Special Issue "Computer Algebra in Scientific Computing"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 January 2019).

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Andreas Weber
Website
Guest Editor
Institute for Computer Science, Bonn University, Endenicher Allee 19A, D-53115 Bonn, Germany
Interests: computer animation; physics-based modelling; computer algebra; life science informatics; hybrid modelling
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Special Issue Information

Dear Colleagues,

Although scientific computing is very often associated with numeric computations, the use of computer algebra methods in scientific computing has obtained considerable attention in the last two decades. Computer algebra methods are especially suitable for parametric analysis of the key properties of systems arising in scientific computing. The in general expression-based computational answers provided by these methods are very appealing, as they directly relate properties to parameters and speed up testing and tuning of mathematical models through all their possible behaviours. Thus, it is no surprise that a conference series with the name, Computer Algebra in Scientific Computing, is now going into its 20th year. Moreover, the development of computer algebra methods for scientific computing is also widely covered in conferences like the International Symposium on Symbolic and Algebraic Computation (ISSAC) or Applications of Computer Algebra (ACA). The topics addressed in this Special Issue are in the spirit of these conference series and cover all the basic areas of scientific computing as they benefit from the application of computer algebra methods, especially in the following topics:

  • algebraic and semi-algebraic computations;
  • symbolic-numeric methods for differential, differential-algebraic and difference equations;
  • homotopy, perturbation and series methods;
  • tropical and polyhedral methods;
  • complexity of algebraic algorithms;
  • automated reasoning in algebra and geometry;
  • applications of computer algebra in the natural sciences and engineering.

Prof. Dr. Andreas Weber
Guest Editor

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Keywords

  • Algebraic computations
  • Semi-algebraic computations
  • Methods using differential algebra
  • Homotopy computations
  • Tropical and polyhedral methods
  • Complexity of algebraic algorithms

Published Papers (8 papers)

