Volume 29, Issue 5

Let A be a precomplete class (a maximal clone) in k -valued logic and T ( A ) be the family of all closed classes (under superposition) in partial k -valued logic that contain A . A simple test is put forward capable of finding out from a partial order defining the precomplete class A of monotone functions whether the family T ( A ) is finite or infinite. This completes the solution of the problem of finiteness of T ( A ) for all precomplete classes of k -valued logic. The proof depends on new families of closed classes founded by the author of the present paper.

We find a relation between the W -intersection matrix (which characterizes the degree of “nonhomomorphy”) of a transformation and the difference distribution table and the correlation matrix (which characterize the degree nonlinearity of a transformation). An upper estimate for the dimension of a subspace invariant under almost bent functions is put forward. A formula for evaluation of the W -intersection matrix of a composition of two transformations is obtained.

The paper contains an extension of Malyshev theorem for n -ary quasigroups with a right or left weak invertibility property to the case of strongly dependent n -ary operations. As a corollary a new proof of Gluskin–Hosszú theorem for strongly dependent n -ary semigroups is obtained.

The paper is concerned with systems of generators of permutation groups on Cartesian products of residue rings. Each separate permutation from the system of generators is constructed on the basis of additions, is characterized by the local action, and leaves fixed the major parts of the components of the element being transformed. A criterion of 2-transitivity of the generated permutation group is given in the form of the strong connectedness of the digraph which corresponds to the system of generators and which is defined on the set of numbers of residue rings in the Cartesian product. Necessary and sufficient conditions under which this group contains an alternating group are formulated.

We consider a special function ρ ( H ) of the subset H of n -dimensional vector linear space over the field K . This function is used in the estimates of accuracy of the Poisson approximation for the distribution of the number of solutions of systems of random equations and random inclusions over K . For the case when K = GF (2) and H is a sphere or ball (in the Hamming metric) in {0, 1} n we obtain explicit and approximate formulas for ρ ( H ) for sufficiently large values of n .

The following results are proved: any nonconstant Boolean function may be implemented by an irredundant circuit of gates in the basis { x & y , x , x ⊕ y ⊕ z } admitting a single fault detection test of length at most 2 with respect to arbitrary stuck-at faults at outputs of gates, there exists a six-place Boolean function ψ such that any nonconstant Boolean function may be implemented by an irredundant circuit of gates in the basis { ψ } admitting a single diagnostic test of length at most 3 with respect to arbitrary stuck-at faults at outputs of gates.

We study the frequencies of tuples in linear recurring sequences (LRS) of vectors over Galois rings. By means of an estimate of an exponential sum some nontrivial bounds on the frequencies of elements in LRS are derived. It is shown that these bounds are in some cases sharper than known results.

Given a numerical semigroup S , a nonnegative integer a and m ∈ S ∖ {0}, we introduce the set C ( S , a , m ) = { s + aw ( s mod m ) | s ∈ S }, where { w (0), w (1), ⋯, w ( m – 1)} is the Apéry set of m in S . In this paper we characterize the pairs ( a , m ) such that C ( S , a , m ) is a numerical semigroup. We study the principal invariants of C ( S , a , m ) which are given explicitly in terms of invariants of S . We also characterize the compositions C ( S , a , m ) that are symmetric, pseudo-symmetric and almost symmetric. Finally, a result about compliance to Wilf’s conjecture of C ( S , a , m ) is given.