Transfinite cascades

Abstract

Just as the natural numbers can be extended to the transfinite ordinals, so too can ladder networks and cascaded three-terminal networks be extended beyond infinity-so long as appropriate assumptions are imposed on their parameters. The author establishes the existence of such transfinite, linear or nonlinear, resistive, electrical networks. In particular, there are cascades of three-terminal networks whose nodes are sequentially numbered form the input to the output first by the natural numbers and then by transfinite ordinals all the way out to omega /sup p/, where rho is any natural number and omega is the first transfinite ordinal. Upon specifying an appropriate source or load at the input (node 0) and also at the output (node omega /sup p/), one obtains as a result a unique set of voltages and currents along the nodes of the cascade. This implies, in turn, that an output load far beyond infinity can be perceived by an observer at the input of the cascade. In fact, these ideas can be extended to cascades that reach still further, for example, out to omega /sup w/ and beyond. All this is a natural extension of a certain kind of infinite ladder network whose load at infinity is perceptible to an observer at the input to the ladder. Such ladder networks arise in practical applications as models of various physical phenomena taking place in infinite domains.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>