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作者 Ostrovskii, Mikhail I
書名 Metric Embeddings : Bilipschitz and Coarse Embeddings into Banach Spaces
出版項 Berlin/Boston : De Gruyter, Inc., 2013
©2013
國際標準書號 9783110264012 (electronic bk.)
9783110263404
book jacket
說明 1 online resource (372 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
系列 De Gruyter Studies in Mathematics Ser. ; v.49
De Gruyter Studies in Mathematics Ser
附註 Intro -- Preface -- 1 Introduction: examples of metrics, embeddings, and applications -- 1.1 Metric spaces: definitions and main examples -- 1.2 Types of embeddings: isometric, bilipschitz, coarse, and uniform -- 1.2.1 Isometric embeddings -- 1.2.2 Bilipschitz embeddings -- 1.2.3 Coarse and uniform embeddings -- 1.3 Probability theory terminology and notation -- 1.4 Applications to the sparsest cut problem -- 1.5 Exercises -- 1.6 Notes and remarks -- 1.6.1 To Section 1.1 -- 1.6.2 To Section 1.2 -- 1.6.3 To Section 1.3 -- 1.6.4 To Section 1.4 -- 1.6.5 To exercises -- 1.7 On applications in topology -- 1.8 Hints to exercises -- 2 Embeddability of locally finite metric spaces into Banach spaces is finitely determined. Related Banach space theory -- 2.1 Introduction -- 2.2 Banach space theory: ultrafilters, ultraproducts, finite representability -- 2.2.1 Ultrafilters -- 2.2.2 Ultraproducts -- 2.2.3 Finite representability -- 2.3 Proofs of the main results on relations between embeddability of a locally finite metric space and its finite subsets -- 2.3.1 Proof in the bilipschitz case -- 2.3.2 Proof in the coarse case -- 2.3.3 Remarks on extensions of finite determination results -- 2.4 Banach space theory: type and cotype of Banach spaces, Khinchin and Kahane inequalities -- 2.4.1 Rademacher type and cotype -- 2.4.2 Kahane-Khinchin inequality -- 2.4.3 Characterization of spaces with trivial type or cotype -- 2.5 Some corollaries of the theorems on finite determination of embeddability of locally finite metric spaces -- 2.6 Exercises -- 2.7 Notes and remarks -- 2.8 Hints to exercises -- 3 Constructions of embeddings -- 3.1 Padded decompositions and their applications to constructions of embeddings -- 3.2 Padded decompositions of minor-excluded graphs -- 3.3 Padded decompositions in terms of ball growth -- 3.4 Gluing single-scale embeddings
3.5 Exercises -- 3.6 Notes and remarks -- 3.7 Hints to exercises -- 4 Obstacles for embeddability: Poincaré inequalities -- 4.1 Definition of Poincaré inequalities for metric spaces -- 4.2 Poincaré inequalities for expanders -- 4.3 Lp-distortion in terms of constants in Poincaré inequalities -- 4.4 Euclidean distortion and positive semidefinite matrices -- 4.5 Fourier analytic method of getting Poincaré inequalities -- 4.6 Exercises -- 4.7 Notes and remarks -- 4.8 A bit of history of coarse embeddability -- 4.9 Hints to exercises -- 5 Families of expanders and of graphs with large girth -- 5.1 Introduction -- 5.2 Spectral characterization of expanders -- 5.3 Kazhdan's property (T) and expanders -- 5.4 Groups with property (T) -- 5.4.1 Finite generation of SLn(ℤ) -- 5.4.2 Finite quotients of SLn(ℤ) -- 5.4.3 Property (T) for groups SLn(ℤ) -- 5.4.4 Criterion for property (T) -- 5.5 Zigzag products -- 5.6 Graphs with large girth: basic definitions -- 5.7 Graph lift constructions and ℓ1-embeddable graphs with large girth -- 5.8 Probabilistic proof of existence of expanders -- 5.9 Size and diameter of graphs with large girth: basic facts -- 5.10 Random constructions of graphs with large girth -- 5.11 Graphs with large girth using variational techniques -- 5.12 Inequalities for the spectral gap of graphs with large girth -- 5.13 Biggs's construction of graphs with large girth -- 5.14 Margulis's 1982 construction of graphs with large girth -- 5.15 Families of expanders which are not coarsely embeddable one into another -- 5.16 Exercises -- 5.17 Notes and remarks -- 5.17.1 Bounds for spectral gaps -- 5.17.2 Graphs with very large spectral gaps -- 5.17.3 Some more results and constructions -- 5.18 Hints to exercises -- 6 Banach spaces which do not admit uniformly coarse embeddings of expanders -- 6.1 Banach spaces whose balls admit uniform embeddings into L1
