分形分析(英文版)

本书讨论了分形分析，这是一个发展很快的数学分支，主要研究分形的动态特性，如分形的热扩散及分形结构材料的振动。本书从基本的自相似集合的几何学原理开始，逐步对新近的研究成果加以讨论，其中包括拉普拉斯算子的特征值和特征函数的性质、自相似集合热核的渐近特性等。 阅读本书只需具备高等分析学、普通拓扑学和测度论的基础知识。本书特别适合分析和概率论专业的研究生和研究人员阅读，同时本书也非常适合作为与分形有关的研究生课程的补充教材。 Introduction1GeometryofSelf-SimilarSets1.1Constructionofself-similarsets1.2Shiftspaceandself-similarsets1.3Serf-similarstructure1.4Self-similarmeasure1.5Dimensionofself-similarsets1.6Connectivityofself-similarsetsNotesandreferencesExercises2AnalysisonLimitsofNetworks2.1DirichletformsandLaplaciansonafiniteset2.2SequenceofdiscreteLaplacians2.3Resistanceformandresistancemetric2.4DirichletformsandLaplaciansonlimitsofnetworksNotesandreferencesExercises3ConstructionofLaplaciansonP.C.F.Serf-SimilarStructures3.1Harmonicstructures3.2Harmonicfunctions3.3Topologygivenbyeffectiveresistance3.4Dirichletformsonp.e.f.serf-similarsets3.5Greensfunction3.6Greensoperator3.7Laplacians3.8NestedfractalsNotesandreferencesExercises4EigenvaluesandEigenfunctionsofLaplacians4.1Eigenvaluesandeigenfunctions4.2Relationbetweendimensions4.3Localizedeigenfunctions4.4Existenceoflocalizedeigenfunctions4.5EstimateofeigenfunctionsNotesandreferences5HeatKernels5.1Constructionofheatkernels5.2Parabolicmaximumprinciple5.3AsymptoticbehavioroftheheatkernelsNotesandreferencesAppendixAAdditionalFactsA.1SecondeigenvalueofAiA.2GeneralboundaryconditionsA.3ProbabilisticapproachBMathematicalBackgroundB.1Self-adjointoperatorsandquadraticformsB.2SemigroupsB.3DirichletformsandtheNashinequalityB.4TherenewaltheoremBibliographyIndexofNotationIndex

字数:版次：1-1 开本:16开

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