Ukrainian Mathematical Journal最新文献

We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into an Urysohn countably compact topological space.

Let 𝕂 be an algebraically closed field of characteristic zero, let 𝕂[x₁,…,x_n] be the polynomial algebra, and let W_n(𝕂) be the Lie algebra of all 𝕂-derivations on 𝕂[x₁,…,x_n]. For any derivation D with linear components, we describe the centralizer of D in W_n(𝕂) and propose an algorithm for finding the generators of this centralizer regarded as a module over the ring of constants of the derivation D in the case where D is a basic Weitzenböck derivation. In a more general case where a finitely generated integral domain A over the field 𝕂 is considered instead of the polynomial algebra 𝕂[x₁,…,x_n] and D is a locally nilpotent derivation on A, we prove that the centralizer C_DerA(D) of D in the Lie algebra DerA of all 𝕂-derivations on A is a “large” subalgebra of Der A. Specifically, the rank of C_DerA(D) over A is equal to the transcendence degree of the field of fractions Frac(A) over the field 𝕂.

Based on the Steklov operator, we consider a modulus of smoothness for functions in some Banach function spaces, which can be not translation invariant, and establish its main properties. A constructive characterization of the Lipschitz class is obtained with the help of the Jackson-type direct theorem and the inverse theorem on trigonometric approximation. As an application, we present several examples of related (weighted) function spaces.

For a nonlinear autonomous boundary-value problem posed for an ordinary differential equation in the critical case, we establish constructive conditions for its solvability and propose a scheme for the construction of solutions based on the use of Adomian’s decomposition method.

We extend the concept of generalized weakly demicompact and relatively weakly demicompact operators on linear relations and present some outstanding results. Moreover, we address the theory of Fredholm and upper semi-Fredholm relations and make an attempt to establish connections with these operators.

We consider linear differential equations of the first order with delayed arguments in a Banach space. We establish conditions for the operator coefficients necessary for the existence of exponential dichotomy on the real axis. It is proved that the analyzed differential equation is equivalent to a difference equation in a certain space. It is shown that, under the conditions of existence and uniqueness of a solution bounded on the entire real axis, the condition of exponential dichotomy is also satisfied for any known bounded function. We also deduce the explicit formula for projectors, which form this dichotomy in the case of a single delay.

Based on the topological dual space ({mathcal{F}}_{theta }^{*}left({mathcal{S}{prime}}_{mathbb{C}}right)) of the space of entire functions with θ-exponential growth of finite type, we introduce a generalized stochastic Bernoulli–Wick differential equation (or a stochastic Bernoulli equation on the algebra of generalized functions) by using the Wick product of elements in ({mathcal{F}}_{theta }^{*}left({mathcal{S}{prime}}_{mathbb{C}}right)). This equation is an infinite-dimensional analog of the classical Bernoulli differential equation for stochastic distributions. This stochastic differential equation is solved and exemplified by several examples.

A relationship between the S-colocalization of an object and the Adams cocompletion of the same object in a complete small 𝒰 -category (𝒰 is a fixed Grothendieck universe) is established together with a specific set of morphisms S.

The paper is essentially motivated by the demonstrated potential for applications of the presented results in numerous widespread research areas, such as the mathematical, physical, engineering, and statistical sciences. The main object is to introduce and investigate a class of fractional integral operators involving a certain general family of multiindex Mittag-Leffler functions in their kernel. Among other results obtained in the paper, we establish several interesting expressions for the composition of well-known fractional integral and fractional derivative operators, such as (e.g.) the Riemann–Liouville fractional integral and fractional derivative operators, the Hilfer fractional derivative operator, and the above-mentioned fractional integral operator involving the general family of multiindex Mittag-Leffler functions in its kernel. Our main result is a generalization of the results obtained in earlier investigations in this field. We also present some potentially useful integral representations for the product of two members of the general family of multiindex Mittag-Leffler functions in terms of the well-known Fox–Wright hypergeometric function _pΨ _q with p numerator and q denominator parameters.