http://dspace.nbuv.gov.ua:80/handle/123456789/188471 2026-04-22T14:32:35Z http://dspace.nbuv.gov.ua:80/handle/123456789/188559 Comaximal factorization in a commutative Bezout ring Zabavsky, B.V.; Romaniv, O.; Kuznitska, B.; Hlova, T. We study an analogue of unique factorization rings in the case of an elementary divisor domain. 2020-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/188558 Modules with minimax Cousin cohomologies Vahidi, A. Let R be a commutative Noetherian ring with non-zero identity and let X be an arbitrary R-module. In this paper, we show that if all the cohomology modules of the Cousin complex for X are minimax, then the following hold for any prime ideal p of R and for every integer n less than X—the height of p: (i) the nth Bass number of X with respect to p is finite; (ii) the nth local cohomology module of Xp with respect to pRp is Artinian. 2020-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/188557 On small world non-Sunada twins and cellular Voronoi diagrams Ustimenko, V. Special infinite families of regular graphs of unbounded degree and of bounded diameter (small world graphs) are considered. Two families of small world graphs Gi and Hi form a family of non-Sunada twins if Gi and Hi are isospectral of bounded diameter but groups Aut(Gi) and Aut(Hi) are nonisomorphic. We say that a family of non-Sunada twins is unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. If all Gi and Hi are edge-transitive we have a balanced family of small world non-Sunada twins. We say that a family of non-Sunada twins is strongly unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. We use term edge disbalanced for the family of non-Sunada twins such that all graphs Gi and Hi are edge-intransitive. We present explicit constructions of the above defined families. Two new families of distance-regular—but not distance-transitive—graphs will be introduced. 2020-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/188556 On growth of generalized Grigorchuk's overgroups Samarakoon, S.T. Grigorchuk’s Overgroup Ĝ, is a branch group of intermediate growth. It contains the first Grigorchuk’s torsion group G of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of G. The group G, corresponding to the sequence (012)∞ = 012012 . . ., is a member of the family {Gω|ω ∈ Ω = {0, 1, 2}ᴺ} consisting of groups of intermediate growth when sequence ω is not eventually constant. Following this construction, we define the family { Ĝω, ω ∈ Ω} of generalized overgroups. Then Ĝ = Ĝ (012)∞ and Gω is a subgroup of Ĝω for each ω ∈ Ω. We prove, if ω is eventually constant, then Ĝω is of polynomial growth and if ω is not eventually constant, then Ĝω is of intermediate growth. 2020-01-01T00:00:00Z