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Books > Science & Mathematics > Science: general issues > History of science
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Transfiniteness - For Graphs, Electrical Networks, and Random Walks (Hardcover, 1996 ed.)
Loot Price: R1,750
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Transfiniteness - For Graphs, Electrical Networks, and Random Walks (Hardcover, 1996 ed.)
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"What good is a newborn baby?" Michael Faraday's reputed response
when asked, "What good is magnetic induction?" But, it must be
admitted that a newborn baby may die in infancy. What about this
one- the idea of transfiniteness for graphs, electrical networks,
and random walks? At least its bloodline is robust. Those subjects,
along with Cantor's transfinite numbers, comprise its ancestry.
There seems to be general agreement that the theory of graphs was
born when Leonhard Euler published his solution to the "Konigsberg
bridge prob lem" in 1736 [8]. Similarly, the year of birth for
electrical network theory might well be taken to be 184 7, when
Gustav Kirchhoff published his volt age and current laws [ 14].
Ever since those dates until just a few years ago, all infinite
undirected graphs and networks had an inviolate property: Two
branches either were connected through a finite path or were not
connected at all. The idea of two branches being connected only
through transfinite paths, that is, only through paths having
infinitely many branches was never invoked, or so it appears from a
perusal of various surveys of infinite graphs [17], [20], [29],
[32]. Our objective herein is to explore this idea and some of its
ramifications. It should be noted however that directed graphs
having transfinite paths have appeared in set theory [6, Section 4.
General
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