We employ tools from the fields of symbolic computation and satisfiability checking---namely, computer algebra systems and SAT solvers---to study the Williamson conjecture from combinatorial design theory and increase the bounds to which Williamson matrices have been enumerated. In particular, we completely enumerate all Williamson matrices of orders divisible by 2 or 3 up to and including 70. We find one previously unknown set of Williamson matrices of order 63 and construct Williamson matrices in every even order up to and including 70. This extended abstract outlines a preprint currently under submission [4].
{"title":"The SAT+CAS paradigm and the Williamson conjecture","authors":"Curtis Bright, I. Kotsireas, Vijay Ganesh","doi":"10.1145/3313880.3313889","DOIUrl":"https://doi.org/10.1145/3313880.3313889","url":null,"abstract":"We employ tools from the fields of symbolic computation and satisfiability checking---namely, computer algebra systems and SAT solvers---to study the Williamson conjecture from combinatorial design theory and increase the bounds to which Williamson matrices have been enumerated. In particular, we completely enumerate all Williamson matrices of orders divisible by 2 or 3 up to and including 70. We find one previously unknown set of Williamson matrices of order 63 and construct Williamson matrices in every even order up to and including 70. This extended abstract outlines a preprint currently under submission [4].","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"93 1","pages":"82-84"},"PeriodicalIF":0.0,"publicationDate":"2019-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75439628","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 a poster and software demonstration regarding the calculation of indefinite integrals, or anti-derivatives, when parameters are present.
我们提出了一个海报和软件演示关于计算不定积分，或不定积分，当参数存在。
{"title":"Comprehensive anti-derivatives and parametric continuity","authors":"Robert M Corless, D. J. Jeffrey, D. R. Stoutemyer","doi":"10.1145/3282678.3282680","DOIUrl":"https://doi.org/10.1145/3282678.3282680","url":null,"abstract":"We present a poster and software demonstration regarding the calculation of indefinite integrals, or anti-derivatives, when parameters are present.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"108 1","pages":"32-33"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79218008","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 a software that allows the user to compute embeddings between arbitrary finite fields. It is written using Julia and C, as part of Nemo and Flint[7, 6]. The software ensures compatibility between the embeddings, so that the user can work with multiple fields and have coherent results independently of where the computations are made and of where the results are expressed.
{"title":"Lattices of compatibly embedded finite fields in nemo/flint","authors":"L. Feo, H. Randriambololona, É. Rousseau","doi":"10.1145/3282678.3282682","DOIUrl":"https://doi.org/10.1145/3282678.3282682","url":null,"abstract":"We present a software that allows the user to compute embeddings between arbitrary finite fields. It is written using Julia and C, as part of Nemo and Flint[7, 6]. The software ensures compatibility between the embeddings, so that the user can work with multiple fields and have coherent results independently of where the computations are made and of where the results are expressed.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"200 1","pages":"38-41"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76975257","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 previously proposed the interval-symbol method with correct zero rewriting (ISCZ method) to reduce the amount of exact computations to obtain the exact results by aid of floating-point computations. Recently we have presented new ideas for reducing time and memory of executing the ISCZ method. In this paper, we apply the new ISCZ method to the Δ-LLL algorithm, which is a generalization of the Lenstra-Lenstra-Lovász (LLL) lattice reduction algorithm. By Maple experiments, we confirm its superiority over the original ISCZ method, and in the irrational case we show its great effect on the Δ-LLL algorithm in the sense that it is much more efficient than the purely exact approach.
