Band 12, Heft 2

In the symmetric semigroup ℑ k of all mappings from the set E k into E k , k ≥ 3, we consider alternating monoids, that is, monoids which contain all even permutations on E k . For each monoid T ∈ ℑ k , we define the set of all functions of P k which preserve graphs of all permutations of T (the clone determined by the monoid T). We describe all clones determined by alternating monoids.

We introduce the notions of an absolutely non-homomorphic function, a minimal function (farthest from homomorphisms) and a bent function, and prove that the class of bent functions coincides with the class of absolutely non-homomorphic functions, a function is uniquely determined by the distances to homomorphisms with shifts, and that in the primary case the bent functions are absolutely minimal.

We give a complete description of the polynomials f(x) with integer coefficients such that the period of the recurring sequence u i+l ≡ f(u i ) (mod m) is equal tom.

For a wide class of multicriteria (vector) optimisation problems with a finite set of vector constraints, basing on the additive method of aggregating special monotone functions of partial criteria, we obtain necessary and sufficient conditions of efficiency (Pareto-optimality) of a solution.

We consider the problem on constructing a binary search tree for an arbitrary set of binary words, which has found a wide use in informatics, biology, mineralogy, and other fields. It is known that the problem on constructing the tree of minimal cost is NP-complete; hence the problem arises to find simple algorithms which allow us to construct trees close to the optimal ones. In this paper we demonstrate that even simplest algorithm yields search trees which are close to the optimal ones in average, and prove that the mean number of nodes checked in the optimal tree differs from the natural lower bound, the binary logarithm of the number of words, by no more than 1.04.

In this paper we study random variables which characterise collections of segments in an equiprobable polynomial scheme related by the H-equivalence. We give an upper bound for the variation distance between the distribution of the random variable ξ k (H) equal to the number of collections of H -equivalent segments and the Poisson distribution. We present sufficient conditions for the convergence of the distribution functions of the number of H -equivalent segments in the triangular array scheme of equiprobable polynomial trials to the normal law, the Poisson law, and the compound Poisson law.

The automorphism groups, the anti-isomorphism classes and the generators of the groups of inner permutations for loops of order six are found.