Volume 18, Issue 2

We consider approximations of functions of k -valued logic by polynomials with various accuracies and find estimates of the ranks and the lengths of the approximating polynomials depending of the accuracy of approximation.

We consider the class of almost symmetric Boolean functions. For any function of this class, the values on all tiers except the second one coincide with the values of a monotone symmetric function with threshold 3. The values on the second tier are arbitrary. We study realisation of functions of this class by & ∨ &- formulas over the basis {&, ∨}. We obtain a sharp bound for the minimum complexity of the functions of this class (the function of minimum complexity is explicitly written out) and an asymptotic estimate of complexity of a monotone symmetric function with threshold 3 which is maximal in order of complexity in the class under consideration.

In this study we suggest methods for obtaining lower bounds for complexity of linear Boolean operators (and the related matrices) in two models of circuits of depth 2. In the first model, only linear elements with arbitrary number of inputs are allowed, whereas in the second one, any Boolean elements can be present. These methods are applicable to matrices with sufficiently large Hamming distance between the rows, for example, to the Hadamard matrices.

We consider the known problem of decoding functions of Boolean algebra, that is, we need to completely restore the table of values of a function defined on an n -dimensional Boolean cube B n using its values on some subset of B n . If the function belongs to some narrower class than the class of all functions of n variables (for example, to the set of monotone or threshold functions), then only vectors of a subset of n -dimensional Boolean cube can be required to completely determine the function. In the paper, we consider the class of functions which split the n -dimensional Boolean cube into cube faces. An asymptotic estimate for complexity of decoding functions that belong to any subclass of this class with fixed structure is obtained.

For some χ 2 -type statistics used to find dependencies between random variables in a sequence under study, we obtain limit theorems in the case where the sample size tends to infinity.

We estimate the probability P k,m that, as k vertices of the unit cube I m = {0, 1} m are randomly chosen, their convex hull is a polyhedron whose graph is complete. In particular, we establish that, as n → ∞, the probability P k(m),m tends to one if k ( m ) = O (2 (m/6) ) and P k(m),m tends to zero if k ( m ) ≥ (3/2) m . The results given in this paper, first, to a great extent explain why the intractable discrete problems are so widely spread, and second, support the well-known Gale's hypothesis published in 1956.

Latin squares C , D of order n are called pseudo-orthogonal if any two rows of the matrices C and D have exactly one common element. We give conditions for existence of families consisting of t pseudo-orthogonal Latin squares of order n . It is proved that the intersection number of a complete r -partite graph equals n 2 if and only if there exists a family consisting of r – 2 pairwise pseudo-orthogonal Latin squares of order n . It is proved that if , where prols( n ) is the maximum t such that there exists a set of t pseudo-orthogonal Latin squares of order n , then the intersection number of the graph is equal to n 2 . Applications of the obtained results to calculating the intersection number of some graphs are given.

A system of generators of a group consisting of three involutions, two of which commute, is called a Mazurov triple. We describe algorithms for finding in an explicit form the Mazurov triples of one of the sporadic Monsters, the finite simple group B , and for constructing a Hamiltonian cycle in the Cayley graph of the finite group with Mazurov triple. We give examples of Hamiltonian cycles in the Cayley graphs of some groups.

In the paper, Malcev rings are defined. The class of Malcev rings strictly contains rings of Malcev–Neumann series, formal skew Laurent series rings, and formal pseudo-differential operator rings. In the paper, ring-theoretic properties of Malcev rings are studied. It turns out that rings of Malcev–Neumann series, formal skew Laurent series rings, and formal pseudo-differential operator rings have similar ring-theoretic properties related to the existence of the filtration with respect to the lowest degree of the series.