Mathematical Analysis of Evolution, Information, and Complexity

504 Pages·2009·4.35 MB·English

MathematicalAnalysisofEvolution,
Information,andComplexity
Editedby
WolfgangArendtand
WolfgangP.Schleich
Foradditionalinformation
reagardingthistopic,pleaserefer also
to thefollowingpublications
Bruß,D.,Leuchs,G.(eds.)
Lectures onQuantum Information
2007
ISBN978-3-527-40527-5
Audretsch,J.(ed.)
EntangledWorld
TheFascinationofQuantumInformationandComputation
2006
ISBN978-3-527-40470-4
Stolze,J.,Suter,D.
Quantum Computing
AShortCoursefromTheorytoExperiment
2004
ISBN978-3-527-40438-4
Mathematical Analysis of Evolution,
Information, and Complexity
Edited by
Wolfgang Arendt and Wolfgang P. Schleich
WILEY-VCH Verlag GmbH & Co. KGaA
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Prof.Dr.WolfgangP.Schleich detailsorotheritemsmayinadvertentlybe
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Dept.ofQuantumPhysics
[e-mail]
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V
Contents
Preface XV
ListofContributors XIX
Prologue XXIII
WolfgangArendt,DelioMugnoloandWolfgangSchleich
1 Weyl’sLaw 1
WolfgangArendt,RobinNittka,WolfgangPeter,FrankSteiner
1.1 Introduction 1
1.2 ABriefHistoryofWeyl’sLaw 2
1.2.1 Weyl’sSeminalWorkin1911–1915 2
1.2.2 TheConjectureofSommerfeld(1910) 5
1.2.3 TheConjectureofLorentz(1910) 7
1.2.4 BlackBodyRadiation:FromKirchhofftoWien’sLaw 8
1.2.5 BlackBodyRadiation:Rayleigh’sLaw 10
1.2.6 BlackBodyRadiation:Planck’sLawandtheClassicalLimit 12
1.2.7 BlackBodyRadiation:TheRayleigh–Einstein–JeansLaw 14
1.2.8 FromAcousticstoWeyl’sLawandKac’sQuestion 18
1.3 Weyl’sLawwithRemainderTerm.I 19
1.3.1 TheLaplacianontheFlatTorusT2 19
1.3.2 TheClassicalCircleProblemofGauss 20
1.3.3 TheFormulaofHardy–Landau–Voronoï 21
1.3.4 TheTraceFormulaontheTorusT2andtheLeadingWeylTerm 22
1.3.5 SpectralGeometry:InterpretationoftheTraceFormula
ontheTorusT2inTermsofPeriodicOrbits 24
1.3.6 TheTraceoftheHeatKernelond-DimensionalTori
andWeyl’sLaw 25
1.3.7 GoingBeyondWeyl’sLaw:OnecanHearthePeriodicOrbits
oftheGeodesicFlowontheTorusT2 27
1.3.8 TheSpectralZetaFunctionontheTorusT2 28
1.3.9 AnExplicitFormulafortheRemainderTerminWeyl’sLaw
ontheTorusT2andfortheCircleProblem 29
VI Contents
1.3.10 TheValueDistributionoftheRemainderTerm
intheCircleProblem 32
1.3.11 AConjectureontheValueDistributionoftheRemainderTerm
inWeyl’sLawforIntegrableandChaoticSystems 34
1.4 Weyl’sLawwithRemainderTerm.II 38
1.4.1 TheLaplace–BeltramiOperatorond-DimensionalCompactRiemann
ManifoldsMdandthePre-TraceFormula 38
1.4.2 TheSumRulefortheAutomorphicEigenfunctionsonMd 39
1.4.3 Weyl’sLawonMdanditsGeneralizationbyCarleman 40
1.4.4 TheSelbergTraceFormulaandWeyl’sLaw 42
1.4.5 TheTraceoftheHeatKernelonM2 44
1.4.6 TheTraceoftheResolventonM2andSelberg’sZetaFunction 45
1.4.7 TheFunctionalEquationforSelberg’sZetaFunctionZ(s) 48
1.4.8 AnExplicitFormulafortheRemainderTerminWeyl’sLawonM2
andtheHilbert–PolyaConjectureontheRiemannZeros 49
1.4.9 ThePrimeNumberTheorem
vs.thePrimeGeodesicTheoremonM2 51
1.5 GeneralizationsofWeyl’sLaw 52
1.5.1 Weyl’sLawforRobinBoundaryConditions 52
1.5.2 Weyl’sLawforUnboundedQuantumBilliards 53
1.6 AProofofWeyl’sFormula 54
1.7 CanOneHeartheShapeofaDrum? 59
1.8 DoesDiffusionDeterminetheDomain? 63
References 64
2 SolutionsofSystemsofLinearOrdinaryDifferentialEquations 73
WernerBalser,ClaudiaRöscheisen,FrankSteiner,EricSträng
2.1 Introduction 73
2.2 TheExponentialAnsatzofMagnus 76
2.3 TheFeynman–DysonSeries,
andMoreGeneralPerturbationTechniques 78
2.4 PowerSeriesMethods 80
