The Princeton Companion to Applied Mathematics

1031 Pages·2015·41.726 MB·English

by  Nicholas J. Higham (Editor), Mark R. Dennis (Editor), Paul Glendinning (Editor), Paul A. Martin (Editor), Fadil Santosa (Editor), Jared Tanner (Editor)

The Princeton Companion to Applied Mathematics

The Princeton Companion to
Applied Mathematics
editor
Nicholas J. Higham
The University of Manchester
associate editors
Mark R. Dennis
University of Bristol
Paul Glendinning
The University of Manchester
Paul A. Martin
Colorado School of Mines
Fadil Santosa
University of Minnesota
Jared Tanner
University of Oxford
PrincetonUniversityPress
PrincetonandOxford
Copyright©2015byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,
41WilliamStreet,Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,
6OxfordStreet,Woodstock,OxfordshireOX201TW
press.princeton.edu
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LibraryofCongressCataloging-in-PublicationData
ThePrincetoncompaniontoappliedmathematics/editor,
NicholasJ.Higham,TheUniversityofManchester;
associateeditors,MarkR.Dennis,UniversityofBristol
[andfourothers].
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-0-691-15039-0(hardcover:alk.paper)
1.Algebra.2.Mathematics.3.Mathematicalmodels.
I.Higham,NicholasJ.,1961–editor.II.Dennis,MarkR.,
editor.III.Title:Companiontoappliedmathematics.
IV.Title:Appliedmathematics.
QA155.P752015
510—dc23 2015013024
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinLucidaBright
Projectmanagement,compositionandcopyediting
byT&TProductionsLtd,London
Printedonacid-freepaper ∞
PrintedintheUnitedStatesofAmerica
12345678910
Contents
Preface ix II.25 MarkovChains 116
Contributors xiii II.26 ModelReduction 117
II.27 MultiscaleModeling 119
II.28 NonlinearEquationsandNewton’sMethod 120
II.29 OrthogonalPolynomials 122
Part I Introduction to Applied
II.30 Shocks 122
Mathematics
II.31 Singularities 124
II.32 TheSingularValueDecomposition 126
I.1 WhatIsAppliedMathematics? 1
II.33 TensorsandManifolds 127
I.2 TheLanguageofAppliedMathematics 8
II.34 UncertaintyQuantification 131
I.3 MethodsofSolution 27
II.35 VariationalPrinciple 134
I.4 Algorithms 40
II.36 WavePhenomena 134
I.5 GoalsofAppliedMathematicalResearch 48
I.6 TheHistoryofAppliedMathematics 55
Part III Equations, Laws, and
Part II Concepts Functions of Applied
Mathematics
II.1 Asymptotics 81
II.2 BoundaryLayer 82 III.1 Benford’sLaw 135
II.3 ChaosandErgodicity 82 III.2 BesselFunctions 137
II.4 ComplexSystems 83 III.3 TheBlack–ScholesEquation 137
II.5 ConformalMapping 84 III.4 TheBurgersEquation 138
II.6 ConservationLaws 86 III.5 TheCahn–HilliardEquation 138
II.7 Control 88 III.6 TheCauchy–RiemannEquations 139
II.8 Convexity 89 III.7 The Delta Function and Generalized
II.9 DimensionalAnalysisandScaling 90 Functions 139
II.10 TheFastFourierTransform 94 III.8 TheDiffusionEquation 142
II.11 FiniteDifferences 95 III.9 TheDiracEquation 142
II.12 TheFinite-ElementMethod 96 III.10 Einstein’sFieldEquations 144
II.13 Floating-PointArithmetic 96 III.11 TheEulerEquations 146
II.14 FunctionsofMatrices 97 III.12 TheEuler–LagrangeEquations 147
II.15 FunctionSpaces 99 III.13 TheGammaFunction 148
II.16 GraphTheory 101 III.14 TheGinzburg–LandauEquation 148
II.17 Homogenization 103 III.15 Hooke’sLaw 149
II.18 HybridSystems 103 III.16 TheKorteweg–deVriesEquation 150
II.19 IntegralTransformsandConvolution 104 III.17 TheLambertW Function 151
II.20 IntervalAnalysis 105 III.18 Laplace’sEquation 155
II.21 InvariantsandConservationLaws 106 III.19 TheLogisticEquation 156
II.22 TheJordanCanonicalForm 112 III.20 TheLorenzEquations 158
II.23 KrylovSubspaces 113 III.21 MathieuFunctions 159
II.24 TheLevelSetMethod 114 III.22 Maxwell’sEquations 160
vi Contents
III.23 TheNavier–StokesEquations 162 IV.36 InformationTheory 545
III.24 ThePainlevéEquations 163 IV.37 AppliedCombinatoricsandGraphTheory 552
III.25 TheRiccatiEquation 165 IV.38 CombinatorialOptimization 564
III.26 Schrödinger’sEquation 167 IV.39 AlgebraicGeometry 570
III.27 TheShallow-WaterEquations 167 IV.40 GeneralRelativityandCosmology 579
III.28 TheSylvesterandLyapunovEquations 168
