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TRANSFINITENESS - FOR
GRAPHS, ELECTRICAL NETWORKS AND RANDOM WALKS |
| SHORT DESCRIPTION OF THE BOOK Transfinite numbers were invented by Cantor over a hundred years ago, and they profoundly affected the development of twentieth-century mathematics. Despite this century-old introduction of transfiniteness into mathematics, its implications for graphs has only been examined during the past decade. This book is devoted exclusively to transfinite graphs and their ramifications for several related topics, such as electrical networks, discrete potential theory, and random walks. A transfinite graph is one wherein at least two nodes are connected through infinite paths but not through any finite path. Such a structure arises when two infinite graphs are joined at their infinite extremities by ``1-nodes.'' In fact, infinitely many graphs may be so joined, and the result may have infinite extremities in a more general sense. The latter may in turn be joined by ``2-nodes.'' These ideas can be extended recursively through the countable cardinals to obtain a hierarchy of transfinite graphs. Transfiniteness introduces radically new constructs and expands graphs and their related topics far beyond their conventional domains. For example, a random walk may now ``wander through infinity.'' Also, a lacuna in the conventional theory of infinite electrical networks can only be closed by connecting resistors to ``different parts of infinity.'' In general, many solved problems of conventional graphs and networks reopen into largely unexplored research areas, and, on the other hand, the researcher of the transfinite is confronted by questions having no counterparts in conventional theories. Chapter Headings (with brief descriptions): Chap. 1: Introduction. (How transfinite networks arise naturally from conventional infinite networks.) Chap. 2: Transfinite Graphs. (Their definitions and the hierarchy of transfiniteness.) Chap. 3: Connectedness. (The nontransitivity of transfinite connectedness, sufficient conditions restoring transitivity, and the cardinality of the branch set.) Chap. 4: Finitely Structured Transfinite Graphs. ( Properties of transfinite graphs that resemble locally finite conventionally infinite graphs. The existence of one-way transfinite paths - a generalization of Konig's lemma. Existence of spanning trees.) Chap. 5: Transfinite Electrical Networks. (Fundamental theorems on the existence and uniqueness of voltage-current regimes. Conditions under which Kirchhoff's laws hold and unique node voltages exist.) Chap. 6. Permissively Finitely Structured Networks. (A transfinite extension of discrete potential theory. Conditions ensuring maximum principles for node voltages.) Chap. 7. Transfinite Random Walks. (Wandering through infinity. The analysis is based on the Nash-Williams rule relating random walks to electrical networks.) Appendix A. Ordinal and Cardinal Numbers. Appendix B. Summable Series. Appendix C. Irreducible and Reversible Markov Chains. |
