This self-contained book examines results on transfinite graphs and
networks achieved through a continuing research effort during the
past several years. These new results, covering the mathematical
theory of electrical circuits, are different from those presented
in two previously published books by the author, Transfiniteness
for Graphs, Electrical Networks, and Random Walks and Pristine
Transfinite Graphs and Permissive Electrical Networks.
Two initial chapters present the preliminary theory summarizing
all essential ideas needed for the book and will relieve the reader
from any need to consult those prior books. Subsequent chapters are
devoted entirely to novel results and cover:
* Connectedness ideas---considerably more complicated for
transfinite graphs as compared to those of finite or conventionally
infinite graphs----and their relationship to hypergraphs
* Distance ideas---which play an important role in the theory of
finite graphs---and their extension to transfinite graphs with more
complications, such as the replacement of natural-number distances
by ordinal-number distances
* Nontransitivity of path-based connectedness alleviated by
replacing paths with walks, leading to a more powerful theory for
transfinite graphs and networks
Additional features include:
* The use of nonstandard analysis in novel ways that leads to
several entirely new results concerning hyperreal operating points
for transfinite networks and hyperreal transients on transfinite
transmission lines; this use of hyperreals encompasses for the
first time transfinite networks and transmission lines containing
inductances and capacitances, in addition to resistances
* A useful appendix with concepts from nonstandard analysis used
in the book
* May serve as a reference text or as a graduate-level textbook
in courses or seminars
Graphs and Networks: Transfinite and Nonstandard will appeal to
a diverse readership, including graduate students, electrical
engineers, mathematicians, and physicists working on infinite
electrical networks. Moreover, the growing and presently
substantial number of mathematicians working in nonstandard
analysis may well be attracted by the novel application of the
analysis employed in the work.
ISBN 0-8176-4292-7
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