2. Articole

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    ABSTRACT COMPLEXES, THEIR HOMOLOGIES AND APPLICATIONS
    (Institutul de Matematică şi Informatică al AŞM, 2010) Cataranciuc, Sergiu; Soltan, Petru
    The complex of multi-ary relations Kn is defined in a more natural way than it was defined in [18, 58, 59]. The groups of homologies and co-homologies of this complex over the group of integer numbers are constructed. The methods used for these constructions are for the most part analogous with classical methods [2,32,52], but sometimes they are based on methods from [18,44,58]. The importance and originality consist in application of the multi-ary relations of a set of objects in construction of homologies. This allows to extend areas of theoretical researches and non-trivial practical applications in a lot of directions. Other abstract structures, which are developed in a natural way from generalized complex of multi-ary relations are also examined. New notions such as the notions of abstract quasi-simplex and its homologies, the complex of abstract simplexes and the complex of the n-dimensional abstract cubes are introduced.
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    ABSTRACT COMPLEXES, THEIR HOMOLOGIES AND APPLICATIONS
    (Institutul de Matematică şi Informatică al AŞM, 2010) Cataranciuc, Sergiu; Soltan, Petru
    The complex of multi-ary relations Kn is defined in a more natural way than it was defined in [18, 58, 59]. The groups of homologies and co-homologies of this complex over the group of integer numbers are constructed. The methods used for these constructions are for the most part analogous with classical methods [2,32,52], but sometimes they are based on methods from [18,44,58]. The importance and originality consist in application of the multi-ary relations of a set of objects in construction of homologies. This allows to extend areas of theoretical researches and non-trivial practical applications in a lot of directions. Other abstract structures, which are developed in a natural way from generalized complex of multi-ary relations are also examined. New notions such as the notions of abstract quasi-simplex and its homologies, the complex of abstract simplexes and the complex of the n-dimensional abstract cubes are introduced.
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    THE EULER TOUR OFN-DIMENSIONAL MANIFOLDWITH POSITIVE GENUS
    (Institutul de Matematică şi Informatică al AŞM, 2008) Cataranciuc, Sergiu; Bujac-Leisz, Mariana; Soltan, Petru
    In the paper [1] it is proved that abstract cubicn-dimensional torus[2] possesses a directed Euler tour of the same dimension. The result prompts toa new (virtual) device for transmission and reception of information. In the presentpaper it is shown that every abstract cubicn-dimensional manifold without borders, ofpositive genus possesses an-dimensional directed Euler tour. This result has practicalapplication.
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    THE MULTIDIMENSIONAL DIRECTED EULER TOUROF CUBIC MANIFOLD
    (Institutul de Matematică şi Informatică al AŞM, 2006) Bujac, Mariana
    n the paper [3] we tried to generalize the problem of existence of a di-rected (n−1)-dimensional Euler tour for the abstract directedn-dimensional manifold,which is acomplex of multi-ary relations[5], namely by means of abstract simplexes.In the paper [3] we show the existence of such kind of tour onlyfor manifolds of odddimension because we have not enough conditions to do more. In the present paperwe will show conditions of existence for a directed Euler tour of abstract manifoldswith even dimensions. In this purpose, we will introduce some new definitions whichpermit us to define manifolds by so-calledabstract cubes.