We propose a new framework for Conley index theory. The main feature of our approach is that we do not use the notion pairs. introduce, instead, notions compactifiable subsets and neighbourhoods, formulate prove basic results in theory using these notions. treat both discrete time case continuous case.

We demonstrate, for the first time, high-speed circuits that generate partitions on a set S of n objects. We offer two versions. In the first, partitions are produced in lexicographical order in response to successive clock pulses. In the second, an index input determines the set partition produced. Such circuits are needed in the hardware implementation of the optimum distribution of tasks to ...

By Doob’s theorem, supermartingales indexed by the natural numbers decompose into the difference of a uniformly integrable martingale and an increasing process. The relative ease of working with increasing processes rather than supermartingales explains the prominent role of this result in stochastic analysis and in the theory of stochastic integration. Meyer [19] then proved that, under the us...