Volume 26, Issue 6

The conjugacy word problem in the tree product of free groups with a cyclic amalgamation is solved in the positive. This result generalizes the known result obtained by S. Lipschutz for the free product of two free groups with cyclic amalgamation. Solution of the main problem involves proving the solvability of the problem of intersection of a finitely generated subgroup of a given class of groups with a cyclic subgroup belonging to the factor of the main group; the solvability of the problem of intersection of a coset of a finitely generated subgroup with a cyclic subgroup belonging to a free factor.

We give a survey of the enumeration of independent sets in graphs and of some related problems.

It is assumed that the input data for scheduling a set of customers are given as a bipartite graph in some system of units. We consider the problem of composing a schedule of smallest length under the condition of continuous work with no downtime of each unit and their simultaneous actuation. Conditions are obtained for a partition of the edge set of a graph into matchings to form a schedule of the required form.

The sequence of n random (0, 1)-variables X 1 , …, X n is considered, with θ n of these variables distributed equiprobable and the others take the value 1 with probability p (0 < p < 1, p ≠ 1/2) θ n is a random variable taking values 0, 1, …, n ). On the assumption that n → ∞ and under certain conditions imposed on p , θ n and X k , k = 1,..., n , several limit theorems for the sum Sn=∑k=1nXk${S_n} = \sum _{k = 1}^n{X_k}$. The results are of interest in connection with steganography and statistical analysis of sequences produced by random number generators.

Let A be a Bezout ring without non-central idempotents. If A is a right or left Rickartian ring, then A is an Hermitian ring. If A is an exchange ring, then every rectangular matrix over A is diagonalizable.