradigm, and if it is programmed carefully, it could be used with various data sets.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"25 1","pages":"117-120"},"PeriodicalIF":0.0,"publicationDate":"2016-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81745484","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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