Mathematical and Computational Modeling: With Applications in Natural and Social Sciences, Engineering, and the Arts

336 Pages·2015·4.26 MB·English

by  Roderick Melnik

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MATHEMATICAL AND COMPUTATIONAL
MODELING
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PUREANDAPPLIEDMATHEMATICS
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MATHEMATICAL AND
COMPUTATIONAL
MODELING
With Applications in Natural and
Social Sciences, Engineering,
and the Arts
Editedby
RODERICKMELNIK
WilfridLaurierUniversity
Waterloo,Ontario,Canada
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Copyright©2015byJohnWiley&Sons,Inc.Allrightsreserved
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LibraryofCongressCataloging-in-PublicationData:
Mathematicalandcomputationalmodeling:withapplicationsinnaturalandsocialsciences,engineering,
andthearts/editedbyRoderickMelnik,WilfridLaurierUniversity,Waterloo,Ontario,Canada.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-85398-6(cloth)
1. Mathematicalmodels. I. Melnik,Roderick,editor.
QA401.M39262015
511(cid:2).8—dc23
2014043267
Setin10/12ptsTimesLtStdbySPiPublisherServices,Pondicherry,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
1 2015
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Teacherswhoeducatechildrendeservemorehonorthanparentswhomerelygave
birth;forbarelifeisfurnishedbytheone,theotherensuresagoodlife.
(Aristotle)
Tomyparentswhoweremyfirstandmostdevotedteachers.
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CONTENTS
LISTOFCONTRIBUTORS xiii
PREFACE xv
SECTION1 INTRODUCTION 1
1 UniversalityofMathematicalModelsinUnderstandingNature,
Society,andMan-MadeWorld 3
RoderickMelnik
1.1 HumanKnowledge,Models,andAlgorithms, 3
1.2 LookingintotheFuturefromaModelingPerspective, 7
1.3 WhatThisBookIsAbout, 10
1.4 ConcludingRemarks, 15
References, 16
SECTION2 ADVANCEDMATHEMATICALANDCOMPUTATIONAL
MODELSINPHYSICSANDCHEMISTRY 17
2 MagneticVortices,AbrikosovLattices,andAutomorphic
Functions 19
IsraelMichaelSigal
2.1 Introduction, 19
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viii CONTENTS
2.2 TheGinzburg–LandauEquations, 20
2.2.1 Ginzburg–Landauenergy, 21
2.2.2 Symmetriesoftheequations, 21
2.2.3 Quantizationofflux, 22
2.2.4 Homogeneoussolutions, 22
2.2.5 TypeIandTypeIIsu√perconductors, 23
2.2.6 Self-dualcaseκ=1/ 2, 24
2.2.7 Criticalmagneticfields, 24
2.2.8 Time-dependentequations, 25
2.3 Vortices, 25
2.3.1 n-vortexsolutions, 25
2.3.2 Stability, 26
2.4 VortexLattices, 30
2.4.1 Abrikosovlattices, 31
2.4.2 ExistenceofAbrikosovlattices, 31
2.4.3 Abrikosovlatticesasgauge-equivariantstates, 34
2.4.4 Abrikosovfunction, 34
2.4.5 Commentsontheproofsofexistenceresults, 35
2.4.6 StabilityofAbrikosovlattices, 40
2.4.7 Functionsγ (τ),δ>0, 42
δ
2.4.8 Keyideasofapproachtostability, 45
2.5 Multi-VortexDynamics, 48
2.6 Conclusions, 51
Appendix2.A Parameterizationoftheequivalenceclasses[L], 51
Appendix2.B Automorphyfactors, 52
References, 54
3 Numerical Challenges in a Cholesky-Decomposed Local Correlation
QuantumChemistryFramework 59
DavidB.Krisiloff,JohannesM.Dieterich,FlorianLibisch,
andEmilyA.Carter
3.1 Introduction, 59
3.2 LocalMRSDCI, 61
3.2.1 MRSDCI, 61
3.2.2 Symmetricgroupgraphicalapproach, 62
3.2.3 Localelectroncorrelationapproximation, 64
3.2.4 Algorithmsummary, 66
3.3 NumericalImportanceofIndividualSteps, 67
3.4 CholeskyDecomposition, 68
3.5 TransformationoftheCholeskyVectors, 71
3.6 Two-ElectronIntegralReassembly, 72
3.7 IntegralandExecutionBuffer, 76
3.8 SymmetricGroupGraphicalApproach, 77
3.9 SummaryandOutlook, 87
References, 87