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GRAPHS AND NETWORKS:
TRANSFINITE AND NONSTANDARD |
| SHORT DESCRIPTION OF THE BOOK This self-contained book examines new results on transfinite graphs and networks achieved through research over several years after the publication of two prior books by the author: ``Transfiniteness for Graphs, Electrical Networks, and Random Walks,'' and ``Pristine Transfinite Graphs and Permissive Electrical Networks.'' Two initial chapters present preliminary theory summarizing all the needed ideas previously presented, relieving the reader of any need to consult those prior books. Subsequent chapters are devoted entirely to novel results and cover: * Connectedness ideas and their relationship to hypergraphs * Distance ideas 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 * The use of nonstandard analysis in novel ways that leads to several transfinite electrical networks, wherein Kirchhoff's laws are satified by hyperreal-valued currents and voltages; this use of hyperreals encompasses for the first time transfinite networks containing inductances and capacitances, in addition to resistances. * The use of nonstandard analysis also yields hyperreal-valued waves and diffusions on transfinite extensions of transmission lines and cables. *Enlargements yielding nonstandard graphs and networks as distinct from standard networks with hyperreal operating points * An appendix summarizing concepts from nonstandard analysis used in the book |
