Expansion Mapping Theorem for Four Mappings Satisfying Common Limit in the Range Property in b-Metric Space

In the present paper, for four weakly compatible mappings in pairs, an expansion mapping theorem has been developed in b-metric space, meeting common limit range property. We proved this theorem without using the b-metric space's completeness condition. The result is an extension and generalization of several metric space results available. To confirm the finding, a suitable example is also discussed.

ON THE COMPLETENESS CONDITION IN NONPARAMETRIC INSTRUMENTAL PROBLEMS

The notion of completeness between two random elements has been considered recently to provide identification in nonparametric instrumental problems. This condition is quite abstract, however, and characterizations have been obtained only in special cases. This paper considers a nonparametric model between the two variables with an additive separability and a large support condition. In this framework, different versions of completeness are obtained, depending on which regularity conditions are imposed. This result allows one to establish identification in an instrumental nonparametric regression with limited endogenous regressor, a case where the control variate approach breaks down.

COMMENTS ON UNITARITY IN THE ANTIFIELD FORMALISM

The local completeness condition was introduced in the analysis of the locality of the gauge fixed action for gauge systems. This condition expresses that the gauge transformations and the reducibility coefficients should be described in such a way that they contain as few derivatives of the gauge parameters as possible. We show here that this condition not only guarantees that the gauge fixed action is local in space-time (as proved previously), but also that the antifield formalism leads to a unitary theory.

Another note on Kempisty's generalized continuity

Under a fairly mild completeness condition on spacesYandZwe show that everyx-continuous functionf:X×Y×Z→Mhas a “substantial” setC(f)of points of continuity. Some odds and ends concerning a related earlier result shown by the authors are presented. Further, a generalization ofS. Kempisty's ideas of generalized continuity on products of finitely many spaces is offered. As a corollary from the above results, a partial answer toM. Talagrand's problem is provided.