Том 106, № 3-4 (2019)

We consider a parametric family of nonautonomous dynamical systems continuously depending on a parameter from some metric space. For any such family, the topological entropy of its dynamical systems is studied as a function of the parameter from the point of view of the Baire classification of functions.

Let f(x) be a complex polynomial of degree n. We associate f with a ℂ-vector space W(f) that consists of complex polynomials p(x) of degree at most n — 2 such that f(x) divides f”(x)p(x) — f’(x)p’(x). The space W(f) first appeared in Yu. G. Zarhin’s work, where a problem concerning dynamics in one complex variable posed by Yu. S. Ilyashenko was solved. In this paper, we show that W(f) is nonvanishing if and only if q(x)² divides f(x) for some quadratic polynomial q(x). In that case, W(f) has dimension (n — 1) — (n₁ + n₂ + 2N₃) under certain conditions, where n_i is the number of distinct roots of f with multiplicity i and N₃ is the number of distinct roots of f with multiplicity at least 3.

Lower and upper bounds are obtained for the size ζ(n, r, s, k) of a minimum system of common representatives for a system of families of k-element sets. By ζ(n, r, s, k) wemean themaximum (over all systems Σ = {M₁, …, M_r} of sets M_i consisting of at least s subsets of {1, …, n} of cardinality not exceeding k) of the minimum size of a system of common representatives of Σ. The obtained results generalize previous estimates of ζ(n, r, s, 1).

In this paper, by introducing the notion of γ-convex set, we distinguish a wider class of discrete control systems in which the global maximum principle holds. A new type of variation of control for such classes of discrete control systems is proposed and stronger global maximum principle and second-order optimality condition expressed in terms of a singular control of new type are obtained. Generalizing the notion of the relative interior of sets, we obtain an optimality condition for discrete systems in the form of an equality, which we call Pontryagin’s equation.

Norms of images of operators of multiplier type with an arbitrary generator are estimated by using best approximations of periodic functions of one variable by trigonometric polynomials in the scale of the spaces L_p, 1 < p < +∞. A Bernstein-type inequality for the generalized derivative of the trigonometric polynomial generated by an arbitrary generator ψ, sufficient constructive ψ-smoothness conditions, estimates of best approximations of ψ-derivatives, estimates of best approximations of ψ-smooth functions, and an inverse theorem of approximation theory for the generalized modulus of smoothness generated by an arbitrary periodic generator are obtained as corollaries.

The purpose of this paper is to establish a local uncertainty inequality for arbitrary connected, simply connected nilpotent Lie groups. This allows us to prove a couple of global uncertainty inequalities. In the nilpotent case, this type of result is only obtained for the Heisenberg group.

Suppose that N = 2ⁿ and N1 = 2^n-1, where n is a natural number. Denote by ℂ_N the space of complex N-periodic sequences with standard inner product. For any N-dimensional complex nonzero vector (b₀, b₁,..., b_N-1) satisfying the condition

\({\left| {{b_l}} \right|^2} + {\left| {{b_{l + {N_1}}}} \right|^2} \leq \frac{2}{{{N^2}}},\;\;\;l = 0,1,...,{N_1} - 1,\)

The main result of the paper is that the Lifshits-Krein trace formula cannot be generalized to the case of functions of noncommuting self-adjoint operators. To prove this, we show that, for pairs (A₁, B₁) and (A₂, B₂) of bounded self-adjoint operators with trace class differences A₂-A₁ and B₂-B₁, it is impossible to estimate the modulus of the trace of the difference f (A₂, B₂) - f (A₁, B₁) in terms of the norm of f in the Lipschitz class.

We establish an embedding theorem for spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space in spaces of the same type and obtain some closely related results.

The paper is devoted to the study of the approximation properties of Fourier sums in terms of the modified Meixner polynomials m_n,N^α(x), n = 0,1,..., which generate, for α > -1, an orthonormal system on the grid Ω_δ = {0, δ, 2δ,...} with weight

\({\rho _N}(x) = {e^{ - x}}\frac{{\Gamma (Nx + \alpha + 1)}}{{\Gamma (Nx + 1)}}{(1 - {e^{ - \delta }})^{\alpha + 1}},\;\;\;\;\text{where}\;\;\delta = \frac{1}{N},\;N \geq 1.\)

The main attention is paid to the derivation of a pointwise estimate for the Lebesgue function λ_n,N^α(x) of Fourier sums in terms of the modified Meixner polynomials for x ∈ [θ_n/2, ∞) and θ_n = 4n + 2α + 2.

A subspace I_f(X) of the space of idempotent probability measures on a given compact space X is constructed. It is proved that if the initial compact space X is contractible, then I_f(X) is a contractible compact space as well. It is shown that the shapes of the compact spaces X and I_f(X) are equal. It is also proved that, given a compact space X, the compact space I_f(X) is an absolute neighborhood retract if and only if so is X.

The definition of a B-derivative is based on the notion of generalized Poisson shift; this derivative coincides, up to a constant, with the singular Bessel differential operator. We introduce the fractional powers of a B-derivative by analogy with fractional Marchaud and Weyl derivatives. We prove statements on the coincidence of these derivatives for the classes of even smooth integrable functions. We obtain analogs of Bernstein’s inequality for B-derivatives of integer and fractional order in the space of even Schlömilch j-polynomials with sup-norm and L_p^γ-norm (the Lebesgue norm with power weight x^γ, γ > 0). The resulting estimates are sharp and define the norms of powers of the Bessel operator in the spaces of even Schlömilch j-polynomials.

We study the solvability of nonlocal boundary-value problems for singular parabolic equations of higher order in which the coefficient of the time derivative belongs to a space L_p of spatial variables and possesses a certain smoothness with respect to time. No constraints are imposed on the sign of this coefficient, i.e., the class of equations also contains parabolic equations with varying time direction. We obtain conditions guaranteeing the solvability of boundary-value problems in weighted Sobolev spaces and the uniqueness of the solutions.

Singular blow-up regimes are studied for a wide class of second-order quasilinear parabolic equations. Energy methods are used to obtain exact (in a certain sense) estimates of the final profile of the generalized solution near the blow-up time depending on the rate of increase of the global energy of this solution.