III.29 TheThin-FilmEquation 169
III.30 TheTricomiEquation 170
Part V Modeling
III.31 TheWaveEquation 171
V.1 The Mathematics of Adaptation
(OrtheTenAvatarsofVishnu) 591
Part IV Areas of Applied V.2 Sport 598
V.3 Inerters 604
Mathematics
V.4 MathematicalBiomechanics 609
V.5 MathematicalPhysiology 616
IV.1 ComplexAnalysis 173
V.6 CardiacModeling 623
IV.2 OrdinaryDifferentialEquations 181
V.7 ChemicalReactions 627
IV.3 PartialDifferentialEquations 190
V.8 DivergentSeries:TamingtheTails 634
IV.4 IntegralEquations 200
V.9 FinancialMathematics 640
IV.5 PerturbationTheoryandAsymptotics 208
V.10 PortfolioTheory 648
IV.6 CalculusofVariations 218
V.11 BayesianInferenceinAppliedMathematics 658
IV.7 SpecialFunctions 227
V.12 A Symmetric Framework with Many
IV.8 SpectralTheory 236
Applications 661
IV.9 ApproximationTheory 248
V.13 GranularFlows 665
IV.10 Numerical Linear Algebra and Matrix
V.14 ModernOptics 673
Analysis 263
V.15 NumericalRelativity 680
IV.11 ContinuousOptimization(Nonlinearand
LinearProgramming) 281 V.16 TheSpreadofInfectiousDiseases 687
IV.12 NumericalSolutionofOrdinaryDifferential V.17 TheMathematicsofSeaIce 694
Equations 293 V.18 NumericalWeatherPrediction 705
IV.13 NumericalSolutionofPartialDifferential V.19 TsunamiModeling 712
Equations 306 V.20 ShockWaves 720
IV.14 ApplicationsofStochasticAnalysis 319 V.21 Turbulence 724
IV.15 InverseProblems 327
IV.16 ComputationalScience 335
IV.17 DataMiningandAnalysis 350 Part VI Example Problems
IV.18 NetworkAnalysis 360
IV.19 ClassicalMechanics 374 VI.1 Cloaking 733
IV.20 DynamicalSystems 383 VI.2 Bubbles 735
IV.21 BifurcationTheory 393 VI.3 Foams 737
IV.22 SymmetryinAppliedMathematics 402 VI.4 InvertedPendulums 741
IV.23 QuantumMechanics 411 VI.5 InsectFlight 743
IV.24 Random-MatrixTheory 419 VI.6 TheFlightofaGolfBall 746
IV.25 KineticTheory 428 VI.7 AutomaticDifferentiation 749
IV.26 ContinuumMechanics 446 VI.8 KnottingandLinkingofMacromolecules 752
IV.27 PatternFormation 458 VI.9 RankingWebPages 755
IV.28 FluidDynamics 467 VI.10 SearchingaGraph 757
IV.29 Magnetohydrodynamics 476 VI.11 EvaluatingElementaryFunctions 759
IV.30 EarthSystemDynamics 485 VI.12 RandomNumberGeneration 761
IV.31 EffectiveMediumTheories 500 VI.13 OptimalSensorLocationintheControlof
IV.32 MechanicsofSolids 505 Energy-EfficientBuildings 763
IV.33 SoftMatter 516 VI.14 Robotics 767
IV.34 ControlTheory 523 VI.15 Slipping,Sliding,Rattling,andImpact:
IV.35 SignalProcessing 533 NonsmoothDynamicsandItsApplications 769
Contents vii
VI.16 FromtheN-BodyProblemtoAstronomyand VII.19 AirportBaggageScreeningwithX-Ray
DarkMatter 771 Tomography 866
VI.17 TheN-BodyProblemandtheFastMultipole VII.20 MathematicalEconomics 868
Method 775 VII.21 MathematicalNeuroscience 873
VI.18 TheTravelingSalesmanProblem 778 VII.22 SystemsBiology 879
VII.23 CommunicationNetworks 883
VII.24 TextMining 887
VII.25 VotingSystems 891
Part VII Application Areas
VII.1 AircraftNoise 783
VII.2 AHybridSymbolic–NumericApproachto Part VIII Final Perspectives
GeometryProcessingandModeling 787
VII.3 Computer-Aided Proofs via Interval VIII.1 MathematicalWriting 897
Analysis 790 VIII.2 HowtoReadandUnderstandaPaper 903
VII.4 ApplicationsofMax-PlusAlgebra 795 VIII.3 HowtoWriteaGeneralInterestMathematics
VII.5 EvolvingSocialNetworks,Attitudes,and Book 906
Beliefs—andCounterterrorism 800 VIII.4 Workflow 912
VII.6 ChipDesign 804 VIII.5 ReproducibleResearchintheMathematical
VII.7 ColorSpacesandDigitalImaging 808 Sciences 916
VII.8 MathematicalImageProcessing 813 VIII.6 ExperimentalAppliedMathematics 925
VII.9 MedicalImaging 816 VIII.7 TeachingAppliedMathematics 933
VII.10 CompressedSensing 823 VIII.8 MediatedMathematics:Representations
VII.11 Programming Languages: An Applied ofMathematicsinPopularCultureand
MathematicsView 828 WhyTheseMatter 943
VII.12 High-PerformanceComputing 839 VIII.9 MathematicsandPolicy 953
VII.13 Visualization 843
VII.14 Electronic Structure Calculations
(SolidStatePhysics) 847
Index 963
VII.15 FlamePropagation 852
VII.16 ImagingtheEarthUsingGreen’sTheorem 857
VII.17 RadarImaging 860
VII.18 ModelingaPregnancyTestingKit 864 Colorplatesfollowpage364

