Vol 104, No 1-2 (2018)

Given a function h analytic in the unit disk D, we study the density in the space A(D) of functions analytic inside D of the set S(h,E) of sums of the form Σ_k λ_kh(λ_kz) with parameters λ_k ∈ E, where E is a compact subset of \(D\). It is proved, in particular, that if the compact set E “surrounds” the point 0 and all Taylor coefficients of the function h are nonzero, then S(h,E) is dense in A(D).

The paper studies the additive structure of the algebra F⁽⁷⁾, i.e., a relatively free associative countably generated algebra with the identity [x₁,..., x₇] = 0 over an infinite field of characteristic ≠ 2, 3. First, the space of proper multilinear polynomials in this algebra is investigated. As an application, estimates for the codimensions c_n = dimF_n⁽⁷⁾ are obtained, where F_n⁽⁷⁾ stands for the subspace of multilinear polynomials of degree n in the algebra F⁽⁷⁾.

The Kantorovich optimal transport problemwith a density constraint onmeasures on an infinite-dimensional space is considered. In this setting, the admissible transport plan is nonnegative and majorized by a given constraint function. The existence and the uniqueness of a solution of this problem are proved.

The paper is devoted to the study of conformally flat Lie groups with left-invariant (pseudo) Riemannianmetric of an algebraic Ricci soliton. Previously conformally flat algebraic Ricci solitons on Lie groups have been studied in the case of small dimension and under an additional diagonalizability condition on the Ricci operator. The present paper continues these studies without the additional requirement that the Ricci operator be diagonalizable. It is proved that any nontrivial conformally flat algebraic Ricci soliton on a Lie group must be steady and have Ricci operator of Segrè type {(1... 1 2)} with a unique eigenvalue (equal to 0).

The notion of a distributed-order Hilfer–Prabhakar derivative is introduced, which reduces in special cases to the existing notions of fractional or distributed-order derivatives. The stability of two classes of distributed-order Hilfer–Prabhakar differential equations, which are generalizations of all distributed or fractional differential equations considered previously, is analyzed. Sufficient conditions for the asymptotic stability of these systems are obtained by using properties of generalized Mittag-Leffler functions, the final-value theorem, and the Laplace transform. Stability conditions for such systems are introduced by using a new definition of the inertia of a matrix with respect to the distributed-order Hilfer–Prabhakar derivative.

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.

A method for constructing solutions of the Hom-Yang–Baxter equations is presented. Thus, methods yields a so-called α-involutory solution of the Hom-Yang–Baxter equation for every monoidal Hom-(co)algebra structure on a space. Characterizations for solutions of Hom-Yang–Baxter equations arising from monoidal Hom-(co)algebra structures are given, and a monoidal Hom-(co)algebra structure which produces such a solution is constructed.

In this paper, a Dirichlet-to-Neumann operator related to the Cauchy problem for the gradient operator with data on a part of the boundary is defined. To this end, a nonlinear relaxation of this problem, which is a mixed boundary problem of Zaremba type for the p-Laplace equation, is considered.

A β-matrix model with singular potential is described. A global asymptotic of the density of eigenvalues or the statistical density is obtained by using the equilibrium measure method. The large n-limit density of eigenvalues generalizes Wigner’s semicircle law.

An approach to the construction of a basis of the Lazard ring by using the coefficients of a universal isogeny is described and examples of the application of this approach are given.

Estimates of the Fourier transform of measures concentrated on analytic hypersurfaces containing a damping factor are considered. The solution of the Sogge and Stein problem on the optimal decrease of the Fourier transform of measures with a damping factor for the particular class of analytic surfaces of three-dimensional space is given.

Asymptotic formulas as x→∞ are obtained for a fundamental system of solutions to equations of the form

, where p is a locally integrable function representable as , and q is a distribution such that q = σ^(k) for a fixed integer k, 0 ≤ k ≤ n, and a function σ satisfying the conditions \(\sigma \in {L^1}\left( {1,\infty } \right)ifk < n,\)\(\left| \sigma \right|\left( {1 + \left| r \right|} \right)\left( {1 + \left| \sigma \right|} \right) \in {L^1}\left( {1,\infty } \right)ifk = n\) . Similar results are obtained for functions representable as , for fixed k, 0 ≤ k ≤ n, where the functions r and s satisfy certain integral decay conditions. Theorems on the deficiency index of the minimal symmetric operator generated by the differential expression l(y) (for real functions p and q) and theorems on the spectra of the corresponding self-adjoint extensions are also obtained. Complete proofs are given only for the case n = 1.

A logarithmic asymptotics for the behavior with respect to n of the exact constant in Bernstein’s inequality for the Weyl derivative of positive noninteger order of trigonometric polynomials of order n in the space L₀ is obtained. It turns out that the order in n of the behavior of this constant for positive noninteger orders of the derivatives has exponential growth in contrast to the power growth in the well-studied case of classical derivatives of positive integer order.

It is proved that, for each number p > 1, there exists a function L¹[0, 1] whose Fourier–Walsh series is quasiuniversal with respect to subseries-signs in the class L^p[0, 1] in the sense of L^p-convergence.

We show that Preiss–Thomson’s AS-integral is inconsistent with Burkill’s SCPintegral, James’ P²-integral, and Denjoy’s totalization T_2s over the class of functions with continuous primitives.