Volume 21, Issue 3

In this paper, we continue our studies concerning consistent sequences of tests in discrete probabilistic statistical schemes. We introduce the concept of a ‘prohibition’ in a discrete probabilistic scheme and give a series of examples where ‘prohibitions’ arise. In statistical problems, tests exist whose critical sets are completely determined by means of ‘prohibitions.’ We find necessary and sufficient conditions for existence of consistent sequences of tests in discrete schemes where all critical sets are defined by means of ‘prohibitions.’

We consider an asynchronous double stochastic flow with initiation of superfluous events (a generalised asynchronous flow), which is a mathematical model of information flows in computer networks, communication systems, etc. We study the stationary mode of the flow. We find the probability density p ( τ ) of the length of the interval between events in the flow and the joint probability density p ( τ 1 , τ 2 ) of the lengths of two neighbouring intervals. We show that the generalised asynchronous flow is a correlated flow in the general case. We find conditions for the flow to become recursive or to degenerate into an elementary one.

We consider the class of all n n single-valued mappings of an n -element set into itself. Assuming that all such mappings have the same probabilities equal to n –n , we investigate the distribution of the random variable ν n equal to the number of connected components in such random mapping. We obtain asymptotic estimates of the probability P { ν n = N } as n, N → ∞ in such a way that the ratio N /ln n does not tend to 0 and infinity. In the case where as n → ∞, we obtain a complete asymptotic expansion of this probability. A similar expansion is obtained for the probability P { ξ n = N }, where ξ n is the random variable equal to the number of cycles in a permutation randomly and equiprobably chosen from the set of all n ! permutations of degree n , and also for the probability P { θ n = N }, where θ n is the number of blocks in a random partition of a set with n elements.

We consider sets of graphs which are closed under the symmetric difference and renaming of vertices. We prove that for any n there are at most 14 such sets consisting of graphs with n vertices.

It is proved that the automorphism group of any commutative Moufang loop Q is an extension of the group F (1) consisting of all automorphisms of the loop Q which induce the identity mapping onto the factor-loop Q / A ( Q ) of Q by means of the automorphism group of the abelian group Q / A ( Q ). We investigate the structure of the group F (1) in the cases where the loop Q is either centrally nilpotent, or finitely generated, or is a ZA -loop.

In the context of the approach given in Shannon's research [Bell Syst. Tech. J. 28: 656–715, 1949], of interest are the mixing properties of operations in a finite field, in particular, the action of each operation on cosets with respect to the action of the second operation. The good mixing properties give a possibility to construct on this base good cryptographic algorithms.

We suggest a new way to construct the structure of all closed classes of two-valued logic. In contrast to classical approaches, in this research the functions of two-valued logic are auxiliary objects and the construction starts from the set of predicates.

In this paper we study the average complexity of optimal algorithms solving the problem of searching for identical objects (PSIO) for random databases. We describe and investigate classes of PSIOs for which the average complexity as a function of the database volume grows logarithmically. For such classes of problems we find exact asymptotics of the complexity. We also study the case where the complexity of the optimal algorithm averaged over the class of problems is bounded and construct a class of PSIOs such that the average complexity of the optimal algorithm is an unbounded function growing slower than the logarithm.

In this paper we suggest a combined method for solving systems of Boolean polynomial equations based on testing a part of variables and using monomial compatibility criteria for dropping out some values of the tested variables. The complexity of the method is estimated for various parameters of the system.