|
PRISTINE TRANSFINITE GRAPHS
AND PERMISSIVE ELECTRICAL NETWORKS |
| PREFACE Georg Cantor's invention of transfinite numbers occurred over one hundred years ago and profoundly influenced the development of twentieth-century mathematics. In fact, it led to a thorough examination of the foundations of mathematics. In contrast to the theory of numbers, graph theory remained on ``this side of infinity'' until a decade ago. An initial embryonic idea concerning ``connections at infinity'' was introduced in 1975, but it was only after 1987 that transfinite graphs and networks were investigated on a continuing basis. This has enriched the theories of graphs and networks with radically new constructs and research problems. In general, many solved problems have now reopened transfinitely and comprise a largely unexplored research area. Moreover, there are problems concerning transfinite graphs and networks having no counterparts in conventional theories. Transfinite graphs and networks have already been examined in two prior books. ``Infinite Electrical Networks'' (Cambridge University Press, 1991) discussed them at an early stage of development, and then ``Transfiniteness for Graphs, Electrical Networks, and Random Walks'' (Birkhauser-Boston, 1996) explored them in great generality. So, why should a third book on this recently conceived subject be offered? Simply because much progress has been achieved in the intervening years in two ways. First, a variety of new results have been obtained. Second, a much simpler rendition of the subject has been devised. This books reports on that continuing research and presents results beyond those appearing in those prior books. Those prior works aimed for generality and struggled with a variety of difficulties resulting from the inherent complexity of that subject. A principal objective of the present rendition is to provide a much simpler exposition, sacrificing some generality but capturing the essential ideas of transfiniteness for graphs and networks. On the other hand, such simplification enables the establishment of a variety of new results, such as a generalization of Minty's powerful theory for nonlinear monotone networks to transfinite networks. One complicating facet of transfinite graphs is that transfinite nodes can contain nodes of lower ranks. By assuming away such nodes, we can disentangle much of transfinite graph theory. Graphs wherein no node contains a node of lower rank will be called ``pristine.'' Actually, no generality is lost so far as connectedness ideas are concerned because nodes can be removed from nodes of higher ranks through ``extraction paths,'' thereby rendering any transfinite graph into a pristine one. A second difficulty with the prior exposition of transfinite graphs and electrical networks was that transfinite nodes were constructed in a strictly graph-theoretic manner that completely ignored their suitability for the flow of electrical currents. With regard to electrical networks, a transfinite node serves no purpose if electrical current cannot flow through it. The presence of such useless nodes caused unnecessary trouble. Our present approach avoids such bother by constructing transfinite graphs in a special way to ensure that only those transfinite nodes that can transmit current need be considered. Transfinite electrical networks having such graphs will be called ``permissive.'' This book is organized as follows. After an introductory chapter, pristine transfinite graphs are defined in Chapter 2. Our definitions of transfinite paths are more concise and much simpler than those in the previously mentioned books. Furthermore, ``sections'' and ``subsections'' now coincide, and there is therefore no need to consider ``subsections.'' Also, there no longer are arrow-omega nodes. Such modifications carry over to Chapter 3 wherein transfinite graphs are explored. Our extended discussion of nu-sequences in Section 3.4 of ``Transfiniteness - for Graphs, Electrical Networks, and Random Walks,'' used in the examination of transitivity for $\nu$-connectedness, is now entirely avoided. The short but important Section 3.5 in this book defines ``local finiteness'' for transfinite graphs, a condition we often use. A new result is presented in Section 3.6, which extends transfinitely the idea of an ``end'' of a conventionally infinite graph. Starting with Chapter 4, we turn to electrical networks. Our new approach to transfinite networks is now based on certain metric spaces that account for the distribution of resistances throughout the network. Distances between nodes are measured by metrics, one for each rank, which pick out those extremities, called ``terminals,'' that are accessible to electrical currents. ``Permissive transfinite nodes'' are then constructed by shorting together terminals. Chapter 5 examines current-voltage regimes in linear networks based on Tellegen's equation. This was the approach adopted in previously mentioned two books, but now much simplification is achieved. There is no longer any need for basic currents; loop currents alone will do. Moreover, node voltages always exist and are unique, and Kirchhoff's laws are always satisfied. None of this was true previously. Chapter 6 presents an entirely new theory for nonlinear transfinite networks based on Minty's classical theory for finite monotone networks and Calvert's generalization to conventionally infinite networks. The theory is founded on Kirchhoff's laws, no use being made of Tellegen's equation. Some classical results on finite nonlinear networks are needed and presented in Sections 6.2 through 6.4: Minty's colored-graph theorem, Wolaver's no-gain property, and the earliest theory of nonlinear networks due to Duffin. The Minty-Calvert theorem follows in Section 6.5. Then, our new theory of transfinite monotone networks appears in Sections 6.6 through 6.10, and this is related in Section 6.11 to the Tellegen-based theory of Chapter 5. Chapter 7 takes up maximum principles for node voltages in linear transfinite networks, and Chapter 8 examines random walks on such networks. Here, too, much simplification is achieved. Because transfinite nodes were defined in strictly graph-theoretic terms in prior books, some severe assumptions had to be imposed in order to derive maximum principles and random walks. A more natural and concise approach to these subjects is achieved in this book as a result of two previously mentioned facts: our permissive transfinite nodes are amenable to electrical regimes and the complications arising from nonpristine nodes are now avoided. One other distinction between ``Transfiniteness - for Graphs, Electrical Networks, and Random Walks'' and this book is worth mentioning. In order to achieve the said maximum principles and transfinite random walks, a quite complicated structure, referred to as ``permissively finitely structured networks,'' was imposed in the prior book. For the same purpose, a simpler structure defined by Conditions 5.3-1 herein is imposed in this book. Neither structure subsumes the other. In this way, we have distinct results. Some knowledge of functional analysis is needed for a comprehension of the analytical parts of this book, but nothing beyond the most commonly known facts concerning metric and Hilbert spaces is required. For the sake of specificity, we refer the reader to some popular textbooks when citing various standard ideas and theorems, but there are indeed many other textbooks that can be so used. Finally, not much is needed from Cantor's theory of transfinite numbers; if need be, the reader might refer to the concise survey in Appendix A of the ``Transfiniteness ---'' book or to any standard text on set theory. Scientific books commonly contain mistakes and misprints despite the best efforts of their authors. Surely, such is the case for this book, too. As corrections are discovered, they will be listed as Errata available on the Internet at www.ece.sunysb.edu/~zeman. Errata for the prior books, ``Infinite Electrical Networks'' and ``Transfiniteness for Graphs, Electrical Networks, and Random Walks,'' are also available there. A.H. Zemanian Stony Brook, New York |
