Computer Algebra in Scientific Computing

162 Pages·2019·4.088 MB·English

by  Andreas Weber (editor)

Computer Algebra in
Scientific Computing
Edited by
Andreas Weber
Printed Edition of the Special Issue Published in Mathematics
www.mdpi.com/journal/mathematics
Computer Algebra in
Scientific Computing

Computer Algebra in
Scientific Computing
SpecialIssueEditor
AndreasWeber
MDPI•Basel•Beijing•Wuhan•Barcelona•Belgrade
SpecialIssueEditor
AndreasWeber
BonnUniversity
Germany
EditorialOffice
MDPI
St.Alban-Anlage66
4052Basel,Switzerland
This is a reprint of articles from the Special Issue published online in the open access journal
Mathematics (ISSN 2227-7390) from 2018 to 2019 (available at: https://www.mdpi.com/journal/
mathematics/specialissues/ComputerAlgebra)
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Contents
AbouttheSpecialIssueEditor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Prefaceto”ComputerAlgebrainScientificComputing”. . . . . . . . . . . . . . . . . . . . . . . ix
Mohammadali Asadi, Alexander Brandt, Robert H. C. Moir and Marc Moreno
Maza Algorithms and Data Structures for Sparse Polynomial Arithmetic
Reprintedfrom:Mathematics2019,7,441,doi:10.3390/math7050441 . . . . . . . . . . . . . . . . . 1
XiaojieDouandJin-SanCheng
AHeuristicMethodforCertifyingIsolatedZerosofPolynomialSystems
Reprintedfrom:Mathematics2018,6,166,doi:10.3390/math6090166 . . . . . . . . . . . . . . . . . 30
MarioAlbertandWernerM.Seiler
ResolvingDecompositionsforPolynomialModules
Reprintedfrom:Mathematics2018,6,161,doi:10.3390/math6090161 . . . . . . . . . . . . . . . . . 48
Valery Antonov, Wilker Fernandes, Valery G. Romanovski and Natalie L. Shcheglova
First Integrals of the May–Leonard Asymmetric System
Reprintedfrom:Mathematics2019,7,292,doi:10.3390/math7030292 . . . . . . . . . . . . . . . . . 65
Erhan G¨ulera nd O¨mer Kis¸i
Dini-Type Helicoidal Hypersurfaces with Timelike Axis in Minkowski 4-Space E41
Reprintedfrom:Mathematics2019,7,205,doi:10.3390/math7020205 . . . . . . . . . . . . . . . . . 80
ErhanGu¨ler,O¨merKis¸iandChristosKonaxis
ImplicitEquationsoftheHenneberg-TypeMinimalSurfaceintheFour-DimensionalEuclidean
Space
Reprintedfrom:Mathematics2018,6,279,doi:10.3390/math6120279 . . . . . . . . . . . . . . . . . 88
FarnooshHajati,AliIranmaneshandAbolfazlTehranian
ACharacterizationofProjectiveSpecialUnitaryGroupPSU(3,3)andProjectiveSpecialLinear
GroupPSL(3,3)byNSE
Reprintedfrom:Mathematics2018,6,120,doi:10.3390/math6070120 . . . . . . . . . . . . . . . . . 98
MauriceR.Kibler
QuantumInformation:ABriefOverviewandSomeMathematicalAspects
Reprintedfrom:Mathematics2018,6,273,doi:10.3390/math6120273 . . . . . . . . . . . . . . . . . 108
v

About the Special Issue Editor
Andreas Weber (Prof. Dr.) studied mathematics and computer science at the Universities of
Tu¨bingen, Germany, and Boulder, Colorado, U.S.A. He was awarded his MS in Mathematics
(Dipl.-Math)in1990andhisPh.D.(Dr.rer.nat.)incomputersciencefromtheUniversityofTu¨bingen
in1993.From1995to1997,hewasawardedascholarshipfromDeutscheForschungsgemeinschaftto
conductresearchasapostdoctoralfellowattheComputerScienceDepartment,CornellUniversity.
From 1997 to 1999 he was a member of the Symbolic Computation Group at the University of
Tu¨bingen, Germany. From 1999 to 2001, he was a member of the research group Animation and
ImageCommunicationattheFraunhoferInstitutforComputerGraphics. HehasbeenProfessorof
computerscienceattheUniversityofBonn,Germany,sincehisappointmentinApril2001. Hehas
servedasChairoftheDepartmentofComputerSciencefrom2014to2016. Duringhisacademic
career, hehaswrittenmorethan100papersforjournalsandrefereedconferenceproceedingsand
hasbeenthefirstsupervisorof9completedPh.D.thesesandover70master’sandbachelor’stheses.
Hehasservedasareviewerformorethan60differentjournalsandconferences.In2013,hehasbeen
awardedtheTeachingAwardoftheUniversityofBonn.
vii

Preface to ”Computer Algebra in Scientific Computing”
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 expression-based computational answers
generally 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
behaviors. The articles contained in this book cover a broad range of topics in the context of computer
algebra in scientific computing. At the core of many computer algebra methods are algorithms for
multivariate polynomials, and the first article on “Algorithms and Data Structures for Sparse
Polynomial Arithmetic” is at the essence of this core, giving 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. “A Heuristic Method for
Certifying Isolated Zeros of Polynomial Systems” deals with the fundamental problem of certifying
the isolated zeros of polynomial systems. Computing Gr¨obner bases and other kind of bases is
another core of computer algebra. In “Resolving Decompositions for Polynomial Modules”, the
authors deal with a fundamental task in “computational commutative algebra and algebraic
geometry”, namely, the determination of free resolutions for polynomial modules. They introduce
the novel concept of resolving decomposition of a polynomial module as a combinatorial structure
that allows for the effective construction of free resolutions and provide a unifying framework for
recent results involving different types of bases. The analysis of certain invariants of a dynamical
system—which are at the heart of many problems in scientific computing—is another major area for
computer algebra research. In the article “First Integrals of the May–Leonard Asymmetric System”,
an important system arising in the life sciences is investigated, which is given by a quadratic system
of the Lotka–Volterra type depending on six parameters. The authors look for subfamilies admitting
invariant algebraic surfaces of degree two, and then for some such subfamilies, they construct first
integrals of the Darboux type, identifying the systems with one first integral or with two independent
first integrals. A problem based in physics, namely “Minkowski 4-space”, is treated in the article
“Dini-Type Helicoidal Hypersurfaces with Timelike Axis in Minkowski 4-Space”. The authors
consider Ulisse Dini-type helicoidal hypersurfaces with timelike axis in Minkowski 4-space, and by
calculating the Gaussian and mean curvatures of the hypersurfaces, they demonstrate some special
symmetries for the curvatures when they are flat and minimal. In the article “Implicit Equations of
the Henneberg-Type Minimal Surface in the Four-Dimensional Euclidean Space” the authors find
implicit algebraic equations of the Henneberg-type minimal surface of values (4,2). The exciting field
of quantum computing has also lead to several problems in computer algebra. In “Quantum
Information: A Brief Overview and Some Mathematical Aspects”, not only is a review of the main
ideas behind quantum computing and quantum information presented, but the focus is also 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.
Andreas Weber
Special Issue Editor
ix