Том 101, № 1-2 (2017)

The problem of the completeness of the system of analytic functions of the form ∪_k=0²{[W(zδ^k)]³ⁿ}_n=0^∞, where n = 0, 1,..., k = 0, 1, 2, and δ = exp(2πi/3), in A(D) is solved.

In the paper, it is proved that the distribution of a measurable polynomial on an infinite-dimensional space with log-concave measure is absolutely continuous if the polynomial is not equal to a constant almost everywhere. A similar assertion is proved for analytic functions and for some other classes of functions. Properties of distributions of norms of polynomial mappings are also studied. For the space of measurable polynomial mappings of a chosen degree, it is proved that the L¹-norm with respect to a log-concave measure is equivalent to the L¹-norm with respect to the restriction of the measure to an arbitrarily chosen set of positive measure.

Necessary and sufficient conditions under which a matrix-valued function of a complex argument is inverse to a weakly hyperbolic or a hyperbolic pencil are established. For hyperbolic pencils, a constructive description of the inverse functions in terms of their partial fraction expansion with matrix coefficients is presented.

The three-dimensional nonlinear system of ordinary differential equations modeling the functioning of the simplest oscillatory genetic network, the so-called repressilator, is considered. The existence, asymptotics, and stability of the relaxation periodicmotion in this system are studied.

We prove that the Cartesian product of octahedra B_1,∞^n,m = B₁ⁿ ×···× B₁ⁿ (m factors) is poorly approximated by spaces of half dimension in the mixed norm: d_N/2(B_1,∞^n,m, ℓ_2,1^n,m) ≥ cm, N = mn. As a corollary, we find the order of linear widths of the Hölder–Nikol’skii classes H_p^r(T^d) in the metric of L_q in certain domains of variation of the parameters (p, q).

A one-dimensional generalization of the Riemann–Hilbert problem from the Riemann sphere to an elliptic curve is considered. A criterion for its positive solvability is obtained and the explicit form of all possible solutions is found. As in the spherical case, the solutions turn out to be isomonodromic.

The paper presents some coercive a priori estimates of the solution of the Dirichlet problem for the linear Stokes equation relating vorticity and the stream function of an axially symmetric flow of an incompressible fluid. This equation degenerates on the axis of symmetry. The method used to obtain the estimates is based on a differential substitution transforming the Stokes equation into the Laplace equation and on the subsequent transition from cylindrical to Cartesian coordinates in three-dimensional space.

The effect of small constantly acting random perturbations of white noise type on a dynamical system with locally stable fixed point is studied. The perturbed system is considered in the form of Itô stochastic differential equations, and it is assumed that the perturbation does not vanish at a fixed point. In this case, the trajectories of the stochastic system issuing from points near the stable fixed point exit from the neighborhood of equilibrium with probability 1. Classes of perturbations such that the equilibrium of a deterministic system is stable in probability on an asymptotically large time interval are described.

The rings over which every square matrix is representable as a sum of a nilpotent matrix and a q-potent matrix, where q is a positive integer power of a prime, are studied. As consequences, matrix analogs of the little Fermat theorem are obtained.

For an algebra A, denote by V_A(n) the dimension of the vector space spanned by the monomials whose length does not exceed n. Let T_A(n) = V_A(n) − V_A(n − 1). An algebra is said to be boundary if T_A(n) − n < const. In the paper, the normal bases are described for algebras of slow growth or for boundary algebras. Let L be a factor language over a finite alphabet A. The growth function T_L(n) is the number of subwords of length n in L. We also describe the factor languages such that T_L(n) ≤ n + const.

The paper presents the proof of the uniqueness theorem formultiple series in the Franklin systemthat converge inmeasure and whosemajorant of cubic partial sums with numbers 2^ν satisfies a certain necessary condition. This result is new in the one-dimensional case as well.

Recent results of S. Dartyge, C. Mauduit, and A. Sárközy concerning the problem of the number of squares among the elements of a finite field with constraints on the coefficients of its basis expansion are strengthened.

An algorithm for the spectral analysis of nonautonomous systems of differential equations on the semiaxis whose matrix can be presented as the sum of exponential-type matrices is developed. This method, which is based on a version of the splitting method, allows us to prove a theorem stating that the initial system is almost reducible to a simpler equivalent system and to formulate a sufficient condition for the asymptotic stability and the stability of its trivial solution.

In the paper, the semigroup of weak limits of the powers of an infinite transformation of rank one of Chacon type is completely described.

An asymptotic theorem important for the study of many power series with multiplicative coefficients is proved. Examples of concrete series to which the theorem can be applied are given. It is shown that power series of many classical arithmetic sequences can be expanded in asymptotic series as the variable tends to the roots of unity along the radii of the unit circle.

For arbitrary Riemann integrable functions f and irrational numbers θ ∈ (0, 1), we obtain estimates of the error R_n(f, θ) of the quadrature formula

\(\int_0^1 {f\left( x \right)dx = \frac{1}{n}\sum\limits_{k = 1}^n {f\left( {\left\{ {k\theta } \right\}} \right) - {R_n}\left( {f,\theta } \right)} } \)

We study sets with at most two-valued metric projection in Banach spaces. We show that a two-dimensional Banach space is smooth if and only if every point of the convex hull of an arbitrary closed set with at most two-valued metric projection lies on a segment with endpoints in that set.