Digital circuit designers have developed highly reliable techniques for associating automata with certain classes of electrical circuits described by smooth differential equations. However, the usual explanations of their methods do not give a general context for the process nor do they shed much light on alternative possibilities for constructing such associations. Because noise is ever-present and because reliability is of paramount importance, the association must be continuous in a relevant topology. In this talk we will discuss a reasonably general topological setting for the problem of associating an automaton with a smooth dynamical system. In our approach homotopy theory and de Rham cohomology play a key role in achieving "tokenization without discontinuity." We will also discuss models for passing from the continuous to the discrete as appropriate for other familiar applications such as A/D converters, decoding algorithms involving soft decoding, estimation of discrete random variables in continuous noise, etc. The solution to these problems appears to involve discontinuity in a more fundamental way.

Material used during the talk

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1998-1999 Mathematics in Biology