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CNAA / Theses / 2007 / May /

On the Abstract Manifolds, Multidimensional Euler Directed Tour and their Applications


Author: Mariana Bujac
Degree:doctor of physics and mathematics
Speciality: 01.01.09 - Mathematical cybernetics and operation research
Year:2007
Scientific adviser: Petru Soltan
doctor habilitat, professor, Moldova State University
Institution: Moldova State University
Scientific council: DH 30-01.01.09
Moldova State University

Status

The thesis was presented on the 30 May, 2007
Approved by NCAA on the 14 June, 2007

Abstract

Adobe PDF document0.34 Mb / in romanian

Keywords

Euler-Poincare characteristic, cycle, coefficient of incidence, coherent, cubic complex, - complex of multi-ary relations, directed Euler tour (Euler tour), abstract cube, normal cubiliaj, face, border, genus, group of homologies (homology group), isomorphism, Posthumus' problem, cartesian oriented product, skeleton, abstract simplex, - torus, vacuum, abstract manifold, cubic manifold, spherical manifold (abstract sphere),

Summary

By analogy with results of classification of compact bidimensional surfaces without borders and the notion of compact multidimensional combinatorial manifold without borders, we apply the investigations of the complex of multi-ary relations Kn to define the abstract multidimensional oriented manifold without borders. We indicate the abstract manifolds' classification by their genus.

We construct an infinite sequence of manifolds of the same dimension n. By some constructive methods, we prove that for an arbitrary manifold of dimension n, there is a manifold in the above-mentioned sequence, such that this one and that arbitrary have the same Euler-Poincare characteristic. So we define the classification, where the manifold existed in above-mentioned sequence is a representative of a class.

Using abstract simplex or abstract cube we generalize the notion of multidimensional Euler tour for an abstract manifold. The notion of abstract manifold of even dimension which is defined by simplexes and the notion of abstract cubic manifold are equivalent if the simplexes of manifold determine a cubic manifold.

We prove that in the set of abstract multidimensional manifolds without borders only torus possesses the property of normal cubiliaj. Torus is autodual because of this property.

The theoretical investigations given in the thesis, in a large measure, are done to indicate the conditions of optimization in the transmission of information from the Posthumus' problem. In other words, it is necessary to generalize the classical device from the mentioned above problem. We indicate that n-dimensional torus admits a cover by cubes of dimension m, 0 < m < n, it means that torus has a directed Euler tour of dimension m. So we can change the drums from the Posthumus' problem by two real tangential differential and 2-dimensional tori, which admits circular channels of any length in a compressed space.