Ukrainian Mathematical Journal最新文献

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.

With the help of Clebsch’s potentials, we propose a Bateman–Luke-type variational principle for a boundary- value problem with a free (unknown) interface between two ideal compressible barotropic fluids (liquid and gas) admitting rotational flows.

We introduce a new Diophantine inequality with prime numbers. Let (1<c<frac{10}{9}.) We show that, for any fixed θ > 1, every sufficiently large positive number N, and a small constant ε > 0, the tangent inequality

has a solution in prime numbers p₁, p₂, and p₃.

Investigation of the theory of complex functions is one of the most fascinating aspects of the theory of complex analytic functions of one variable. It has a huge impact on all areas of mathematics. Numerous mathematical concepts are explained when viewed through the theory of complex functions. Let (fleft(zright)in A, fleft(zright)=z+{sum }_{nge 2}^{infty }{a}_{n}{z}^{n},) be an analytic function in an open unit disc U = {z : |z| < 1, z ∈ ℂ} normalized by f(0) = 0 and f′(0) = 1. For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where r is a positive integer of order ({2}^{-r}left(0<{2}^{-r}le frac{1}{2}right).) By using subordination, we propose a criterion for f(z) ∈ S^*[a^r, b^r]. The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study their regional transformations. In addition, the properties of convexity are examined for f(z) ∈ A.

We give a short proof of a weighted version of the discrete Hardy inequality. This includes the known case of classical monomial weights with optimal constant. The proof is based on the ideas of the short direct proof given recently in [P. Lefèvre, Arch. Math. (Basel), 114, No. 2, 195–198 (2020)].

By using the q-Jackson integral and some elements of the q-harmonic analysis associated with the q-Hankel transform, we introduce and study a q-analog of the Hankel–Stockwell transform. We present some properties from harmonic analysis (Plancherel formula, inversion formula, reproducing kernel, etc.). Furthermore, we establish a version of Heisenberg’s uncertainty principles. Finally, we study the q-Hankel–Stockwell transform on a subset of finite measure.