http://dspace.nbuv.gov.ua:80/handle/123456789/150395 2026-04-15T05:00:38Z http://dspace.nbuv.gov.ua:80/handle/123456789/155748 Transformations of (0,1] preserving tails of Δμ-representation of numbers Isaieva, T.M.; Pratsiovytyi, M.V. In the paper, classes of continuous strictly increasing functions preserving ``tails'' of Δμ-representation of numbers are constructed. Using these functions we construct also continuous transformations of (0,1]. We prove that the set of all such transformations is infinite and forms non-commutative group together with an composition operation. 2016-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/155747 Amply (weakly) Goldie-Rad-supplemented modules Mutlu, F.T. Let R be a ring and M be a right R-module. We say a submodule S of M is a \textit{(weak) Goldie-Rad-supplement} of a submodule N in M, if M=N+S, (N∩S≤Rad(M)) N∩S≤Rad(S) and Nβ∗∗S, and M is called amply (weakly) Goldie-Rad-supplemented if every submodule of M has ample (weak) Goldie-Rad-supplements in M. In this paper we study various properties of such modules. We show that every distributive projective weakly Goldie-Rad-Supplemented module is amply weakly Goldie-Rad-Supplemented. We also show that if M is amply (weakly) Goldie-Rad-supplemented and satisfies DCC on (weak) Goldie-Rad-supplement submodules and on small submodules, then M is Artinian. 2016-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/155746 Hamming distance between the strings generated by adjacency matrix of a graph and their sum Ganagi, A.B.; Ramane, H.S. Let A(G) be the adjacency matrix of a graph G. Denote by s(v) the row of the adjacency matrix corresponding to the vertex v of G. It is a string in the set Zn2 of all n-tuples over the field of order two. The Hamming distance between the strings s(u) and s(v) is the number of positions in which s(u) and s(v) differ. In this paper the Hamming distance between the strings generated by the adjacency matrix is obtained. Also HA(G), the sum of the Hamming distances between all pairs of strings generated by the adjacency matrix is obtained for some graphs. 2016-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/155745 Generalized norms of groups Drushlyak, M.G.; Lukashova, T.D.; Lyman, F.M. In this survey paper the authors specify all the known findings related to the norms of the group and their generalizations. Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of infinite subgroups, the norm of infinite Abelian subgroups and the norm of other systems of Abelian subgroups. 2016-01-01T00:00:00Z