Special Issue "Computer Algebra in Scientific Computing"

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

Deadline for manuscript submissions: closed (31 January 2019)

Special Issue Editor

Guest Editor
Prof. Dr. Andreas Weber

Multimedia, Simulation and Virtual Reality Group, Institute of Computer Science II, University of Bonn, Bonn, Germany
Website | E-Mail
Interests: computer algebra; computer animation; motion analysis; computational biology; physics-based modeling

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 (6 papers)

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Research

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 (registering DOI)
Received: 31 January 2019 / Revised: 11 February 2019 / Accepted: 13 February 2019 / Published: 22 February 2019
PDF Full-text (239 KB)
Abstract
We consider Ulisse Dini-type helicoidal hypersurfaces with timelike axis in Minkowski 4-space E14. 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)
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
Received: 18 October 2018 / Revised: 20 November 2018 / Accepted: 22 November 2018 / Published: 25 November 2018
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Abstract
Considering the Weierstrass data as (ψ,f,g)=(2,1-z-m,zn), we introduce a two-parameter family of Henneberg-type minimal surface that we call Hm,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)
Figures

Figure 1

Open AccessArticle Quantum Information: A Brief Overview and Some Mathematical Aspects
Mathematics 2018, 6(12), 273; https://doi.org/10.3390/math6120273
Received: 23 October 2018 / Revised: 14 November 2018 / Accepted: 20 November 2018 / Published: 22 November 2018
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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)
Open AccessArticle A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems
Mathematics 2018, 6(9), 166; https://doi.org/10.3390/math6090166
Received: 12 July 2018 / Revised: 24 August 2018 / Accepted: 3 September 2018 / Published: 11 September 2018
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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)
Open AccessArticle Resolving Decompositions for Polynomial Modules
Mathematics 2018, 6(9), 161; https://doi.org/10.3390/math6090161
Received: 12 July 2018 / Revised: 3 September 2018 / Accepted: 4 September 2018 / Published: 7 September 2018
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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)
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
Received: 17 May 2018 / Revised: 27 June 2018 / Accepted: 29 June 2018 / Published: 10 July 2018
PDF Full-text (271 KB) | HTML Full-text | XML Full-text
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)
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