Volume 24, Issue 6

The paper is concerned with a scheme of m-dependent allocation of non-equiprobable groups of particles into cells. It is assumed that the number of groups and the number of cells tend to infinity and are of the same order. Asymptotic formulas for expectation and variance are obtained and sufficient conditions for the asymptotic normality of the number of empty cells in this scheme are formulated.

The paper is concerned with critical branching processes in a Markov random environment. A conditional functional limit theorem for the number of particles in a process and a conditional invariance principle are proved. The asymptotic tail behaviour for the distributions of the maximum and the total number of particles in a process is found.

Let F n be the class of all Boolean n-argument functions and let M n ⊂F n be the class of functions that do not admit implicents of three or less essential variables. Next, let n (3) max (m) be the largest n for which the class M n contains a function of weight m. It will be shown that n (3) max (11) = 5, n (3) max (12) = 11, n (3) max (13) = 11 (heretofore, the exact values were known only for m ≤ 10). The tensor product of matrices is used to define the binary operation ξ:F n1 × F n2 → F n1-n2 . Moreover, it is shown that if one of the arguments ξ belongs to M n1 and the other is a Boolean function f (x 1 , ..., x n2 ) such that f(x 1 , ...,x n2 )⋁f(x̄ 1 , ...,x̄ n2 ) Belongs to M n2 , then the result of the operation always belongs to M n1-n2 . With relaxed conditions, the result of ξ may fail to belong to M n1-n2 . The operation ξ is used in the algorithm for construction of a Boolean functions without three-argument implicents with arbitrary small ratio of the weight of the function to the number of its variables.

The additive complexity of a nondegenerate matrix of size n is the minimum number of additions in a chain of elementary transformations over rows required to reduce the matrix to the identity one. It is shown that if the order of the field tends to infinity, then almost all matrices are of maximum possible additive complexity (n−1)n. The matrices of additive complexity (n − 1)n are shown to be MDS-matrices.

This work is concerned with the random forest formed by the trajectories of a homogeneous Galton- Watson branching process with the Poisson distribution of the number of offsprings that begins with N particles. For the subset of trajectories containing at most non-root n nodes, the limit distributions of the maximum size of a tree as N, n → ∞ are obtained for various types of the behaviour of the offsprings distribution parameter.

We consider the problem of substring search in a set of strings. The problem is the following: given a set of strings and an arbitrary substring, list all strings from the set that contain this substring.We describe search algorithms and obtain lower bounds for the running time and for the memory volume required by the fastest algorithms