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1 online resource (372 pages) 

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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 KahaneKhinchin 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 minorexcluded graphs  3.3 Padded decompositions in terms of ball growth  3.4 Gluing singlescale 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 Lpdistortion 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 ℓ1embeddable 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 Nonembeddability result  6.5 Examples of nonreflexive 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 Expanderlike structures implying coarse nonembeddability 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, qconvexity and psmoothness  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 almostextension  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 Testspace 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 testspace characterization of superreflexivity: binary trees  9.5.2 Further results on testspaces  9.5.3 Further results on the Ribe program  9.5.4 Nonlocal 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 

The series is devoted to the publication of monographs and highlevel textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the nonspecialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob 

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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


Electronic books

