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TRANSFINITENESS FOR GRAPHS, ELECTRICAL NETWORKS, AND RANDOM WALKS |
| TABLE OF CONTENTS Preface Chapter 1. Introduction 1.1 Notations and Terminology 1.2 Conventional Graphs---A Poor Start 1.3 0-Graphs 1.4 Why Transfiniteness? Chapter 2. Transfinite Graphs 2.1 1-Graphs 2.2 mu-Graphs 2.3 omega-arrow-Graphs 2.4 omega-Graphs 2.5 Graphs of Higher Ranks Chapter 3. Connectedness 3.1 Transfinite Connectedness 3.2 A Transitivity Criterion for Transfinite Connectedness 3.3 About Nodes and Paths 3.4 nu-Sequences 3.5 Another Transitivity Criterion for Transfinite Connectedness 3.6 The Cardinality of the Branch Set Chapter 4. Finitely Structured Transfinite Graphs 4.1 Subsections and Cores 4.2 A Generalization of Konig's Lemma 4.3 Isolating Sets and Cuts 4.4 Contractions 4.5 Finitely Structured nu-Graphs 4.6 Spanning Trees Chapter 5. Transfinite Electrical Networks 5.1 Kirchhoff's Laws, Tellegen's Equation, and Ohm's Law 5.2 The Voltage-Current Regime 5.3 Conditions Legitimizing Kirchhoff's Laws 5.4 The Regime Induced by a Pure Current Source 5.5 Node Voltages Chapter 6. Permissively Finitely Structured Networks 6.1 Permissivley Finitely Structured nu-Networks 6.2 Excitations by Pure Sources 6.3 Maximum Principles for Node Voltages and Branch Currents Chapter 7. Transfinite Random Walks 7.1 The Nash-Williams Rule 7.2 Transfinite Walks 7.3 Transfinite Random Walks---The General Idea 7.4 Reaching a Bordering Node 7.5 Leaving a beta-Node 7.6 From a beta-Node to a beta-Adjacent (beta+)-Node 7.7 Wandering on a nu-Network Appendix A: Ordinal and Cardinal Numbers Appendix B: Summable Series Appendix C: Irreducible and Reversible Markov Chains Bibliography Index of Symbols Index |
