Author	Zemanian, Armen H. author
Title	Graphs and Networks [electronic resource] : Transfinite and Nonstandard / by Armen H. Zemanian
Imprint	Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2004
Connect to	http://dx.doi.org/10.1007/978-0-8176-8178-4
Descript	XII, 202 p. online resource

This self-contained book examines results on transfinite graphs and networks achieved through a continuing research effort during the past several years. These new results, covering the mathematical theory of electrical circuits, are different from those presented in two previously published books by the author, Transfiniteness for Graphs, Electrical Networks, and Random Walks and Pristine Transfinite Graphs and Permissive Electrical Networks. Two initial chapters present the preliminary theory summarizing all essential ideas needed for the book and will relieve the reader from any need to consult those prior books. Subsequent chapters are devoted entirely to novel results and cover: * Connectedness ideas---considerably more complicated for transfinite graphs as compared to those of finite or conventionally infinite graphs----and their relationship to hypergraphs * Distance ideas---which play an important role in the theory of finite graphs---and their extension to transfinite graphs with more complications, such as the replacement of natural-number distances by ordinal-number distances * Nontransitivity of path-based connectedness alleviated by replacing paths with walks, leading to a more powerful theory for transfinite graphs and networks Additional features include: * The use of nonstandard analysis in novel ways that leads to several entirely new results concerning hyperreal operating points for transfinite networks and hyperreal transients on transfinite transmission lines; this use of hyperreals encompasses for the first time transfinite networks and transmission lines containing inductances and capacitances, in addition to resistances * A useful appendix with concepts from nonstandard analysis used in the book * May serve as a reference text or as a graduate-level textbook in courses or seminars Graphs and Networks: Transfinite and Nonstandard will appeal to a diverse readership, including graduate students, electrical engineers, mathematicians, and physicists working on infinite electrical networks. Moreover, the growing and presently substantial number of mathematicians working in nonstandard analysis may well be attracted by the novel application of the analysis employed in the work.

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 % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGafqyYdCNbaSaaaaa!3CAA! $$ \vec \omega $$-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 โ{128}{156}Extremitiesโ{128}{157} to โ{128}{156}Endsโ{128}{157}? -- 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 % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGafqyYdCNbaSaaaaa!3CAA! $$ \vec \omega $$-Wgraphs and % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGafqyYdCNbaSaaaaa!3CAA! $$ \vec \omega $$-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 Kirchhoffโ{128}{153}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 Kirchhoffโ{128}{153}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 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 -- A SomeElements of Nonstandard Analysis -- B The Fibonacci Numbers -- C A Laplace Transform for an Artificial RC Cable -- References -- Index of Symbols