Author	Zemanian, Armen H. author
Title	Pristine Transfinite Graphs and Permissive Electrical Networks [electronic resource] / by Armen H. Zemanian
Imprint	Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2001
Connect to	http://dx.doi.org/10.1007/978-1-4612-0163-2
Descript	XI, 183 p. online resource

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

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 $$ \mathop{\omega }\limitŝ{ \to } $$-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 Kรถnigโ{128}{153}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 $$ \mathop{\omega }\limitŝ{ \to } $$, ?, and Higher -- 5 Linear Networks; Tellegen Regimes -- 5.1 A Tellegen-Type Fundamental Theorem -- 5.2 Node Voltages -- 5.3 Transfinite Current Flowsโ{128}{148}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 Mintyโ{128}{153}s Colored-Graph Theorem -- 6.3 Wolaverโ{128}{153}s No-Gain Property -- 6.4 Duffinโ{128}{153}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 Adjacent Bordering Nodes -- 8.7 Wandering on a v-Network -- References -- Index of Symbols