6.2 Banach spaces not admitting coarse embeddings of expander families, using interpolation -- 6.3 Banach space theory: a characterization of reflexivity -- 6.4 Some classes of spaces whose balls are not uniformly embeddable into L1 -- 6.4.1 Stable metric spaces and iterated limits -- 6.4.2 Non-embeddability result -- 6.5 Examples of non-reflexive spaces with nontrivial type -- 6.6 Exercises -- 6.7 Notes and remarks -- 6.8 Hints to exercises -- 7 Structure properties of spaces which are not coarsely embeddable into a Hilbert space -- 7.1 Expander-like structures implying coarse non-embeddability into L1 -- 7.2 On the structure of locally finite spaces which do not admit coarse embeddings into a Hilbert space -- 7.3 Expansion properties of metric spaces not admitting a coarse embedding into a Hilbert space -- 7.4 Exercises -- 7.5 Notes and remarks -- 7.6 Hints to exercises -- 8 Applications of Markov chains to embeddability problems -- 8.1 Basic definitions and results on finite Markov chains -- 8.2 Markov type -- 8.3 First application of Markov type to embeddability problems: Euclidean distortion of graphs with large girth -- 8.4 Banach space theory: renormings of superreflexive spaces, q-convexity and p-smoothness -- 8.4.1 Definitions and duality -- 8.4.2 Pisier theorem on renormings of uniformly convex spaces -- 8.5 Markov type of uniformly smooth Banach spaces -- 8.6 Applications of Markov type to lower estimates of distortions of embeddings into uniformly smooth Banach spaces -- 8.7 Exercises -- 8.8 Notes and remarks -- 8.9 Hints to exercises -- 9 Metric characterizations of classes of Banach spaces -- 9.1 Introduction -- 9.2 Proof of the Ribe theorem through Bourgain's discretization theorem -- 9.2.1 Proving Bourgain's discretization theorem. Preliminary step: it suffices to consider spaces with differentiable norm
9.2.2 First step: picking the system of coordinates -- 9.2.3 Second step: construction of a Lipschitz almost-extension -- 9.2.4 Third step: further smoothing of the map using Poisson kernels -- 9.2.5 Poisson kernel estimates and proofs of Lemmas 9.14 and 9.15 -- 9.3 Test-space characterizations -- 9.3.1 More Banach space theory: superreflexivity -- 9.3.2 Characterization of superreflexivity in terms of diamond graphs -- 9.4 Exercises -- 9.5 Notes and remarks -- 9.5.1 Another test-space characterization of superreflexivity: binary trees -- 9.5.2 Further results on test-spaces -- 9.5.3 Further results on the Ribe program -- 9.5.4 Non-local properties -- 9.6 Hints to exercises -- 10 Lipschitz free spaces -- 10.1 Introductory remarks -- 10.2 Lipschitz free spaces: definition and properties -- 10.3 The case where dX is a graph distance -- 10.4 Lipschitz free spaces of some finite metric spaces -- 10.5 Exercises -- 10.6 Notes and remarks -- 10.7 Hints to exercises -- 11 Open problems -- 11.1 Embeddability of expanders into Banach spaces -- 11.2 Obstacles for coarse embeddability of spaces with bounded geometry into a Hilbert space -- 11.2.1 The main problem -- 11.2.2 Comments -- 11.3 Embeddability of graphs with large girth -- 11.4 Coarse embeddability of a Hilbert space into Banach spaces -- Bibliography -- Author index -- Subject index
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
鏈接 Print version: Ostrovskii, Mikhail I. Metric Embeddings : Bilipschitz and Coarse Embeddings into Banach Spaces Berlin/Boston : De Gruyter, Inc.,c2013 9783110263404
主題 Banach spaces.;Lipschitz spaces.;Stochastic partial differential equations
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