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 |
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 โExtremitiesโ to โEndsโ? -- 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 Kirchhoffโs 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 Kirchhoffโs 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