# Recent And Current Research Problems

## Miscellaneous Theoretical Problems

### Locality-Invariance Theorem:

Given a processor with vista function W(d) and invarinace group Inv(p), the following is always true. Any u in Inv(p) is conformal on the vista function: W(u(d)) = u(W(d)) " d Î D. This theorem is very significant as it brings out an intimate relationship between two notions that have been hitherto considered entirely unrelated. This might yield interesting results when the implications of the theorem that each invariance transformation will yield mutually isomorphic isolation topologies is investigated further.

### Quasinclusion Based Set Theory:

A definition of quasinclusion as a generalisation of subinclusion (as developed in the thesis for sets of equal rank) now applicable to any two sets of arbitrary rank is the starting point of the development. Quasi-union and quasi-intersection may be defined along lines similar to their conventional counterparts, except for using quasinclusion rather than ordinary inclusion as the basis. It turns out that the resulting operations have very interesting properties, such as the non-uniqueness of the quasi-union and quasi-intersection. Thus also emerges quasi-equivalence, the weak counterpart of set equality. The quasi-power set is then defined as the set of all quasi-subsets. The quasi-power set includes as a subset the ordinary power set. Among questions presently under investigation is a conjecture that the cardinality of the quasi-equivalence partition of the quasi-power set is the same as that of the power set, and that one member of the power set is included in each class of the partition.

### Hypoprocessing:

Maps on the power sets of various (higher) orders of the signal space are termed as hypoprocessors. The processing effected by these entities is qualitatively different from that achieved by ordinary processors, which are their special instances. A hypoprocessor of the kth kind is distinguished by the nonexistence of all preservance topologies of order less than k. Possible questions of interest in this connection include a depiction of processors as relations on the signal space rather than as functions; a theory of hypoprocessing may yield the mathematical foundation for a study of this kind. The candidate has just recently started work on this area.

### The 'Law Of Nondiminishing Symmetry':

This law is a by-product of the candidate's earlier studies in set theoretic signal processing. It constitutes a sort of negative result regarding the evolution of any system governed by invariant laws. The Nondiminishing Symmetry Law disallows certain kinds of state transitions in such systems, thus constraining the evolution of these systems to remain within a certain class of possible trajectories. When applied to physical systems, the Law permits one to make certain predictions regarding disallowed future states, and importantly, can do so even without exact information regarding the relevant governing laws, beyond their invariance properties. This Law of nondiminishing symmetry was originally formulated only for deterministic systems, but recent work, carried out over the last few months, has discovered a sufficient condition that allows its application on probabilistic systems. In its present form, it stands completed.