Volume 17, Issue 6

The output sequence of a simplest self-controlled 2-linear shift register over the residue ring R = Z 2 n is considered. For a fixed output function we study the rank and the period of the output sequence. In some special cases frequency characteristics of cycles of the first coordinate sequence of the output sequence are considered. It is shown that the rank of the output sequence of the 2-dimensional shift register is much greater than the rank of the output sequence of a 1-dimensional register of the same length.

We consider the number ξ( A, b | z ) of solutions of a system of random linear equations Ax = b over a finite field K which belong to the set X r ( z ) of the vectors differing from a given vector z in a given number r of coordinates (or in at most a given number of coordinates). We give conditions under which, as the number of unknowns, the number of equations, and the number of noncoinciding coordinates tend to infinity, the limit distribution of the vector (ξ( A, b | z (1) ), …, ξ( A, b | z ( k ) )) (or of the vector obtained from this vector by normalisation or by shifting some components by one) is the k -variate Poisson law. As corollaries we get limit distributions of the variable (ξ( A, b | z (1) , …, z ( k ) ) equal to the number of solutions of the system belonging to the union of the sets X r ( z ( s ) ), s = 1, …, k . This research continues a series of the author's and V. G. Mikhailov's studies.

We investigate a multitype Galton–Watson process in a random environment generated by a sequence of independent identically distributed random variables. Assuming that the associated random walk constructed by the logarithms of the Perron roots of the reproduction mean matrices satisfies Spitzer's condition, we find the asymptotics of the survival probability at time n as n → ∞.

We find an expression of the intersection number of a graph in terms of the minimum number of complete subgraphs that form a covering of the graph. This provides us with a uniform approach to studying properties of the intersection number of a graph. We distinguish the class of graphs for which the intersection number is equal to the least number of cliques covering the graph. It is proved that the intersection number of a complete r -partite graph is equal to the least n such that . It is proved that the intersection number of the graph is equal to the least n such that . Formulas for the intersection numbers of the graphs rC 4 , r Chain(3), r (C 4 + K m ), rW 5 are obtained.

We construct pairs of reciprocal relations which contain as the coefficients three-index combinatorial numbers. We introduce three-index generalisations of a series of known combinatorial numbers.

It is proved that in any commutative semigroup the complexity of transformation of equal terms of length at most n into each other is of order n log n .