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Almost periodic mappings on topological spaces


Author: Pavel Dorin
Degree:doctor of physics and mathematics
Speciality: 01.01.04 - Geometry and topology
Year:2010
Scientific adviser: Mitrofan Cioban
doctor habilitat, professor, Tiraspol State University
Scientific consultant: Laurenţiu Calmuţchi
doctor habilitat, professor, Tiraspol State University
Institution: Tiraspol State University
Scientific council: DH 01-01.01.04
Institute of Mathematics and Computer Science of the ASM

Status

The thesis was presented on the 19 November, 2009
Approved by NCAA on the 11 February, 2010

Abstract

Adobe PDF document0.32 Mb / in romanian

Keywords

almost periodic functions, weakly almost periodic functions, lattice, compactification, algebraical compactification, universal algebra, semigroup of translations, stable pseudometrics

Summary

The thesis is dedicated to the study of the general problems of the theory of extensions of the topological universal algebras from the topological point of view. In particular, the concept of almost periodicity on a universal algebra is investigated.

Objectives of the thesis are: the elaboration of the concept of the almost periodicity on a universal algebra; the investigation of the classes of almost periodic functions on a given universal algebra; the construction of the compact algebraical extensions of topological universal algebras.

In the thesis there are solved the next problems: the elaboration of the methods of research of the spaces of almost periodic functions; the construction of the special compactifications of the n-ary topological groupoids.

There are established: the space of (weakly) almost periodic functions is a Banach space; the space of (weakly) almost periodic mappings is complete; the space of weakly almost periodic functions generates the Bohr compactification proposed by Alfsen and Holm; the space of almost periodic functions is determined by the stable totally bounded pseudometrics.

The basic results are new. The notion of complete compactity is essential. In the thesis, in particular, there are solved some concrete problems arised by J.E.Hart, K.Kunen and A.Cashu. Thee concept of the almost periodicity is based on the notions of the semigroup of translations and the oriented semigroup of translations. This new point of view permits to establish the concepts of almost periodicity and weakly almost periodicity on any universal algebra. For each universal algebra the Weil’s reduction is obtained.