Preface
1 WhatIsTheCompanion? 2 Scope
The Princeton Companion to Applied Mathematics de- Itisdifficulttogiveaprecisedefinitionofappliedmath-
scribes what applied mathematics is about, why it ematics,asdiscussedinwhatisappliedmathemat-
is important, its connections with other disciplines, ics?[I.1]and,fromahistoricalperspective,inthehis-
and some of the main areas of current research. It tory of applied mathematics [I.6]. The Companion
also explains what applied mathematicians do, which treats applied mathematics in a broad sense, and it
includes not only studying the subject itself but also cannot cover all aspects in equal depth. Some parts
writingaboutmathematics,teachingit,andinfluencing of mathematical physics are included, though a full
policymakers. treatmentofmodernfundamentaltheoriesisnotgiven.
TheCompaniondiffersfromanencyclopediainthat Statistics and probability are not explicitly included,
itisnotanexhaustivetreatmentofthesubject,andit althoughanumberofarticlesmakeuseofideasfrom
differs from a handbook in that it does not cover all thesesubjects,andinparticulartheburgeoningareaof
relevant methods and techniques. Instead, the aim is uncertaintyquantification[II.34]bringstogether
to offer a broad but selective coverage that conveys many ideas from applied mathematics and statistics.
the excitement of modern applied mathematics while Applied mathematics increasingly makes use of algo-
also giving an appreciation of its history and the out- rithms and computation, and a number of aspects at
standingchallenges.TheCompanionfocusesontopics theinterfacewithcomputerscienceareincluded.Some
feltbytheeditorstobeofenduringinterest,andsoit parts of discrete and combinatorial mathematics are
shouldremainrelevantformanyyearstocome. alsocovered.
Withonlinesourcesofinformationaboutmathemat-
3 Audience
ics growing ever more extensive, one might ask what
role a printed volume such as this has. Certainly, one The target audience for The Companion is mathe-
can use Google to search for almost any topic in the maticians at undergraduate level or above; students,
bookandfindrelevantmaterial,perhapsonWikipedia. researchers, and professionals in other subjects who
WhatdistinguishesTheCompanion isthatitisaself- use mathematics; and mathematically interested lay
contained, structured reference work giving a consis- readers. Some articles will also be accessible to stu-
tent treatment of the subject. The content has been dentsstudyingmathematicsatpre-universitylevel.
curated by an editorial board of applied mathemati- Prospectiveresearchstudentsmightusethebookto
cianswithawiderangeofinterestsandexperience,the obtainsomeideaofthedifferentareasofappliedmath-
articleshavebeenwrittenbyleadingexpertsandhave ematicsthattheycouldworkin.Researcherswhoreg-
beenrigorouslyeditedandcopyedited,andthewhole ularlyattendseminarsinareasoutsidetheirownspe-
volumeisthoroughlycross-referencedandindexed. cialities should find that the articles provide a gentle
Withineacharticle,theauthorsandeditorshavetried introductiontosomeoftheseareas,makinggoodpre-
hardtoconveythemotivationforeachtopicorconcept orpost-seminarreading.
and the basic ideas behind it, while avoiding unnec- Insolicitingandeditingthearticlestheeditorsaimed
essary detail. It is hoped that The Companion will be tomaximizeaccessibilitybykeepingdiscussionsatthe
seen as a friendly and inspiring reference, containing lowest practical level. A good question is how much
both standard material and more unusual, novel, or of the book a reader should expect to understand.
unexpectedtopics. Of course “understanding” is an imprecisely defined