Volume 16, Issue 3

A set B of integers is called sum-free if for any a, b ∈ B the number a + b does not belong to the set B . Let s ( n ) be the number of sum-free sets in the interval of natural numbers [1, n ]. As shown by Cameron, Erdős, and Sapozhenko, there exist constants c 0 and c 1 such that s ( n ) ~ ( c 0 + 1)2 ⌈ n /2⌉ for even n and s ( n ) ~ ( c 1 + 1)2 ⌈ n /2⌉ for odd n tending to infinity. The constants c 0 and c 1 are usually referred to as the Cameron–Erdős constants. In this paper, we obtain upper and lower bounds for the Cameron–Erdős constants which give the two first decimal places of their exact values.

Let (S, º) be a finite simple group, s i , i = 1, ... , n , be fixed (not necessarily distinct) elements of S , and let E α 1 , E α 2 , ... , E α k +1 be a random realisation of a chain of states of a simple homogeneous irreducible Markov chain with the set of states { E 1 , E 2 , ... , E n }. We study convergence conditions and limit distributions for the sequences of random products of the form η ( k ) = s α 1 º s α 2 º ... º s α k +1 . The convergence conditions are formulated in terms of some homomorphism from the fundamental group of the transition graph of the Markov chain to the structural group of the semigroup S .

We introduce the concept of a determinising automaton which, for every superword taken from a given set fed into its input, beginning with some step, at any time t yields the value of the input word at time t + 1, that is, predicts the input superword. We find a criterion whether a given set of superwords is determinisable, that is, whether for the set there exists a determinising automaton. We give the best (in some sense) method to design a determinising automaton for an arbitrary determinisable set of superwords. For some determinisable sets we present optimal and asymptotically optimal determinising automata.

Protocols to retrieve information which hide the query allows a user to get the desired information bit from a database replicated on several noncommunicating servers in such a way that the administrator of the database knows nothing about the index of the bit the user queries. A protocol is said to be degenerate if the user, as the result of the query, gets the whole database. We find bounds for protocol parameters where the degeneracy can be obviated.

For the problem of interval search on the Boolean cube, we investigate the behaviour of the Shannon function in the class of tree-like circuits called information trees. It is shown that for databases of cardinality coinciding in order with cardinality of the Boolean cube the Shannon function is equal in order to the optimal complexity in the class of balanced tree-like circuits. For databases of cardinality less in order than the cardinality of the Boolean cube we give the asymptotics of the logarithm of the Shannon function.

This research is devoted to estimating the number of Boolean Pascal's triangles of large enough size s containing a given number of ones ξ ≤ ks, k > 0. We demonstrate that any such Pascal's triangle contains a zero triangle whose size differs from s by at most constant depending only on k . We prove that there is a monotone unbounded sequence of rational numbers 0 = k 0 < k 1 < ... such that the distribution of the number of triangles is concentrated in some neighbourhoods of the points k i s . The form of the distribution in each neighbourhood depends not on s but on the residue of s some modulo depending on i ≥ 0.

In this research, we construct combinatorially continuous (neighbour-to-neighbour) mappings of a 64-pixel triangle to a 64-pixel cube and square. The pixels constituting the triangle, cube, and square are, respectively, triangles, cubes, and squares themselves, which form a partition of the initial object. On these objects, various neighbouring relations are considered. With the use of a computer, we construct a mapping of a triangle onto a cube such that any triangular pixels with common side are mapped to overlapping cubical ones. Also with the use of a computer, we establish nonexistence of a mapping of a triangle onto a cube such that any triangular pixels with common side are mapped to cubical ones with common side. Without help of a computer, we construct a mapping of a triangle to a square which maps overlapping triangular pixels to overlapping square ones.

Let be a class of finite groups. An Ω-foliated formation of finite groups with direction φ is called a minimal Ω-foliated non--formation φ, or a Ωφ -critical formation if ⊈ , but all proper Ω-foliated subformations with direction φ in are contained in the class . In this paper we give a complete description of the structure of minimal Ω-foliated non--formations with br -direction φ satisfying the condition φ ≤ φ 3 .

In this paper, we investigate the lattice of all Fitting classes, the lattice of all n -multiply Ω-foliated Fitting classes with direction φ, Ψ 0 ≤ φ, and the lattice of all totally canonical Fitting classes. It is shown that these lattices are algebraic with 1-generated compact elements.

We study the lattices Ω K n and K n of all n -multiply Ω-canonical and n -multiply canonical formations respectively. The main results of the paper are the proofs of -separability of the lattice Ω K n and coincidence of the systems of identities of the lattices K n and K m for different positive integers n and m .