{"title":"Effect of the interval-symbol method with correct zero rewriting on the Δ-LLL algorithm","authors":"Hiroki Nagashima, Kiyoshi Shirayanagi","doi":"10.1145/3282678.3282679","DOIUrl":"https://doi.org/10.1145/3282678.3282679","url":null,"abstract":"We previously proposed the interval-symbol method with correct zero rewriting (ISCZ method) to reduce the amount of exact computations to obtain the exact results by aid of floating-point computations. Recently we have presented new ideas for reducing time and memory of executing the ISCZ method. In this paper, we apply the new ISCZ method to the Δ-LLL algorithm, which is a generalization of the Lenstra-Lenstra-Lovász (LLL) lattice reduction algorithm. By Maple experiments, we confirm its superiority over the original ISCZ method, and in the irrational case we show its great effect on the Δ-LLL algorithm in the sense that it is much more efficient than the purely exact approach.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"4 1","pages":"24-31"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82900267","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}
François Boulier, F. Fages, O. Radulescu, S. Samal, A. Schuppert, W. Seiler, T. Sturm, S. Walcher, A. Weber
The international interdisciplinary SYMBIONT project ranges from mathematics via computer science to systems biology, with a balanced team of researchers from those fields. At the present stage the project has a clear focus on fundamental research on mathematical methods, and prototypes in software. Results are systematically benchmarked against models from computational biology databases. We summarize the motivation and aims for the project, and report on existing results by the consortium and first activities. The project website can be found at www.symbiont-project.org.
{"title":"The SYMBIONT project: symbolic methods for biological networks","authors":"François Boulier, F. Fages, O. Radulescu, S. Samal, A. Schuppert, W. Seiler, T. Sturm, S. Walcher, A. Weber","doi":"10.1145/3313880.3313885","DOIUrl":"https://doi.org/10.1145/3313880.3313885","url":null,"abstract":"The international interdisciplinary SYMBIONT project ranges from mathematics via computer science to systems biology, with a balanced team of researchers from those fields. At the present stage the project has a clear focus on fundamental research on mathematical methods, and prototypes in software. Results are systematically benchmarked against models from computational biology databases. We summarize the motivation and aims for the project, and report on existing results by the consortium and first activities. The project website can be found at www.symbiont-project.org.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"31 1","pages":"67-70"},"PeriodicalIF":0.0,"publicationDate":"2018-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84982898","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}
In the paper "A Chaotic Search for i" ([25]), Strang completely explained the behaviour of Newton's method when using real initial guesses on f(x) = x2 + 1, which has only the pair of complex roots ±i. He explored an exact symbolic formula for the iteration, namely xn = cot (2nθ0), which is valid in exact arithmetic. In this paper, we extend this to to kth order Householder methods, which include Halley's method, and to the secant method. Two formulae, xn = cot (θn-1 + θn-2) with θn-1 = arccot (xn-1) and θn-2 = arccot (xn-2), and xn = cot ((k + 1)nθ0) with θ0 = arccot(x0), are provided. The asymptotic behaviour and periodic character are illustrated by experimental computation. We show that other methods (Schröder iterations of the first kind) are generally not so simple. We also explain an old method that can be used to allow Maple's Fractals[Newton] package to visualize general one-step iterations by disguising them as Newton iterations.
{"title":"Revisiting Gilbert Strang's \"a chaotic search for i\"","authors":"Ao Li, Robert M Corless","doi":"10.1145/3363520.3363521","DOIUrl":"https://doi.org/10.1145/3363520.3363521","url":null,"abstract":"In the paper \"A Chaotic Search for <i>i</i>\" ([25]), Strang completely explained the behaviour of Newton's method when using real initial guesses on <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> + 1, which has only the pair of complex roots ±<i>i</i>. He explored an exact symbolic formula for the iteration, namely <i>x<sub>n</sub></i> = cot (2<sup><i>n</i></sup><i>θ</i><sub>0</sub>), which is valid in exact arithmetic. In this paper, we extend this to to <i>k<sup>th</sup></i> order Householder methods, which include Halley's method, and to the secant method. Two formulae, <i>x<sub>n</sub></i> = cot (<i>θ</i><sub><i>n</i>-1</sub> + <i>θ</i><sub><i>n</i>-2</sub>) with <i>θ</i><sub><i>n</i>-1</sub> = arccot (<i>x</i><sub><i>n</i>-1</sub>) and <i>θ</i><sub><i>n</i>-2</sub> = arccot (<i>x</i><sub><i>n</i>-2</sub>), and <i>x<sub>n</sub></i> = cot ((<i>k</i> + 1)<sup><i>n</i></sup><i>θ</i><sub>0</sub>) with <i>θ</i><sub>0</sub> = arccot(<i>x</i><sub>0</sub>), are provided. The asymptotic behaviour and periodic character are illustrated by experimental computation. We show that other methods (Schröder iterations of the first kind) are generally not so simple. We also explain an old method that can be used to allow Maple's <i>Fractals[Newton]</i> package to visualize general one-step iterations by disguising them as Newton iterations.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"1 1","pages":"1-22"},"PeriodicalIF":0.0,"publicationDate":"2018-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85519073","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}
