Transfinite and Nonstandard
Buch, Englisch, 202 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 341 g
ISBN: 978-0-8176-4292-1
Verlag: Birkhäuser Boston
This self-contained book examines results on transfinite graphs and networks achieved through contiued research effort over 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." Specific topics covered include connectedness ideas, distance ideas, and nontransitivity of connectedness.
The book 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.
Zielgruppe
Research
Autoren/Hrsg.
Fachgebiete
- Mathematik | Informatik Mathematik Operations Research Graphentheorie
- Mathematik | Informatik EDV | Informatik Professionelle Anwendung Computer-Aided Design (CAD)
- Technische Wissenschaften Technik Allgemein Computeranwendungen in der Technik
- Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
- Mathematik | Informatik EDV | Informatik Angewandte Informatik Computeranwendungen in Wissenschaft & Technologie
- Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
- Technische Wissenschaften Elektronik | Nachrichtentechnik Nachrichten- und Kommunikationstechnik Funktechnik
- Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
Weitere Infos & Material
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.