2.4.1 RegularPoints 80
2.4.2 SingularitiesoftheFirstKind 81
2.4.3 SingularitiesofSecondKind 82
2.5 Multi-SummabilityofFormalPowerSeries 84
2.5.1 AsymptoticPowerSeriesExpansions 84
2.5.2 GevreyAsymptotics 85
2.5.3 AsymptoticExistenceTheorems 85
2.5.4 k-Summability 86
2.5.5 Multi-Summability 89
2.5.6 ApplicationstoPDE 90
2.5.7 PerturbedOrdinaryDifferentialEquations 91
2.6 PeriodicODE 92
2.6.1 Floquet–LyapunovTheoremandFloquetTheory 92
Contents VII
2.6.2 TheMathieuEquation 93
2.6.3 TheWhittaker–HillFormula 93
2.6.4 CalculatingtheDeterminant 94
2.6.5 ApplicationstoPDE 94
References 95
3 AScalar–TensorTheoryofGravitywithaHiggsPotential 99
NilsManuelBezares-Roder,FrankSteiner
3.1 Introduction 99
3.1.1 GeneralRelativityandtheStandardModelofParticlePhysics 99
3.1.2 AlternativeTheoriesofGravityandHistoricalOverview 111
3.2 Scalar-TensorTheorywithHiggsPotential 115
3.2.1 LagrangeDensityandModels 115
3.2.2 TheFieldEquations 118
3.2.3 FieldEquationsAfterSymmetryBreakdown 119
3.2.4 Outlook 124
References 131
4 RelatingSimulationandModelingofNeuralNetworks 137
Stefano Cardanobile, Heiner Markert, DelioMugnolo, Günther Palm,
FriedhelmSchwenker
4.1 Introduction 137
4.2 Voltage-BasedModels 138
4.3 ChangingParadigm–FromBiologicalNetworksofNeurons
toArtificialNeuralNetworks 142
4.4 NumericalSimulationofNeuralNetworks 143
4.5 Population-BasedSimulationofLargeSpikingNetworks 148
4.6 SynapticPlasticityandDevelopingNeuralNetworks 152
References 153
5 BooleanNetworksforModelingGeneRegulation 157
ChristianWawra,MichaelKühl,HansA.Kestler
5.1 Introduction 157
5.2 BiologicalBackground 158
5.3 AimsofModeling 160
5.4 ModelingTechniques 161
5.5 ModelingGRNswithBooleanNetworks 162
5.6 DynamicBehaviorofLargeRandomNetworks 165
5.7 InferenceofGeneRegulatoryNetworksfromRealData 169
5.7.1 ProblemDefinition 170
5.7.2 IdentifyingAlgorithms 170
5.7.3 NoisyDataandtheDataFirstApproach 171
5.7.4 AnInformationTheoreticalApproach 174
5.7.5 UsingtheChi-SquareTest
toFindRelationshipsAmongGenes 175
5.8 Conclusion 175
VIII Contents
References 177
6 SymmetriesinQuantumGraphs 181
JensBolte,StefanoCardanobile,DelioMugnolo,RobinNittka
6.1 Symmetries 181
6.2 QuantumGraphs 185
6.3 EnergyMethodsforSchrödingerEquations 186
6.4 SymmetriesinQuantumGraphs 190
6.5 SchrödingerEquationwithPotentials 192
6.6 ConcludingRemarksandOpenProblems 193
References 195
7 DistributedArchitectureforSpeech-ControlledSystems
BasedonAssociativeMemories 197
ZöhreKaraKayikci,DmitryZaykovskiy,HeinerMarkert,WolfgangMinker,
GüntherPalm
7.1 Introduction 197
7.2 SystemArchitecture 199
7.3 FeatureExtractiononMobileDevices 202
7.3.1 ETSIDSRFront-End 202
7.3.1.1 FeatureExtraction 202
7.3.1.2 FeatureCompression 203
7.3.2 ImplementationoftheFront-EndonMobilePhones 204
7.3.2.1 Multi-Threading 204
7.3.2.2 Fixed-PointArithmetic 204
7.3.2.3 ProcessingTimeonRealDevices 205
7.4 SpeechRecognitionSystemsBasedonAssociativeMemory 205
7.4.1 FeaturestoSubwordUnitsConversionusingHMMs 206
7.4.1.1 AcousticModels 206
7.4.1.2 LanguageModelandDictionary 207
7.4.2 SubwordUnitstoWordsConversion
usingNeuralAssociativeMemory 207
7.4.2.1 NeuralAssociativeMemories 207
7.4.2.2 TheNeuralAssociativeMemory-BasedArchitecture
forWordRecognition 209
7.4.2.3 TheFunctionalityoftheArchitecture 210
7.4.2.4 LearningofNewWords 211
7.5 WordstoSemanticsConversionusingAssociativeMemory 211
7.5.1 SpokenWordMemory 212
7.5.2 LanguageParser 213
7.5.3 Ambiguities 214
7.5.4 LearningofNewObjects 215
7.6 SampleSystem/ExperimentalResults 215
7.7 Conclusion 216
References 217