To the validity of constraints in the class of generalized elements

A. G. Chentsov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2014), pp. 90-109

Résumé

The problem of validity of asymptotic constraints is considered. This problem is reduced to a generalized problem in the class of ultrafilters of initial solution space. The above-mentioned asymptotic constraints are associated with the standard component defined by the usual requirement of belonging to a given set. This component corresponds conceptually to Warga construction of exact solutions. At the same time, under validity of above-mentioned constraints, asymptotic regimes realizing the idea of validity of belonging conditions with a “certain index” can arise; however, the fixed set characterizing the standard constraint in terms of inclusion is replaced by a nonempty family. This family often arises due to sequential weakening of the belonging constraint to a fixed set in topological space (often metrizable) for an element dependent on the solution choice. The elements of above-mentioned family are the sets which are defined by belonging of their elements to neighborhoods of the given fixed set. But it is possible that the family defining the asymptotic constraints arises from the very beginning and does not relate to weakening of a standard condition. The paper deals with the general case, for which the set structure of admissible generalized elements is investigated. It is shown that for “well-constructed” generalized problem the standard component of “asymptotic constraints” is responsible for the realization of the insides of above-mentioned set of admissible generalized elements; the particular representation of this topological property is established. Some corollaries of mentioned representation concerning generalized admissible elements not approximable (in topological sense) by precise solutions are obtained.

Détail

Keywords: extension, topological space, ultrafilter.