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Research

Open AccessArticle
Algorithms and Data Structures for Sparse Polynomial Arithmetic
Mathematics 2019, 7(5), 441; https://doi.org/10.3390/math7050441 - 17 May 2019
Abstract
We provide a comprehensive presentation of algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers as implemented in the freely available Basic Polynomial Algebra Subprograms (BPAS) library. We report on an algorithm for sparse [...] Read more.
We provide a comprehensive presentation of algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers as implemented in the freely available Basic Polynomial Algebra Subprograms (BPAS) library. We report on an algorithm for sparse pseudo-division, based on the algorithms for division with remainder, multiplication, and addition, which are also examined herein. The pseudo-division and division with remainder operations are extended to multi-divisor pseudo-division and normal form algorithms, respectively, where the divisor set is assumed to form a triangular set. Our operations make use of two data structures for sparse distributed polynomials and sparse recursively viewed polynomials, with a keen focus on locality and memory usage for optimized performance on modern memory hierarchies. Experimentation shows that these new implementations compare favorably against competing implementations, performing between a factor of 3 better (for multiplication over the integers) to more than 4 orders of magnitude better (for pseudo-division with respect to a triangular set). Full article
(This article belongs to the Special Issue Computer Algebra in Scientific Computing) Printed Edition available
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Open AccessArticle
First Integrals of the May–Leonard Asymmetric System
Mathematics 2019, 7(3), 292; https://doi.org/10.3390/math7030292 - 21 Mar 2019
Cited by 2
Abstract
For the May–Leonard asymmetric system, which is a quadratic system of the Lotka–Volterra type depending on six parameters, we first look for subfamilies admitting invariant algebraic surfaces of degree two. Then for some such subfamilies we construct first integrals of the Darboux type, [...] Read more.
For the May–Leonard asymmetric system, which is a quadratic system of the Lotka–Volterra type depending on six parameters, we first look for subfamilies admitting invariant algebraic surfaces of degree two. Then for some such subfamilies we construct first integrals of the Darboux type, identifying the systems with one first integral or with two independent first integrals. Full article
(This article belongs to the Special Issue Computer Algebra in Scientific Computing) Printed Edition available
Open AccessArticle
Dini-Type Helicoidal Hypersurfaces with Timelike Axis in Minkowski 4-Space E14
Mathematics 2019, 7(2), 205; https://doi.org/10.3390/math7020205 - 22 Feb 2019
Abstract
We consider Ulisse Dini-type helicoidal hypersurfaces with timelike axis in Minkowski 4-space E 1 4 . Calculating the Gaussian and the mean curvatures of the hypersurfaces, we demonstrate some special symmetries for the curvatures when they are flat and minimal. [...] Read more.
We consider Ulisse Dini-type helicoidal hypersurfaces with timelike axis in Minkowski 4-space E 1 4 . Calculating the Gaussian and the mean curvatures of the hypersurfaces, we demonstrate some special symmetries for the curvatures when they are flat and minimal. Full article
(This article belongs to the Special Issue Computer Algebra in Scientific Computing) Printed Edition available
Open AccessArticle
Implicit Equations of the Henneberg-Type Minimal Surface in the Four-Dimensional Euclidean Space
Mathematics 2018, 6(12), 279; https://doi.org/10.3390/math6120279 - 25 Nov 2018
Abstract
Considering the Weierstrass data as ( ψ , f , g ) = ( 2 , 1 - z - m , z n ) , we introduce a two-parameter family of Henneberg-type minimal surface that we call H m , n for [...] Read more.
Considering the Weierstrass data as ( ψ , f , g ) = ( 2 , 1 - z - m , z n ) , we introduce a two-parameter family of Henneberg-type minimal surface that we call H m , n for positive integers ( m , n ) by using the Weierstrass representation in the four-dimensional Euclidean space E 4 . We define H m , n in ( r , θ ) coordinates for positive integers ( m , n ) with m 1 , n - 1 , - m + n - 1 , and also in ( u , v ) coordinates, and then we obtain implicit algebraic equations of the Henneberg-type minimal surface of values ( 4 , 2 ) . Full article
(This article belongs to the Special Issue Computer Algebra in Scientific Computing) Printed Edition available
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Open AccessArticle
Quantum Information: A Brief Overview and Some Mathematical Aspects
Mathematics 2018, 6(12), 273; https://doi.org/10.3390/math6120273 - 22 Nov 2018
Abstract
The aim of the present paper is twofold. First, to give the main ideas behind quantum computing and quantum information, a field based on quantum-mechanical phenomena. Therefore, a short review is devoted to (i) quantum bits or qubits (and more generally qudits), [...] Read more.
The aim of the present paper is twofold. First, to give the main ideas behind quantum computing and quantum information, a field based on quantum-mechanical phenomena. Therefore, a short review is devoted to (i) quantum bits or qubits (and more generally qudits), the analogues of the usual bits 0 and 1 of the classical information theory, and to (ii) two characteristics of quantum mechanics, namely, linearity, which manifests itself through the superposition of qubits and the action of unitary operators on qubits, and entanglement of certain multi-qubit states, a resource that is specific to quantum mechanics. A, second, focus is on some mathematical problems related to the so-called mutually unbiased bases used in quantum computing and quantum information processing. In this direction, the construction of mutually unbiased bases is presented via two distinct approaches: one based on the group SU(2) and the other on Galois fields and Galois rings. Full article
(This article belongs to the Special Issue Computer Algebra in Scientific Computing) Printed Edition available
Open AccessArticle
A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems
Mathematics 2018, 6(9), 166; https://doi.org/10.3390/math6090166 - 11 Sep 2018
Abstract
In this paper, by transforming the given over-determined system into a square system, we prove a necessary and sufficient condition to certify the simple real zeros of the over-determined system by certifying the simple real zeros of the square system. After certifying a [...] Read more.
In this paper, by transforming the given over-determined system into a square system, we prove a necessary and sufficient condition to certify the simple real zeros of the over-determined system by certifying the simple real zeros of the square system. After certifying a simple real zero of the related square system with the interval methods, we assert that the certified zero is a local minimum of sum of squares of the input polynomials. If the value of sum of squares of the input polynomials at the certified zero is equal to zero, it is a zero of the input system. As an application, we also consider the heuristic verification of isolated zeros of polynomial systems and their multiplicity structures. Full article
(This article belongs to the Special Issue Computer Algebra in Scientific Computing) Printed Edition available
Open AccessArticle
Resolving Decompositions for Polynomial Modules
Mathematics 2018, 6(9), 161; https://doi.org/10.3390/math6090161 - 07 Sep 2018
Abstract
We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases. Full article
(This article belongs to the Special Issue Computer Algebra in Scientific Computing) Printed Edition available
Open AccessArticle
A Characterization of Projective Special Unitary Group PSU(3,3) and Projective Special Linear Group PSL(3,3) by NSE
Mathematics 2018, 6(7), 120; https://doi.org/10.3390/math6070120 - 10 Jul 2018
Cited by 1
Abstract
Let G be a finite group and ω(G) be the set of element orders of G. Let kω(G) and mk be the number of elements of order k in G. Let n [...] Read more.
Let G be a finite group and ω(G) be the set of element orders of G. Let kω(G) and mk be the number of elements of order k in G. Let nse(G)={mk|kω(G)}. In this paper, we prove that if G is a finite group such that nse(G) = nse(H), where H=PSU(3,3) or PSL(3,3), then GH. Full article
(This article belongs to the Special Issue Computer Algebra in Scientific Computing) Printed Edition available
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