Beyond Topologies, Part I

Arguments on the need, and usefulness, of going beyond the usual Hausdorff-Kuratowski-Bourbaki, or in short, HKB concept of topology are presented. The motivation comes, among others, from well known {\it topological type processes}, or in short TTP-s, in the theories of Measure, Integration and Ordered Spaces. These TTP-s, as shown by the classical characterization given by the {\it four Moore-Smith conditions}, can {\it no longer} be incorporated within the usual HKB topologies. One of the most successful recent ways to go beyond HKB topologies is that developed in Beattie & Butzmann. It is shown in this work how that extended concept of topology is a {\it particular} case of the earlier one suggested and used by the first author in the study of generalized solutions of large classes of nonlinear partial differential equations

Quantum Exotic PDE's

Following the previous works on the A. Pr\'astaro's formulation of algebraic topology of quantum (super) PDE's, it is proved that a canonical Heyting algebra ({\em integral Heyting algebra}) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prastaro's geometric theory of quantum PDE's is applied to the new category of {\em quantum hypercomplex manifolds}, related to the well-known Cayley-Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE's in this new category of noncommutative manifolds. Finally the extension of the concept of exotic PDE's, recently introduced by A.Pr\'astaro, has been extended to quantum PDE's. Then a smooth quantum version of the quantum (generalized) Poincar\'e conjecture is given too. These results extend ones for quantum (generalized) Poincar\'e conjecture, previously given by A. Pr\'astaro.Comment: 52 page

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200

Estimation for bilinear stochastic systems

Three techniques for the solution of bilinear estimation problems are presented. First, finite dimensional optimal nonlinear estimators are presented for certain bilinear systems evolving on solvable and nilpotent lie groups. Then the use of harmonic analysis for estimation problems evolving on spheres and other compact manifolds is investigated. Finally, an approximate estimation technique utilizing cumulants is discussed