Permissively Finitely Structured Networks

Abstract

The purpose of this chapter is to lay a foundation for a theory of random walks on transfinite networks, a subject that will be explored in Chapter 7. What is needed for this purpose is not only the uniqueness of the node voltages but also a maximum principle for them. The desired principle will assert that in a sourceless subsection every internal node voltage is less than the voltage at some bordering node of the subsection and greater than the voltage at another bordering node of the subsection — except in the case when all the node voltages within the subsection are identical. The uniqueness of the node voltages has already been established under the hypotheses of Theorem 5.5-4 and Corollary 5.5-5. The maximum principle is established in Section 6.3, and for this purpose we will impose stronger conditions than those used for uniqueness. In particular, we will assume that the network NT is not only finitely structured but that the contraction paths can be chosen so that every one of them is permissive. These conditions will also suffice for our theory of transfinite random walks. Some ancillary results that are also established in this chapter are Kirchhoff’s current law for a cut at an isolating set, the feasibility of exciting N^ν by a pure voltage source, the boundedness of all the node voltages by the value of that voltage source, and another maximum principle expressed in terms of currents.