Research

29 pages, 555 KiB

Open AccessArticle

Algorithms and Data Structures for Sparse Polynomial Arithmetic

by Mohammadali Asadi, Alexander Brandt, Robert H. C. Moir and Marc Moreno Maza

Mathematics 2019, 7(5), 441; https://doi.org/10.3390/math7050441 - 17 May 2019

Cited by 8 | Viewed by 4087

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)

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15 pages, 267 KiB

Open AccessArticle

First Integrals of the May–Leonard Asymmetric System

by Valery Antonov, Wilker Fernandes, Valery G. Romanovski and Natalie L. Shcheglova

Mathematics 2019, 7(3), 292; https://doi.org/10.3390/math7030292 - 21 Mar 2019

Cited by 5 | Viewed by 2193

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)

8 pages, 262 KiB

Open AccessArticle

Dini-Type Helicoidal Hypersurfaces with Timelike Axis in Minkowski 4-Space E₁⁴

by Erhan Güler and Ömer Kişi

Mathematics 2019, 7(2), 205; https://doi.org/10.3390/math7020205 - 22 Feb 2019

Cited by 4 | Viewed by 2278

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)

10 pages, 1309 KiB

Open AccessArticle

Implicit Equations of the Henneberg-Type Minimal Surface in the Four-Dimensional Euclidean Space

by Erhan Güler, Ömer Kişi and Christos Konaxis

Mathematics 2018, 6(12), 279; https://doi.org/10.3390/math6120279 - 25 Nov 2018

Cited by 1 | Viewed by 2970

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 \neq 1, n \neq - 1, - m + n \neq - 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)

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40 pages, 461 KiB

Open AccessArticle

Quantum Information: A Brief Overview and Some Mathematical Aspects

by Maurice R. Kibler

Mathematics 2018, 6(12), 273; https://doi.org/10.3390/math6120273 - 22 Nov 2018

Cited by 1 | Viewed by 3502

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)

18 pages, 293 KiB

Open AccessArticle

A Heuristic Method for Certifying Isolated Zeros of Polynomial Systems

by Xiaojie Dou and Jin-San Cheng

Mathematics 2018, 6(9), 166; https://doi.org/10.3390/math6090166 - 11 Sep 2018

Viewed by 2627

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)

17 pages, 346 KiB

Open AccessArticle

Resolving Decompositions for Polynomial Modules

by Mario Albert and Werner M. Seiler

Mathematics 2018, 6(9), 161; https://doi.org/10.3390/math6090161 - 07 Sep 2018

Cited by 1 | Viewed by 2549

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)

10 pages, 271 KiB

Open AccessArticle

A Characterization of Projective Special Unitary Group PSU(3,3) and Projective Special Linear Group PSL(3,3) by NSE

by Farnoosh Hajati, Ali Iranmanesh and Abolfazl Tehranian

Mathematics 2018, 6(7), 120; https://doi.org/10.3390/math6070120 - 10 Jul 2018

Cited by 1 | Viewed by 3050

Abstract

Let G be a finite group and

ω (G)

be the set of element orders of G. Let

k \in ω (G)

and

m_{k}

be the number of elements of order k in G. Let [...] Read more.

Let G be a finite group and

ω (G)

be the set of element orders of G. Let

k \in ω (G)

and

m_{k}

be the number of elements of order k in G. Let

n s e (G) = {m_{k} | k \in ω (G)}

. In this paper, we prove that if G is a finite group such that nse(G) = nse(H), where

H = P S U (3, 3)

or

P S L (3, 3)

, then

G ≅ H

. Full article

(This article belongs to the Special Issue Computer Algebra in Scientific Computing)

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