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© 2001

Pristine Transfinite Graphs and Permissive Electrical Networks

Book

Table of contents

  1. Front Matter
    Pages i-xi
  2. Armen H. Zemanian
    Pages 1-16
  3. Armen H. Zemanian
    Pages 17-28
  4. Armen H. Zemanian
    Pages 29-52
  5. Armen H. Zemanian
    Pages 53-80
  6. Armen H. Zemanian
    Pages 81-103
  7. Armen H. Zemanian
    Pages 105-134
  8. Armen H. Zemanian
    Pages 135-147
  9. Armen H. Zemanian
    Pages 149-171
  10. Back Matter
    Pages 173-183

About this book

Introduction

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.

Keywords

Circuit theory Graph Graph theory Node Operations Research Potential Sim analytical engineering systems theory

Authors and affiliations

  1. 1.Electrical Engineering DepartmentUniversity at Stony BrookStony BrookUSA

Bibliographic information

Industry Sectors
Finance, Business & Banking

Reviews

"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