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Full Description
Scientia Gratiii Scientiae It is now thirteen years since the first book that discusses transfinite graphs and elec trical networks appeared [50]. This was followed by two more books [51] and [54] which compiled results from an ongoing research effort on that subject. Why then is a fourth book, this one, being offered? Simply because still more has been achieved beyond that appearing in those prior books. An exposition of these more recent re sults is the purpose of this book. The idea of transfiniteness for graphs and networks appeared as virgin research territory about seventeen years ago. Notwithstanding the progress that has since been achieved, much more remains to be done-or so it appears. Many conclusions con cerning conventionally infinite graphs and networks can be reformulated as open problems for transfinite graphs and networks. Furthermore, questions peculiar to transfinite concepts for graphs and networks can be suggested. Indeed, these two considerations have inspired the new results displayed herein.
Contents
1 Some Preliminaries.- 1.1 Concerning Symbols and Terminology.- 1.2 Ranks of Transfiniteness.- 2 Transfinite Graphs.- 2.1 Branches or Synonymously (-l)-Graphs.- 2.2 0-Graphs.- 2.3 1-Graphs.- 2.4 ?-Graphs.- 2.5
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$$-Graphs.- 2.6 ?-Graphs.- 2.7 A Concise Characterization of Transfinite Paths and Loops.- 2.8 Graphs of Higher Ranks.- 2.9 Why Not Restrict "Extremities" to "Ends"?.- 3 Connectedness, Trees, and Hypergraphs.- 3.1 Transfinite Connectedness.- 3.2 Transfinite Trees.- 3.3 Hypergraphs from ?-Graphs.- 4 Ordinal Distances in Transfinite Graphs.- 4.1 Natural Sums of Ordinals.- 4.2 Lengths of Paths.- 4.3 Metrizable Sets of Nodes.- 4.4 Distances between Nodes.- 4.5 Eccentricities and Related Ideas.- 4.6 Some General Results.- 4.7 When the Nodes of Highest Rank Are Pristine.- 4.8 The Center Lies in a ?-Block.- 4.9 The Centers of Cycle-free ?-Graphs.- 5 Walk-Based Transfinite Graphs and Networks.- 5.1 0-Walks and 1-Wgraphs.- 5.2 1-Walks, 2-Wgraphs, and 2-Walks.- 5.3 ?-Walks and (? + 1)-Wgraphs.- 5.4
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$$-Walks.- 5.5 ?-Wgraphs and ?- 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.- 6 Hyperreal Currents and Voltages in Transfinite Networks.- 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.- 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 ?2-Lmc.- 7.8 A Hyperreal Wave on an ?2-Line.- 7.9 Transfinite Lines of Higher Ranks.- 7.10 A Hyperreal Diffusion on a Transfinite Cable.- 8 Nonstandard Graphs and Networks.- 8.1 Nonstandard Graphs Defined.- 8.2 Incidencesand 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.- A SomeElements of Nonstandard Analysis.- B The Fibonacci Numbers.- C A Laplace Transform for an Artificial RC Cable.- References.- Index of Symbols.
