A Class of Contracting Stream Operators

In (Tucker, J. V. and Zucker, J. I. (2007) Computability of analog networks. Theoret. Comput. Sci., 371, 115–146; Tucker, J. V. and Zucker, J. I. (2011) Continuity of operators on continuous and discrete time streams. Theoret. Comput. Sci., 412, 3378–3403), Tucker and Zucker present a model for the semantics of analog networks operating on streams from topological algebras. Central to their model is a parametrized stream operator representing the network along with a theory that concerns the existence, uniqueness, continuity and computability of a fixed point of that stream operator. We narrow the scope of this paper from general topological algebras to algebras of streams that assume values only from a Banach space. This restriction facilitates the definition of a fairly broad class of stream operators to which the theory described in the above two papers applies. As a demonstration in their original work, the authors provide two case studies: analog networks that model the behavior of simple mass-spring-damper systems. The case studies showcase the theory well, but they seem to require the imposition of somewhat peculiar conditions on the parameters (the masses, the spring constants and the damping coefficients). The extra conditions—while not catastrophic to the case studies—make them somewhat unsatisfying. We show here that while their original mass–spring–damper models do not fall within our new class, they can be trivially reconfigured into equivalent models that do. This modification obviates the extra conditions on the parameters.