Graphs and discrete Dirichlet spaces [electronic resource]

• 作者： Keller, Matthias.
• 其他作者：
  • Lenz, Daniel.
  • Wojciechowski, Radoslaw K.
  • SpringerLink (Online service)
• 其他題名：
  • Grundlehren der mathematischen Wissenschaften ;
• 出版： Cham : Springer International Publishing :Imprint: Springer
• 叢書名： Grundlehren der mathematischen Wissenschaften,v. 358
• 主題： Graph theory. , Dirichlet forms. , Graph Theory. , Functional Analysis. , Probability Theory and Stochastic Processes. , Global Analysis and Analysis on Manifolds. , Operator Theory.
• ISBN： 9783030814595 (electronic bk.) 、 9783030814588 (paper)
• FIND@SFXID: CGU
• 資料類型: 電子書
• 內容註: Part 0 Prelude -- Chapter 0 Finite Graphs -- Part 1 Foundations and Fundamental Topics -- Chapter 1 Infinite Graphs - Key Concepts -- Chapter 2 Infinite Graphs - Toolbox -- Chapter 3 Markov Uniqueness and Essential Self-Adjointness -- Chapter 4 Agmon-Allegretto-Piepenbrink and Persson Theorems -- Chapter 5 Large Time Behavior of the Heat Kernel -- Chapter 6 Recurrence -- Chapter 7 Stochastic Completeness -- Part 2 Classes of Graphs -- Chapter 8 Uniformly Positive Measure -- Chapter 9 Weak Spherical Symmetry -- Chapter 10 Sparseness and Isoperimetric Inequalities -- Part 3 Geometry and Intrinsic Metrics -- Chapter 11 Intrinsic Metrics: Definition and Basic Facts -- Chapter 12 Harmonic Functions and Caccioppoli Theory -- Chapter 13 Spectral Bounds -- Chapter 14 Volume Growth Criterion for Stochastic Completeness and Uniqueness Class -- Appendix A The Spectral Theorem -- Appendix B Closed Forms on Hilbert Spaces -- Appendix C Dirichlet Forms and Beurling-Deny Criteria -- Appendix D Semigroups, Resolvents and their Generators -- Appendix E Aspects of Operator Theory -- References -- Index -- Notation Index.
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• 摘要註: The spectral geometry of infinite graphs deals with three major themes and their interplay: the spectral theory of the Laplacian, the geometry of the underlying graph, and the heat flow with its probabilistic aspects. In this book, all three themes are brought together coherently under the perspective of Dirichlet forms, providing a powerful and unified approach. The book gives a complete account of key topics of infinite graphs, such as essential self-adjointness, Markov uniqueness, spectral estimates, recurrence, and stochastic completeness. A major feature of the book is the use of intrinsic metrics to capture the geometry of graphs. As for manifolds, Dirichlet forms in the graph setting offer a structural understanding of the interaction between spectral theory, geometry and probability. For graphs, however, the presentation is much more accessible and inviting thanks to the discreteness of the underlying space, laying bare the main concepts while preserving the deep insights of the manifold case. Graphs and Discrete Dirichlet Spaces offers a comprehensive treatment of the spectral geometry of graphs, from the very basics to deep and thorough explorations of advanced topics. With modest prerequisites, the book can serve as a basis for a number of topics courses, starting at the undergraduate level.
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