Graphs and Networks: Transfinite and Nonstandard - Softcover

"The book presents almost everything that has been achieved so far in the relatively new field of transfinite graphs and electrical networks. In the first two chapters, the reader is familiarized with transfinite graphs and with the symbols and notations used in the book.... The last chapter is dedicated to the approach of nonstandard analysis applied to transfinite graphs. The book must be appreciated especially because new results in the field of transfinite graphs are also included. Therefore, I find this book a welcome addition to the literature."   ―Zentralblatt MATH

"For about thirty years Zemanian has been developing a theory of infinite electrical networks. This book is the latest in a series of books...on the subject. The subject is necessarily abstract and sophisticated because infinite objects are the main objects of discourse.... The first few chapters are important not only to remind the reader of the terms, but also to give an improved or alternate treatment of some earlier results.... There does not yet seem to be a large following of researchers in this area, but it seems very attractive and ripe for investigation. It’s intriguing to see the connections between set theory and electrical network problems.... To understand these concepts fully the reader must consult the book under review. The reviewer highly recommends devoting the effort needed to understand these original and surprising concepts."   ―SIAM Review

This self-contained book examines results on transfinite graphs and networks achieved through continued research effort over 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. Specific topics covered include connectedness ideas, distance ideas, and nontransitivity of connectedness.

The book 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.