“9781118853986pre” — 2015/4/21 — 9:11 — page vi — #6 “9781118853986pre” — 2015/4/21 — 9:11 — page i — #1 MATHEMATICAL AND COMPUTATIONAL MODELING “9781118853986pre” — 2015/4/21 — 9:11 — page ii — #2 PUREANDAPPLIEDMATHEMATICS AWileySeriesofTexts,Monographs,andTracts FoundedbyRICHARDCOURANT EditorsEmeriti:MYRONB.ALLENIII,PETERHILTON,HARRY HOCHSTADT,ERWINKREYSZIG,PETERLAX,JOHNTOLAND Acompletelistofthetitlesinthisseriesappearsattheendofthisvolume. “9781118853986pre” — 2015/4/21 — 9:11 — page iii — #3 MATHEMATICAL AND COMPUTATIONAL MODELING With Applications in Natural and Social Sciences, Engineering, and the Arts Editedby RODERICKMELNIK WilfridLaurierUniversity Waterloo,Ontario,Canada “9781118853986pre” — 2015/4/21 — 9:11 — page iv — #4 Copyright©2015byJohnWiley&Sons,Inc.Allrightsreserved PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey PublishedsimultaneouslyinCanada Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor byanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,(978)750-8400,fax (978)750-4470,oronthewebatwww.copyright.com.RequeststothePublisherforpermissionshould beaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken, NJ07030,(201)748-6011,fax(201)748-6008,oronlineathttp://www.wiley.com/go/permissions. LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitable foryoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublishernor authorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnotlimitedto special,incidental,consequential,orotherdamages. Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactour CustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStatesat (317)572-3993orfax(317)572-4002. Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmay notbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourwebsiteat www.wiley.com. 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 “9781118853986pre” — 2015/4/21 — 9:11 — page v — #5 Teacherswhoeducatechildrendeservemorehonorthanparentswhomerelygave birth;forbarelifeisfurnishedbytheone,theotherensuresagoodlife. (Aristotle) Tomyparentswhoweremyfirstandmostdevotedteachers. “9781118853986pre” — 2015/4/21 — 9:11 — page vi — #6 “9781118853986pre” — 2015/4/21 — 11:32 — page vii — #7 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 “9781118853986pre” — 2015/4/21 — 11:32 — page viii — #8 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