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Infinite Electrical Networks : Cambridge Tracts in Mathematics - Armen H. Zemanian

Infinite Electrical Networks

Cambridge Tracts in Mathematics


Published: 29th November 1991
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Over the past two decades a general mathematical theory of infinite electrical networks has been developed. This is the first book to present the salient features of this theory in a coherent exposition. Using the basic tools of functional analysis and graph theory, the author presents the fundamental developments of the past two decades and discusses applications to other areas of mathematics. The first half of the book presents existence and uniqueness theorems for both infinite-power and finite-power voltage-current regimes, and the second half discusses methods for solving problems in infinite cascades and grids. A notable feature is the recent invention of transfinite networks, roughly analogous to Cantor's extension of the natural numbers to the transfinite ordinals. The last chapter is a survey of applications to exterior problems of partial differential equations, random walks on infinite graphs, and networks of operators on Hilbert spaces. The jump in complexity from finite electrical networks to infinite ones is comparable to the jump in complexity from finite-dimensional to infinite-dimensional spaces. Many of the questions that are conventionally asked about finite networks are presently unanswerable for infinite networks, while questions that are meaningless for finite networks crop up for infinite ones and lead to surprising results, such as the occasional collapse of Kirchoff's laws in infinite regimes. Some central concepts have no counterpart in the finite case, as for example the extremities of an infinite network, the perceptibility of infinity, and the connections at infinity.

"...The book would make a fine Graduate level text for Electrical Engineers; it would also be useful for students in applied mathematics as a guide to possible research topics." Andrew Booth, Physics in Canada

Infinite-power regimes
Finite-power regimes: the linear case
Finite-power regimes: the nonlinear case
Transfinite electrical networks
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780521401531
ISBN-10: 0521401534
Series: Cambridge Tracts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 324
Published: 29th November 1991
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 2.2
Weight (kg): 0.65