In [2] we find an exploration of a new mathematical model of the flow in a leaking bucket suitable for beginning students. The model is derived using the non-steady Bernoulli's Principle, and results in a more sophisticated model than the simple ordinary differential equation [EQUATION] derived using the steady Bernoulli's Principle. The simpler model goes sometimes by the name of Torricelli's Law and is very well studied; indeed it is a favourite example in many textbooks. This present paper provides an alternative derivation of the new model that uses an energy balance, and carefully lays out some numerical issues omitted from the treatment in [2]. We also provide an analytic solution in terms of 2F1 hypergeometric functions, which, while possibly unfamiliar to the student, are available to them via computer algebra systems. Even before that solution, an intermediate equation [EQUATION] is derived, which already explains the similarity of the solutions to the more sophisticated model to the ones from the simple Torricelli's Law. This paper gives a useful example for use of a CAS in a classroom setting.
{"title":"Revisiting the discharge time of a cylindrical leaking bucket: or, \"one does not simply call dsolve into mordor.\"","authors":"Robert M Corless, J. Jankowski","doi":"10.1145/3243034.3243035","DOIUrl":"https://doi.org/10.1145/3243034.3243035","url":null,"abstract":"In [2] we find an exploration of a new mathematical model of the flow in a leaking bucket suitable for beginning students. The model is derived using the non-steady Bernoulli's Principle, and results in a more sophisticated model than the simple ordinary differential equation [EQUATION] derived using the steady Bernoulli's Principle. The simpler model goes sometimes by the name of Torricelli's Law and is very well studied; indeed it is a favourite example in many textbooks. This present paper provides an alternative derivation of the new model that uses an energy balance, and carefully lays out some numerical issues omitted from the treatment in [2]. We also provide an analytic solution in terms of 2F1 hypergeometric functions, which, while possibly unfamiliar to the student, are available to them via computer algebra systems. Even before that solution, an intermediate equation [EQUATION] is derived, which already explains the similarity of the solutions to the more sophisticated model to the ones from the simple Torricelli's Law. This paper gives a useful example for use of a CAS in a classroom setting.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"86 1","pages":"1-10"},"PeriodicalIF":0.0,"publicationDate":"2018-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81275115","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 second International conference is organized jointly by Dorodnicyn Computing Center of Federal Research Center "Computer Science and Control" of RAS and Plekhanov Russian University of Economics with a support of the Russian Foundation for Basic Research (project 17-01-20398 17). The first edition of the event [1] (http://www.ccas.ru/ca/conference2016) was held in 2016 at the Dorodnicyn Computing Center in cooperation with Russian University of Friendship of Peoples. The conference website: http://www.ccas.ru/ca/conference.
{"title":"The second conference","authors":"S. Abramov, A. Ryabenko, T. Sadykov","doi":"10.1145/3199652.3199653","DOIUrl":"https://doi.org/10.1145/3199652.3199653","url":null,"abstract":"The second International conference is organized jointly by Dorodnicyn Computing Center of Federal Research Center \"Computer Science and Control\" of RAS and Plekhanov Russian University of Economics with a support of the Russian Foundation for Basic Research (project 17-01-20398 17). The first edition of the event [1] (http://www.ccas.ru/ca/conference2016) was held in 2016 at the Dorodnicyn Computing Center in cooperation with Russian University of Friendship of Peoples.\u0000 The conference website: http://www.ccas.ru/ca/conference.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"411 1","pages":"103-110"},"PeriodicalIF":0.0,"publicationDate":"2018-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76829474","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 study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent (transversal) commuting vector field (see Theorem 2.1). In what follows, we will use the standard correspondence between (polynomial) vector fields and derivations on (polynomial) rings. Let [MATH HERE] be a derivation, where f is a polynomial with coefficients in a field K of zero characteristic. This derivation corresponds to the differential equation ẍ = f(x), which is called a conservative Newton system as it is the expression of Newton's second law for a particle confined to a line under the influence of a conservative force. Let H be the Hamiltonian polynomial for d with zero constant term, that is [MATH HERE]. Then the set of all polynomial derivations that commute with d forms a K[H]-module Md [6, Corollary 7.1.5]. We show that, for every such d, the module Md is of rank 1 if and only if deg f ⩾ 2. For example, the classical elliptic equation ẍ = 6x2 + a, where a ∈ C, falls into this category. For proofs of the results stated in this abstract, see [5].
