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. This process can be
repeated to obtain a hierarchy of transfiniteness whose ranks increase
through the countable ordinals. This idea, which is of recent origin,
has enriched the theories of graphs and networks with radically new
constructs and research problems.
The book provides a more accessible introduction to the subject that,
though sacrificing some generality, captures the essential ideas of
transfiniteness for graphs and networks. Thus, for example, some
results concerning discrete potentials and random walks on transfinite
networks can now be presented more concisely. Conversely, the
simplifications enable the development of many new results that were
previously unavailable.
Topics and features: *A simplified exposition provides an introduction
to transfiniteness for graphs and networks.*Various results for
conventional graphs are extended transfinitely. *Minty's powerful
analysis of monotone electrical networks is also extended
transfinitely.*Maximum principles for node voltages in linear
transfinite networks are established. *A concise treatment of random
walks on transfinite networks is developed. *Conventional theory is
expanded with radically new constructs.
Mathematicians, operations researchers and electrical engineers, in
particular, graph theorists, electrical circuit theorists, and
probabalists will find an accessible exposition of an advanced
subject.
Rezensionen / Stimmen
"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 MATH
Sprache
Verlagsort
Verlagsgruppe
Illustrationen
Dateigröße
ISBN-13
978-1-4612-0163-2 (9781461201632)
DOI
10.1007/978-1-4612-0163-2
Schweitzer Klassifikation
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.
