Vol 39, No 1 (2018)

In this paper it is considered the implementation of ternary logic functions by the circuits of unreliable functional elements in an arbitrary complete finite basis. It is assumed that all the circuit elements pass to fault states independently of each other, and the faults can be arbitrary (for example, inverse or constant). Previously known class of ternary logic functions is extended, the circuits of these functions can be used to raise the reliability of the original circuits. With inverse faults at the outputs of basis elements it is constructively proved using functions of this class (we denote by G it) that a function which differs from any one of the variables can be implemented by a reliable circuit, and the probability of the inverse fault is bounded above by a constant. In particular if the basis under consideration contains at least one of the class G functions then for any function which differs from any one of the variables the constructed circuit is not only reliable, but this one is asymptotically optimal by reliability (we remind that function which is equal to one of the variables can be implemented absolutely reliably, not using functional element).

In this paper we study the differential and integral invariants for the action of the projective group PGL(n + 1) on the group of diffeomorphisms Diff(ℝPⁿ) by conjugations. Cases n = 1 and n = 2 are considered. For n = 1 the algebra of differential invariants is found and the criterion of the local equivalence of two diffeomorphisms is obtained. Also several integral invariants for n = 1 and n = 2 are calculated, the analogy with Calaby integral invariant for the symplectic groups is established.

This paper is aimed at presenting a systematic exposition of the existing now different formulations for the problem of projection of the origin of the Euclidean space onto the convex polyhedron (PPOCP). We have concentrated on the convex polyhedron given as a convex hull of finitely many vectors of the space. We investigated the reduction of the projection program to the problems of quadratic programming, maximin, linear complementarity, and nonnegative least squares. Such reduction justifies the opportunity of utilizing a much more broad spectrum of powerful tools of mathematical programming for solving the PPOCP. The paper’s goal is to draw the attention of a wide range of research at the different formulations of the projection problem.

The article gives an overview of algorithmic properties of Coxeter groups with tree structure. The paper considers free product of two 2-generated of Coxeter groups amalgamated by cyclic subgroup which refers to Coxeter groups with tree structure. The authors proves decidability of conjugacy problem for finite sets of subgroups in this class of groups.

We introduce a spin polarization-scaling map for spin-j particles, whose physical meaning is the decrease of spin polarization along three mutually orthogonal axes. We find conditions on three scaling parameters under which the map is positive, completely positive, entanglement breaking, 2-tensor-stable positive, and 2-locally entanglement annihilating. The results are specified for maps on spin-1 particles. The difference from the case of spin-1/2 particles is emphasized.

In this paper for μ > 0 we study an asymptotic behavior of the sequence defined as T_n(μ) = (τ(n))⁻¹\({\max _{1 \leqslant t \leqslant \left[ {{n^{1/\mu }}} \right]}}\) {τ(n + t)}, where τ(n) denotes the number of natural divisors of given positive integer n. The motivation of this observation is to explore whether τ-function oscillates rapidly.

In this paper it is proved that subantivarieties of an antivariety K form a semigroup with respect to Mal’cev’s multiplication whenever K is an antivariety of algebraic systems whose signature Ω contains only finite number of function symbols.We show that the condition of finiteness of the set of function symbols from Ω is significant. Semigroups of subantivarieties of antivarieties of algebras are characterized.

The paper concerns the development of a constructive method of solving one of the main boundary value problems of Riemann problem type in the class of metaanalytic functions of the second type in the case when a unit circle serves as a boundary

A unit vector field τ in the Euclidean space E³ is considered. Let P be the vector field from the first Aminov’s divergent representation K = div{(R · τ)P} for the total curvature of the second kind K of the field τ. For the field P, an invariant representation of the form P = −rotR* is obtained, where the field R* is expressed in terms of the Frenet basis (τ, ν, β) and the first curvature k and the second curvature κ of the streamlines L_τ of the field τ. Formulas relating the quantities K (or P), κ, τ, ν, and β are derived. Three-dimensional analogs of the conservation law div S_p* = 0 (which is valid for a family of plane curves L_τ) are obtained, where S_p* is the sum of the curvature vectors of the plane curves L_τ and their orthogonal curves L_ν. It is shown that if the field τ is holonomic: 1) the vector field S(τ) from the second Aminov’s divergent representation K = −1/2 div S(τ) can be interpreted as the sum of three curvature vectors of three curves related to surfaces S_τ with the normal τ; 2) the non-holonomicity values of the fields of the principal directions l₁ and l₂ are equal.

An efficient method for factorization of square triangular matrix-functions of arbitrary order is proposed. It generalizes the method by G. N. Chebotarev.

In this article the uniqueness of the solution of Dirichlet boundary value problem for quasiharmonic functions in arbitrary disk T_r⁺ = {z: |z| < r}, where r ≠ 1, is established. Also the non-uniqueness of the solution of this problem in a unit disk is proven.