| TABLE OF CONTENTS
GRAPHS AND NETWORKS:
TRANSFINITE AND NONSTANDARD |
| TABLE OF CONTENTS Preface Chapter 1. Some Preliminaries 1.1 Concerning Symbols and Terminology 1.2 Ranks of Transfiniteness Chapter 2. Transfinite Graphs 2.1 Branches or Synonymously (-1)-Graphs 2.2 0-Graphs 2.3 1-Graphs 2.4 mu-Graphs 2.5 arrow-omega-Graphs 2.6 omega-Graphs 2.7 A Concise Characterization of Transfinite Paths and Loops 2.8 Graphs of Higher Ranks 2.9 Why Not Restrict ``Extremities'' to ``Ends''? Chapter 3. Connectedness, Trees, and Hypergraphs 3.1 Transfinite Connectedness 3.2 Transfinite Trees 3.3 Hypergraphs from nu-Graphs Chapter 4. Ordinal Distances in Transfinite Graphs 4.1 Natural Sums of Ordinals 4.2 Lengths of Paths 4.3 Metrizable Sets of Nodes 4.4 Distances between Nodes 4.5 Eccentricities and Related ideas 4.6 Some General Ideas 4.7 When Nodes of Highest Rank are Pristine 4.8 The Center Lies in a nu-Block 4.9 The Centers of Cycle-Free nu-Graphs Chapter 5. Walk-Based Transfinite Graphs and Networks 5.1 0-Walks and 1-Wgraphs 5.2 1-Walks, 2-Wgraphs, and 2-Walks 5.3 mu-Walks and (mu+1)-Wgraphs 5.4 arrow-omega-Wgraphs and arrow-omega-Walks 5.5 omega-Wgraphs and omega-Walks 5.6 Walk-Based Extremities 5.7 Lengths of Walks 5.8 Wdistances between Wnodes 5.9 Weccentricities and Related Ideas 5.10 Walk-Based Transfinite Electrical Networks 5.11 Tours and Tour Currents 5.12 The Solution Space T 5.13 The Existence of a Unique Current-Voltage Regime 5.14 Kirchhoff's Laws 5.15 The Uniqueness of Wnode Voltages Chapter 6. Hyperreal Currents and Voltages in Transfinite Network 6.1 Two Examples 6.2 Restorable Networks 6.3 Hyperreal Currents and Voltages; a Hyperreal Operating Point 6.4 Eventual Connectedness, Eventual Separability, and Kirchhoff's Laws 6.5 Three Examples Involving Ladder Networks 6.6 Random Walks on Restorable Transfinite Networks 6.7 Appending and Inserting Branches; Buildable Graphs 6.8 Other Ideas: Nonstandard Graphs and Networks Chapter 7. Hyperreal Transients in Transfinite RLC Networks 7.1 Hyperreal Transients on the Hyperreal Time Line 7.2 Hyperreal Transients in Restorable RLC Networks 7.3 A Transfinite RLC Ladder 7.4 A Transfinite Artificial Cable 7.5 A Transfinite Artificial Transmission Line 7.6 Conventionally Infinite, Uniform, Transmission Lines and Cables, and Nonstandard Enlargements 7.7 The omega^2-Line 7.8 A Hyperreal Wave on an omega^2-Line 7.9 Transfinite Lines of Higher Ranks 7.10 A Hyperreal Diffusion on a Transfinite Cable Chapter 8. Nonstandard Graphs and Networks 8.1 Nonstandard Graphs Defined 8.2 Incidences and Adjacencies between Nodes and Branches 8.3 Nonstandard Hyperfinite Paths and Loops 8.4 Connected Nonstandard Graphs 8.5 Nonstandard Subgraphs 8.6 Nonstandard Trees 8.7 Some Numerical Formulas 8.8 Nonstandard 1-Graphs 8.9 A Fundamental Theorem for Nonstandard 1-Networks Appendix A: Some Elements of Nonstandard Analysis Appendix B: The Fibonacci Numbers Appendix C: A Laplace Transform for an Artificial RC Cable Bibliography Index of Symbols Index |