{"title":"Commuting planar polynomial vector fields for conservative newton systems","authors":"Joel Nagloo, A. Ovchinnikov, Peter Thompson","doi":"10.1145/3313880.3313883","DOIUrl":"https://doi.org/10.1145/3313880.3313883","url":null,"abstract":"We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent (transversal) commuting vector field (see Theorem 2.1). In what follows, we will use the standard correspondence between (polynomial) vector fields and derivations on (polynomial) rings. Let [MATH HERE] be a derivation, where <i>f</i> is a polynomial with coefficients in a field <i>K</i> of zero characteristic. This derivation corresponds to the differential equation ẍ = <i>f</i>(<i>x</i>), which is called a conservative Newton system as it is the expression of Newton's second law for a particle confined to a line under the influence of a conservative force. Let <i>H</i> be the Hamiltonian polynomial for d with zero constant term, that is [MATH HERE]. Then the set of all polynomial derivations that commute with d forms a <i>K</i>[<i>H</i>]-module <i>M</i><sub>d</sub> [6, Corollary 7.1.5]. We show that, for every such <i>d,</i> the module <i>M</i><sub>d</sub> is of rank 1 if and only if deg <i>f</i> ⩾ 2. For example, the classical elliptic equation <i>ẍ = 6x<sup>2</sup> + a,</i> where <i>a</i> ∈ C, falls into this category. For proofs of the results stated in this abstract, see [5].","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"194 1","pages":"59-62"},"PeriodicalIF":0.0,"publicationDate":"2018-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77577113","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}
S. A. Haque, X. Li, Farnam Mansouri, M. M. Maza, Davood Mohajerani, W. Pan
The CUDA Modular Polynomial (CUMODP) Library implements arithmetic operations for dense matrices and dense polynomials, primarily with modular integer coefficients. Some operations are available for integer or floating point coefficients. Similar to other software libraries, like CuBLAS 1 targeting Graphics Processing Units (GPUs), CUMODP focuses on efficiency-critical routines and provides them in the form of device functions and CUDA kernels. Hence, these routines are primarily designed to offer GPU support to polynomial system solvers. A bivariate system solver is part of the library, as a proof-of-concept. Its implementation is presented in [10] and it is integrated in Maple's Triangularize command2, since the release 18 of Maple.
{"title":"CUMODP: a CUDA library for modular polynomial computation","authors":"S. A. Haque, X. Li, Farnam Mansouri, M. M. Maza, Davood Mohajerani, W. Pan","doi":"10.1145/3177795.3177799","DOIUrl":"https://doi.org/10.1145/3177795.3177799","url":null,"abstract":"The CUDA Modular Polynomial (CUMODP) Library implements arithmetic operations for dense matrices and dense polynomials, primarily with modular integer coefficients. Some operations are available for integer or floating point coefficients. Similar to other software libraries, like CuBLAS 1 targeting Graphics Processing Units (GPUs), CUMODP focuses on efficiency-critical routines and provides them in the form of device functions and CUDA kernels. Hence, these routines are primarily designed to offer GPU support to polynomial system solvers. A bivariate system solver is part of the library, as a proof-of-concept. Its implementation is presented in [10] and it is integrated in Maple's Triangularize command2, since the release 18 of Maple.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"27 1","pages":"89-91"},"PeriodicalIF":0.0,"publicationDate":"2018-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72620736","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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