http://dspace.nbuv.gov.ua:80/handle/123456789/150327 2026-04-18T21:26:01Z http://dspace.nbuv.gov.ua:80/handle/123456789/155716 On large indecomposable modules, endo-wild representation type and right pure semisimple rings Simson, D. The existence of large indecomposable right Rmodules over a right artinian ring R is discussed in connection with the pure semisimplicity problem and the endo-wildness of the category Mod(R) of right R-modules. Some conjectures and open problems are presented. 2003-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/155714 On the representation of a number as a sum of the k-th powers in an arithmetic progression Prosyanyuk, N.S. In this paper we obtain the asymptotic formula for a natural n ≤ x which representate as a sum of two non-negative k-th powers in an arithmetic progression. 2003-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/155712 Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II Chernousova, Zh.T.; Dokuchaev, M.A.; Khibina, M.A.; Kirichenko, V.V.; Miroshnichenko, S.G.; Zhuravlev, V.N. The main concept of this part of the paper is that of a reduced exponent matrix and its quiver, which is strongly connected and simply laced. We give the description of quivers of reduced Gorenstein exponent matrices whose number s of vertices is at most 7. For 2 ≤ 6 s ≤ 5 we have that all adjacency matrices of such quivers are multiples of doubly stochastic matrices. We prove that for any permutation σ on n letters without fixed elements there exists a reduced Gorenstein tiled order Λ with σ(ε) = σ. We show that for any positive integer k there exists a Gorenstein tiled order Λk with inΛk = k. The adjacency matrix of any cyclic Gorenstein order Λ is a linear combination of powers of a permutation matrix Pσ with non-negative coefficients, where σ = σ(Λ). If A is a noetherian prime semiperfect semidistributive ring of a finite global dimension, then Q(A) be a strongly connected simply laced quiver which has no loops. 2003-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/155701 Flows in graphs and the homology of free categories Husainov, A.A.; Calısıcı, H. We study the R−module of generalized flows in a graph with coefficients in the R−representation of the graph over a ring R with 1 and show that this R−module is isomorphic to the first derived functor of the colimit. We generalize Kirchhoff’s laws and build an exact sequence for calculating the R−module of flows in the union of graphs. 2003-01-01T00:00:00Z