Volume 25, Issue 1

A nonnegative basis of a complete lattice is constructed. The criterion of existence of a nonnegative basis of an arbitrary lattice is proved. The problem of existence of a lattice basis from an arbitrary convex cone is investigated.

We investigate the possibility to use non-associative groupoids in the realization of an open key distribution procedure based on a generalization of the well known Diffie-Hellman algorithm.We prove the existence of non-associative groupoidswhich are simultaneously power commuting and not power-associative.

We consider an approach to the classification of n-variable Boolean bent functions of the nonlinearity degree 3. We utilize the apparatus of bent rectangles introduced by S. V. Agievich. This apparatus was used for the classification of 8-variable Boolean cubic bent functions. The results of our research allow to construct cubic bent functions that depend on an arbitrary even number of variables; the construction is based on well studied quadratic bent functions.

Let G be a permutation group acting on a finite set Ω of cardinality n. The number of orbits of the induced action of G on the set Ω m of all m-element subsets of Ω obeys the trivial estimates |Ω m |/|G| ≤ |Ω m /G| ≤ |Ω m |. In this paper the upper estimate is improved in terms of the minimal degree of the group G or the minimal degree of its subset with small complement. In particular, using the universal estimates obtained by Bochert for the minimal degree of a group and by Babai-Pyber for the order of a group, in terms of n only we demonstrate that if G is a 2-transitive group other than the full symmetric or the alternating groups,mand n are large enough, and the ratio m/n is bounded away from 0 and 1, then |Ω m /G| ≈ |Ω m |/|G|. Similar results hold true for the induced action of G on the set Ω (m) of all m-element multisets with elements drawn from Ω, provided that the ratio m/(m + n) is uniformly bounded away from 0 and 1.

The structure of finite abelian n-ary groups is completely described: it is reduced to the direct product of primary abelian semicyclic n-ary groups up to isomorphism.

We consider a discrete stochastic model of inventory control based on a controlled semi-Markov process. Probabilistic characteristics of the semi-Markov process are found along with characteristics of a stationary cost functional connected with this process. It is proved that an optimal policy of inventory control is a deterministic one. Explicit analitical representation of stationary functional characterising the control quality is obtained. An optimal control problem is reduced to the solution of an extremal problem for a multivariate function.