|
PRISTINE TRANSFINITE GRAPHS
AND PERMISSIVE ELECTRICAL NETWORKS |
| SHORT DESCRIPTION OF THE BOOK A transfinite graph or electrical network is obtained conceptually by connecting conventionally infinite graphs or networks together at their infinite extremities. The resulting transfiniteness is said to be of rank 1. Then, infinitely many such entities of rank 1 can be connected together at their extremities to obtain graphs or networks of rank 2. This process can be continued to obtain a hierarchy of transfiniteness having ranks that progress through the natural numbers and then through the countably infinite ordinals. This idea, which is of recent origin, has enriched the theories of graphs and networks with radically new constructs and research problems. The problems either involve extensions of conventional theories or are fundamentally different from any questions concerning ordinary graphs and networks. The first book devoted exclusively to this idea, ``Transfiniteness for Graphs, Electrical Networks, and Random Walks,'' aimed for generality and struggled with a variety of difficulties resulting from the inherent complexity of the subject. One objective of this book is to provide a much simpler exposition, sacrificing some generality but capturing the essential ideas of transfiniteness for graphs and networks. Thus, for example, discrete potential theory and random walks on transfinite networks can now be presented more concisely. On the other hand, the simplifications now adopted enables the establishment of many new results, which is another objective of this book. For example, Minty's powerful theory for nonlinear monotone networks is now extended to transfinite networks. This book provides a more accessible introduction to transfinite graphs and networks and may be read before undertaking the more general theory presented in the prior book. SOME KEY FEATURES: 1. A simplified exposition providing a more accessible introduction to transfiniteness for graphs and networks. 2. Extends transfinitely Minty's powerful analysis of monotone electrical networks. 3. Provides a concise treatment of random walks on transfinite networks. 4. Extends conventional theory with radically new constructs. 5. A promising area for a new kind of mathematical research. |
