2. Articole
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Item ON THE DIVISION OF ABSTRACT MANIFOLDS IN CUBES(Institutul de Matematică şi Informatică al AŞM, 2006) Bujac, Mariana; Cataranciuc, Sergiu; Soltan, PetruWe prove that in the class of abstract multidimensional manifolds withoutborders only torusVn1of dimensionn≥1 can be divided in abstract cubes with theproperty: every faceImfromVn1is shared by 2n−mcubes,m= 0,1, . . . , n−1. Theabstract torusVn1is realized inEd, n+1≤d≤2n+1, so it results that in the class ofalln-dimensional combinatorial manifolds [1]onlytorus respects this propriety. Torusis autodual because of this propriety.Item THE EULER TOUR OFN-DIMENSIONAL MANIFOLDWITH POSITIVE GENUS(Institutul de Matematică şi Informatică al AŞM, 2008) Cataranciuc, Sergiu; Bujac-Leisz, Mariana; Soltan, PetruIn 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.Item ABSTRACT COMPLEXES, THEIR HOMOLOGIES AND APPLICATIONS(Institutul de Matematică şi Informatică al AŞM, 2010) Cataranciuc, Sergiu; Soltan, PetruThe 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.Item ABSTRACT COMPLEXES, THEIR HOMOLOGIES AND APPLICATIONS(Institutul de Matematică şi Informatică al AŞM, 2010) Cataranciuc, Sergiu; Soltan, PetruThe 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.Item VARIETĂŢI ABSTRACTE MULTIDIMENSIONALE DEGENERATE(CEP USM, 2013) Cataranciuc, SergiuEste definită varietatea degenerată n -dimensională şi sunt studiate unele proprietăţi ale acesteia. Varietăţile sunt considerate închise, conexe, omogene şi fără găuri. Pornind de la complexul generalizat de relaţii multi-are, se face clasificarea varietăţilor degenerate. Rezultatele obţinute sunt similare celor cunoscute pentru varietăţile clasice de genul p [5,11,12], studiate prin intermediul complexului simplu de relaţii multi-are.Item MATRICELE DE RELAŢII MULTI-ARE ŞI NUMERELE PRIME ÎN CRIPTAREA INFORMAŢIEI(CEP USM, 2012) Cataranciuc, Sergiu; Zgureanu, AureliuSe propune un sistem simetric de criptare a informaţiei, numit Crypto 2. Acest sistem se bazează pe utilizarea numerelor prime mari şi pe dezvoltarea lor polinomială. Un rol deosebit revine matricelor multidimensionale, folositepentru reprezentarea mulţimilor de relaţii multi-are. Se descrie algoritmul de criptare/decriptare a informaţiei, Cripto 2,şi se demonstrează că acest algoritm are o complexitate liniară.Item ABOUT RATES OF CONVERGENCE IN THE LIMIT THEOREM(CEP USM, 2011) Munteanu, Bogdan Gheorghe; Leahu, Alexei; Cataranciuc, SergiuÎn lucrare se propune o estimare a ratei de convergenţă a densităţii sumei de variabile aleatoare în raport cu densitatea probabilităţii de repartiţie. Studii similare au fost făcute şi de către alţi matematicieni. Aici putem menţiona lucrarea lui W. Macht şi Wolf W. „On the local central limit theorem” [2], în care aproximarea densităţii probabilităţii unui şir de variabile aleatoare este făcută cu referire la densitatea probabilităţii distribuţiei normale standarde. Principalul rezultat al acestui articol se conţine în teorema 2.1, care stabileşte rata de convergenţă a densităţii probabilităţii unui şir de variabile aleatoare în raport cu densitatea probabilităţii de repartiţie a variantei cu una şi cu trei grade de libertate.Item THE SPECIAL METRICS OF TH E ABSTRACT CUBIC COMPLEX(CEP USM, 2008) Cataranciuc, SergiuSe examinează complexul cubic abstract nK ca un caz particular al G-complexului de relaţii multi-are [2]. Pentru complexul cubic abstract nK se defineşte o funcţie specială ce posedă proprietăţile metricii.Item THE CONVEXITY IN THE COMPLEX OF MULTI-ARY RELATIONS(Romanian Society of Applied & Industrial Mathematics Universitatea din Pitesti, 2015) Braguţă, Galina; Cataranciuc, SergiuFor a complex of multi-ary relations [12] it is defined the concept of (k,m)-chain which is a generalization of the concept of chain known from the graph theory. Using (k,m)- chains it is introduced the concept of the distance function and it is proved that this function generate a convexity in the complex of multi-ary relations. It is operating with the concepts of convexity and convex hull, axiomatically defined by F.Levi [29] and we describe the iterative procedure to construct a convex hull for a subset of elements from the complex of multi-ary relations. [ABSTRACT FROM AUTHOR]Item ON NONTRIVIAL COVERS AND PARTITIONS OF GRAPHS BY CONVEX SETS(Institutul de Matematică şi Informatică al Academiei de Ştiinţe a Moldovei, 2018) Buzatu, Radu; Cataranciuc, SergiuIn this paper we prove that it is NP-complete to decide whet- her a graph can be partitioned into nontrivial convex sets. We show that it can be verified in polynomial time whether a graph can be covered by nontrivial convex sets. Also, we propose a re- cursive formula that establishes the maximum nontrivial convex cover number of a tree.