Band 12, Heft 4

A set A of integers is called sum-free if a + b ∉ A for any a, b ∈ A. For an arbitrary Ɛ > 0, let s Ɛ (n) denote the number of sum-free sets in the segment [(1/4 + Ɛ)n, n]. We prove that for any Ɛ > 0 there exists a constant c = c(Ɛ) such that s Ɛ (n) ≤ c2 n/2 . This research was supported by the Russian Foundation for Basic Research, grant 01-01-00266.

In this paper, we consider the problem of searching for undirected Hamiltonian circuits in the complete graph on n vertices with the use of unconditional edge tests. We prove that the minimal test contains exactly n(n - 3)/2 - [n/3] + 1 edges. We propose an explicit characterisation of all minimal difference sets of edges. This research was supported by the Russian Foundation for Basic Research, grants 02-01-00985 and 00-15-96103, by the Program ‘Universities of Russia,’ and the Federal Program ‘Integration.’

The class of Boolean bijunctive functions is one of the Sheffer classes. The main property which makes investigations of bijunctive functions important is the property that the problem of testing the consistency of a system of equations over a Sheffer class of functions is of a polynomial complexity (see, for example, [1-4]). In this paper, we estimate the number of bijunctive functions containing a given permutation in their inertia groups with respect to the symmetric group. In particular, we describe properties and find the number of bijunctive functions invariant with respect to a unicyclic permutation of the variables.

We give the description of word hyperbolic groups of the form (a, b, t; t -1 at = b, t -1 bt = w(a, b)). The research was supported by the Russian Foundation for Basic Research, grant 00-01-00767.

Let I and O be finite alphabets. For a finite alphabet Ω, let Ω* denote the set of all words of finite lengths over the alphabet Ω. In this paper we give a complete description of all automaton mappings of the set I* into O* which multiply symbol replacement errors in words by a factor not exceeding K. We give a complete description of injective automaton mappings of the set 1* into O* which multiply symbol skip errors by a factor no greater than K. A similar result is obtained for the deletion and insertion metric.

The main result of this paper is a theorem about convergence of the distribution of the number of pairs of H-connected s-tuples in two independent sequences of independent identically distributed variables. The concept of H-connection is a generalisation of the concept of H-equivalence of tuples. We give sufficient conditions for convergence and an explicit estimate of the rate of convergence. We use the local variant of the Chen-Stein method for estimating the accuracy of Poisson approximation for distribution of the set of dependent random indicators. The main results of this paper were announced in [7]. The research was supported by the Russian Foundation for Basic Research, grants 02-01-00266 and 00-15-96136.

We consider a sequence of independent trials, which, under the hypothesis H 0 , is a realisation of some polynomial scheme. We define a family of multivariate chi-square statistics which includes the sequential chi-square test statistics. We formulate conditions which guarantee that the distributions of the statistics under consideration converge to limit distributions under the hypothesis H 0 and the alternatives contigual to H 0 . We find the form of the Laplace transforms of the limit distributions of the statistics of the family. This research was supported by the Russian Foundation for Basic Research, grant 00-15-96136.

We consider maximal branching processes defined by the recurrence relation where ⋁ stands for the operation of taking maximum, ξ m,n , m ≥ 1, n ≥ 0, are independent with distribution function F on ℤ + . We prove limit theorems for stationary distributions of the processes{[Z n (ℕ) } with the distribution functions F (ℕ) (x) = F (ℕ) (x) as ℕ →∞ in the case where F belongs to the domain of attraction of the double exponential law. This research was supported by the Russian Foundation for Basic Research, grant 00-01-00131.

We consider three combinatorial problems covered by generalised allocation scheme: sampling of balls from an urn that contains balls of various colours; random forests; and random permutations with given number of cycles. For these problems, we present limit distributions of the number of pairs of balls of the same colour; of the number of paths in a forest; and of the number of simple (non-closed) paths in the permutation graph respectively. This research was supported by the Russian Foundation for Basic Research, grant 00-01-00233.