Filters and Ultrafilters as Approximate Solutions in the Attainability Problems with Constraints of Asymptotic Character

Abstract problems about attainability in topological spaces are considered. Some nonsequential version of the Warga approximate solutions is investigated: we use filters and ultrafilters of measurable spaces. Attrac- tion sets are constructed. AMS (MOS) subject classification. 46A, 49 K 40.

Constraints of Asymptotic Character and Attainability Problems

The attainability problem with “asymptotic constraints” is considered. Concrete variants of this problem arise in control theory. Namely, we can consider the problem about construction and investigation of attainability domain under perturbation of traditional constraints (boundary and immediate conditions;phase constraints). The natural asymptotic analog of the usual attainability domain is attraction set, for representation of which, the Warga generalized controls can be applied. More exactly, for this, attainability domain in the class of generalized controls is constructed. This approach is similar to methods for optimal control theory (we keep in mind approximate and generalized controls of J. Warga). But, in the case of attainability problem, essential difficulties arise. Namely, here it should be constructed whole set of limits corresponding to different variants of all more precise realization of usual solutions in the sense of constraints validity. Moreover, typically, the above-mentioned control problems are infinite-dimensional. Real possibility for investigation of the arising limit sets is connected with extension of control space. For control problems with geometric constraints on the choice of programmed controls, procedure of this extensions was realized (for extremal problems) by J. Warga. More complicated situation arises in theory of impulse control. It is useful to note that, for investigation of the problem about constraints validity, it is natural to apply asymptotic approach realized in part of perturbation of standard constraints. And what is more, we can essentially generalize self notion of constraints: namely, we can consider arbitrary systems of conditions defined in terms of nonempty families of sets in the space of usual controls. Thus, constraints of asymptotic character arise.

IDEAL STRUCTURE OF UNIFORM ROE ALGEBRAS OVER SIMPLE CORES

This paper characterizes ideal structure of the uniform Roe algebra B*(X) over simple cores X. A necessary and sufficient condition for a principal ideal of B*(X) to be spatial is given and an example of non-spatial ideal of B*(X) is constructed. By establishing an one-one correspondence between the ideals of B* (X) and the ω-filters on X, the maximal ideals of B*(X) are completely described by the corona of the Stone-Cech compactification of X.

Statistical Convergence and Measure Convergence Generated by a Single Statistical Measure

The purpose of this paper is to discuss those kinds of statistical convergence, in terms of F , or ideal Z-convergence, which are equivalent to measure convergence defined by a single statistical measure. We prove a number of characterizations of a single statistical measure μ-convergence by using properties of its corresponding quotient Banach space l∞/l∞ （Iμ）. We also show that the usual sequential convergence is not equivalent to a single measure convergence.

Norming Subspaces Isomorphic to l_1

Norming subspaces are studied widely in the duality theory of Banach spaces. These subspaces are applied to the Borel and Baire classifications of the inverse operators. The main result of this article asserts that the dual of a Banach space X contains a norming subspace isomorphic to l1 provided that the following two conditions are satisfied： （1） X＊ contains a subspace isomorphic to l1; and （2） X＊ contains a separable norming subspace.