Transfinite Electrical Networks

Abstract

An electrical network is a graph whose branches are assigned certain analytical structures, which in turn are described by certain physically motivated laws. Since graphs have now been extended transfinitely, the transfinite extension of electrical networks beckons. Toward this we proceed, but, as with graphs, there are surprises in store for us. The laws that govern finite resistive networks, namely Kirchhoff’s voltage and current laws, break when stretched transfinitely. A more fundamental principle will be needed to bring transfinite networks to heel. Another peculiarity is that a pure voltage source or a pure current source cannot in general be applied to any two nodes of a purely resistive network, in contrast to finite networks with positive resistances in all branches. We will find ourselves searching for special kinds of transfinite networks that allow arbitrary applications of sources. Still another difficulty is that node voltages may fail to exist throughout a transfinite electrical network, and, even when they do exist, they may not be uniquely determined. This is an obstacle to the development of a theory of random walks on transfinite networks, and we will once again have to restrict our transfinite networks in order to avoid it.