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Pristine Transfinite Graphs and Permissive Electrical Networks

BuchKartoniert, Paperback
183 Seiten
Englisch
Springererschienen am01.11.2012Softcover reprint of the original 1st ed. 2001
A transfinite graph or electrical network of the first rank is obtained conceptually by connecting conventionally infinite graphs and networks together at their infinite extremities.mehr
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Produkt

KlappentextA transfinite graph or electrical network of the first rank is obtained conceptually by connecting conventionally infinite graphs and networks together at their infinite extremities.
Details
ISBN/GTIN978-1-4612-6641-9
ProduktartBuch
EinbandartKartoniert, Paperback
Verlag
Erscheinungsjahr2012
Erscheinungsdatum01.11.2012
AuflageSoftcover reprint of the original 1st ed. 2001
Seiten183 Seiten
SpracheEnglisch
Gewicht312 g
IllustrationenXI, 183 p.
Artikel-Nr.30048021

Inhalt/Kritik

Inhaltsverzeichnis
1 Introduction.- 1.1 Notations and Terminology.- 1.2 Transfinite Nodes and Graphs.- 1.3 A Need for Transfiniteness.- 1.4 Pristine Graphs.- 2 Pristine Transfinite Graphs.- 2.1 0-Graphs and 1-Graphs.- 2.2 ?-Graphs and (? + 1)-Graphs.- 2.3 $$ \mathop{\omega }\limits^{ \to } $$-Graphs and ?-Graphs.- 2.4 Transfinite Graphs of Higher Ranks.- 3 Some Transfinite Graph Theory.- 3.1 Nondisconnectable Tips and Connectedness.- 3.2 Sections.- 3.3 Transfinite Versions of König´s Lemma.- 3.4 Countable Graphs.- 3.5 Locally Finite Graphs.- 3.6 Transfinite Ends.- 4 Permissive Transfinite Networks.- 4.1 Linear Electrical Networks.- 4.2 Permissive 1-Networks.- 4.3 The 1-Metric.- 4.4 The Recursive Assumptions.- 4.5 Permissive (? + l)-Networks.- 4.6 Permissive Networks of Ranks $$ \mathop{\omega }\limits^{ \to } $$, ?, and Higher.- 5 Linear Networks; Tellegen Regimes.- 5.1 A Tellegen-Type Fundamental Theorem.- 5.2 Node Voltages.- 5.3 Transfinite Current Flows-Some Ideas.- 5.4 Current Flows at Natural-Number Ranks.- 5.5 Current Flows at the Rank ?.- 6 Monotone Networks; Kirchhoff Regimes.- 6.1 Some Assumptions.- 6.2 Minty´s Colored-Graph Theorem.- 6.3 Wolaver´s No-Gain Property.- 6.4 Duffin´s Theorem on Operating Points.- 6.5 The Minty-Calvert Theorem.- 6.6 Potentials and Branch Voltages.- 6.7 Existence of a Potential.- 6.8 Existence of an Operating Point.- 6.9 Uniqueness of an Operating Point.- 6.10 Monotones ?-Networks.- 6.11 Reconciling Two Theories.- 7 Some Maximum Principles.- 7.1 Input Resistance Matrices.- 7.2 Some Maximum Principles for Node Voltages.- 8 Transfinite Random Walks.- 8.1 The Nash-Williams Rule.- 8.2 Transfinite Walks.- 8.3 Transfiniteness for Random Walks.- 8.4 Reaching a Bordering Node.- 8.5 Leaving a Bordering Node.- 8.6 Transitions for AdjacentBordering Nodes.- 8.7 Wandering on a v-Network.- References.- Index of Symbols.mehr
Kritik
"Transfinite graphs and networks have already been studied in two prior books by the same author. The purpose of the present volume is twofold. First, it is to include a variety of new results that have since been obtained. Second, it is to present a much simpler exposition of the subject. The previous works aimed at generality with a variety of difficulties resulting from the complexity of the subject. The present much simpler exposition sacrifices some generality, but captures the essential ideas of transfiniteness for graphs and networks.... Overall, the author has done excellent work in presenting this complex subject."

-Mathematical Reviews

"Transfiniteness is an abstract mathematical concept that is extended by the author to the very real world of graphs and networks and the idea of random walks, topics of particular interest to circuits and systems network research.... Mathematicians, operations researchers and electrical engineers, in particular, graph theorists, electrical circuit theorists, and probabalists will find an accessible exposition of an advanced subject."

-Zentralblatt MATHmehr