Volume 4, Issue 2

The dynamics of classical and quantum systems in the presence of noise is usually described in the language of stochastic differential equations. When the system observables comprise a C *-algebra, stochastic evolutions are obtained by solving such equations driven by creation, preservation and annihilation processes on Fock space, with linear maps on the algebra as coefficients. *-Homomorphic evolutions are obtained precisely when the collection of maps has a certain structure; this structure admits a cohomological description. Here we consider equations governing the joint evolution of the system and noise (from input to output) by supposing that the characteristics of the (input) noise processes are given by a group representation. The structure required for *-homomorphic evolution is determined.

We show that the set of injective functions from any uncountable cardinal less than the continuum into the real numbers is of the second category in the box product topology.

Global existence of solutions of equations of compressible barotropic viscous fluids in a bounded domain with boundary slip condition which are sufficiently close to a rest state is proved. Moreover, stability of the solutions and existence of corresponding stationary solutions are shown too.

In the present paper, a new generalization of Browder fixed point theorem is obtained. As its applications, we obtain some generalized versions of Browder's theorems for quasi-variational inequality and Ky Fan's minimax inequality and minimax principle.

Sufficient conditions are found for oscillation of all solutions to a class of nonlinear impulsive differential equations of first order with retarded argument and fixed moments of impulse effect.

H. Kudō introduced a notion of convergence of a sequence B n of sub- σ -fields. We emphasize a connection with Mosco-convergence of associated subspaces where E is a separable Banach space with separable dual. This enables us to study the convergence of conditional expectation sequences of random sets and therefore to obtain a new version of the multivalued Fatou lemma.

By extending the definition of Dini–Hadamard directional derivatives, higher–order necessary optimality conditions for a nonsmooth extremum problem are established, in the presence of both equality and inequality constraints. The presence of a regularity assumption for the inequality constraints strenghtens the optimality condition.

This paper gives bounds for the uncoupling measures of a stochastic matrix P in terms of its eigenvalues. The proofs are combinatorial. We use the Matrix–Tree Theorem which represents principal minors of I – P as sums of weights of directed forests.

We consider a refined coderivative construction for nonsmooth and set-valued mappings between Banach spaces. This limiting mixed coderivative is different from “normal” coderivatives generated by normal cones/subdifferentials and turns out to be useful for studying some basic propertiers in variational analysis particularly related to Lipschitzian stability. We develop a strong calculus for this coderivative important for various applications.