Том 105, № 1-2 (2019)

An asymptotic method for reducing nonautonomous systems of differential equations with matrices represented as sums of matrix functions of different periods to systems with almost-constant matrices is proposed. On the basis of the proposed method, sufficient conditions for the asymptotic stability of solutions of systems of differential equations of this type are derived.

We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where the only nonmonotone element of the basis is negation was completely described by A. A. Markov: the minimum number of negations sufficient for the realization of an arbitrary Boolean function f (the inversion complexity of the function f) is equal to ⌈log₂(d(f) + 1)⌉, where d(f) is the maximum (over all increasing chains of sets of values of the variables) number of changes of the function value from 1 to 0. In the present paper, this result is generalized to the case of the computation of Boolean functions over an arbitrary basis B of prescribed form. It is shown that the minimum number of nonmonotone functions sufficient for computing an arbitrary Boolean function f is equal to ⌈log₂(d(f)/D(B) +1)⌉, where D(B) = max d(ω); the maximum is taken over all nonmonotone functions ω of the basis B.

It is known that there exists a finitely generated residually finite group (for short, a residually F-group) the extension by which of some finite group is not a residually F-group. In the paper, it is shown that, nevertheless, every extension of a finite group by a finitely generated residually F-group is a Hopf group, and every extension of a center-free finite group by a finitely generated residually F-group is a residually F-group. If a finitely generated residually F-group G is such that every extension of an arbitrary finite group by G is a residually F-group, then a descending HNN-extension of the group G also has the same property, provided that it is a residually F-group.

Tangent-valued forms, tangent and cotangent vectors of the first and the second order are considered. For an affine connection, second-order tangent-valued (vertical and horizontal) forms determining linear operators in the second-order tangent and cotangent spaces are constructed.

Estimates of the norms of spaces associated to weighted first-order Sobolev spaces with various weight functions and summation parameters are established. As the main technical tool, boundedness criteria for the Hardy–Steklov integral operator with variable limits of integration in Lebesgue spaces on the real axis are used.

The aim of this paper is to classify unknotted ribbons in the plane and on the sphere up to regular isotopy.

The present paper deals with the classification of multivariate holomorphic function germs that are equivariant simple under representations of cyclic groups. We obtain a complete classification of such function germs of two and three variables for all possible nontrivial ℤ₃-actions. Our main classification methods generalize those used for the classification of simple germs in the nonequivariant case.

A particular class of estimates related to the Nelson–Erdős–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric ℓ₁, certain series of distance graphs of small dimensions are considered. Independence numbers of such graphs are estimated by using the linear-algebraic method and combinatorial observations. This makes it possible to obtain certain lower bounds for the chromatic numbers of the spaces mentioned above and, for each case, specify a series of graphs leading to the strongest results.

Differential inequalities for polynomials generalizing the well-known Smirnov, Rahman, Schmeisser, and Bernstein inequalities are obtained.

Nonsingular endomorphisms of the m-torus \(\mathbb{T}\)^m, m ≥ 2, which are C¹ perturbations of linear hyperbolic endomorphisms are considered. Sufficient conditions for such maps to be hyperbolic (i.e., belong to the class of Anosov endomorphisms) are found.

In this paper, we introduce uniform Lotka–Volterra operators and construct Lyapunov functions for them. We establish that the ergodic averages associated with operators of such kind diverge.

A new correspondence between the solutions of theminimal surface equation in a certain 3-dimensional Riemannian warped product and the solutions of the maximal surface equation in a 3-dimensional standard static space-time is given. This widely extends the classical duality between minimal graphs in 3-dimensional Euclidean space and maximal graphs in 3-dimensional Lorentz–Minkowski space-time. We highlight the fact that this correspondence can be restricted to the respective classes of entire solutions. As an application, a Calabi–Bernstein-type result for certain static standard space-times is proved.

Generalized Chebyshev polynomials are introduced and studied in this paper. They are applied to obtain a lower bound for the sup-norm on the closed interval for nonzero polynomials with integer coefficients of arbitrary degree.

It is well known that the classical and Sobolev wave fronts were extended to nonequivalent global versions by the use of the short-time Fourier transform. In this very short paper, we give complete characterizations of the former wave front sets in terms of the short-time Fourier transform.