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By Armen H. Zemanian
Cambridge Tracts in Mathematics
(No. 101)
Publisher: Cambridge University Press
Print Publication Year:1991
Online Publication Date:February 2012
Online ISBN:9780511895432
Hardback ISBN:9780521401531
Paperback ISBN:9780521063395
Book DOI: http://dx.doi.org/10.1017/CBO9780511895432
Subjects: Discrete Mathematics Information Theory and Coding , Algorithmics, Complexity, Computer Algebra, Computational Geometry
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.
Reviews:
pp. i-iv
pp. v-vi
pp. vii-xi
pp. xii-xii
pp. 1-29
2 - Infinite-power Regimes: Read PDF
pp. 30-60
3 - Finite-power Regimes: The Linear Case: Read PDF
pp. 61-107
4 - Finite-power Regimes: The Nonlinear Case: Read PDF
pp. 108-138
5 - Transfinite Electrical Networks: Read PDF
pp. 139-157
pp. 158-214
pp. 215-266
pp. 267-293
pp. 294-302
pp. 303-304
pp. 305-308