| TABLE OF CONTENTS
PRISTINE TRANSFINITE GRAPHS
AND PERMISSIVE ELECTRICAL NETWORKS |
| TABLE OF CONTENTS Preface Chapter 1. Introduction 1.1 Notations and Terminology 1.2 Transfinite Nodes and Graphs 1.3 A Need for Transfiniteness 1.4 Pristine Graphs Chapter 2. Pristine Transfinite Graphs 2.1 0-Graphs and 1-Graphs 2.2 mu-Graphs and (mu+1)-Graphs 2.3 arrow-omega-Graphs and omega-Graphs 2.4 Transfinite Graphs of Higher Ranks Chapter 3. Some Transfinite Graph Theory 3.1 Nondisconnectable Tips and Connectedness 3.2 Sections 3.3 Transfinite Versions of Konig's Lemma 3.4 Countable Graphs 3.5 Locally Finite Graphs 3.6 Transfinite Ends Chapter 4. Permissive Transfinite Networks 4.1 Linear Electrical Networks 4.2 Permissive 1-Networks 4.3 The 1-Metric 4.4 The Recursive Assumptions 4.5 Permissive (mu+1)-Networks 4.6 Permissive Networks of Ranks arrow-omega, omega, and Higher Chapter 5. Linear Networks; Tellegen Regimes 5.1 A Tellegen-Type Fundamental Theorem 5.2 Node Voltages 5.3 Transfinite Current Flows - Some Ideas 5.4 Current Flows at Natural-Number Ranks 5.5 Current Flows at the Rank omega Chapter 6. Monotone Networks; Kirchhoff Regimes 6.1 Some Assumptions 6.2 Minty's Colored Graph Theorem 6.3 Wolaver's No-Gain Property 6.4 Duffin's Theorem on Operating Points 6.5 The Minty-Calvert Theorem 6.6 Potentials and Branch Voltages 6.7 Existence of a Potential 6.8 Existence of an Operating Point 6.9 Uniqueness of an Operating Point 6.10 Monotone omega-Networks 6.11 Reconciling Two Theories Chapter 7. Some Maximum Principles 7.1 Input Resistance Matrices 7.2 Some Maximum Principles for Node Voltages Chapter 8. Transfinite Random Walks 8.1 The Nash-Williams Rule 8.2 Transfinite Walks 8.3 Transfiniteness for Random Walks 8.4 Reaching a Bordering Node 8.5 Leaving a Bordering Node 8.6 Transitions for Adjacent Bordering Nodes 8.7 Wandering on a nu-Network Bibliography Index of Symbols Index |
