Showing papers in "Ukrainian Mathematical Journal in 2012"
TL;DR: Some analogies of Dragomir's generalization of the Ostrowski integral inequality have been established in this article for the composite quadrature rule and some sharp inequalities are proved.
Abstract: Some analogs of Dragomir’s generalization of the Ostrowski integral inequality
$$ \left| {\left( {b - a\left[ {\lambda \frac{{f(a) + f(b)}}{2} + \left( {1 - \lambda } \right)f(x)} \right] - \int\limits_a^b {f(t)dt} } \right)} \right| \leqslant \left[ {\frac{{{{\left( {b - a} \right)}^2}}}{4}\left( {\lambda^2 + {{\left( {1 - \lambda } \right)}^2}} \right) + {{\left( {x - \frac{{a + b}}{2}} \right)}^2}} \right]{\left\| {f'} \right\|_\infty } $$
are established. Some sharp inequalities are proved. An application to the composite quadrature rule is provided.
35 citations
TL;DR: In this article, it was shown that every coarse structure on a set X can be determined by some group G of permutations of X and some group ideal $ \mathcal{I} $ on G.
Abstract: We show that every ballean (equivalently, coarse structure) on a set X can be determined by some group G of permutations of X and some group ideal $ \mathcal{I} $
on G. We refine this characterization for some basic classes of balleans (metrizable, cellular, graph, locally finite, and uniformly locally finite). Then we show that a free ultrafilter $ \mathcal{U} $
on ω is a T -point with respect to the class of all metrizable locally finite balleans on ω if and only if $ \mathcal{U} $
is a Q-point. The paper is concluded with a list of open questions.
33 citations
TL;DR: In this paper, the first boundary value problem for a third-order equation with multiple characteristics was considered, and the unique solvability of the problem was proved by the method of energy integrals.
Abstract: We consider the first boundary-value problem for a third-order equation with multiple characteristics u
xxx
− u
yy = f(x, y) in a domain D = {(x, y):0 < x < p, 0 < y < l}. The unique solvability of the problem is proved by the method of energy integrals and its explicit solution is constructed by the method of Green functions.
22 citations
TL;DR: In this paper, a regularization of the formal differential expression of the minimal operator was proposed. But the conditions of convergence for the generalized resolvents of the analyzed operators in norm were not established.
Abstract: We propose a regularization of the formal differential expression
$$ l(y) = {i^m}{y^{(m)}}(t) + q(t)y(t),\quad t \in \left( {a,\,b} \right), $$
of order m ≥ 3 by quasiderivatives. It is assumed that the distribution coefficient q has the antiderivative $ Q \in L\left( {\left[ {a,\,b} \right];\mathbb{C}} \right) $
. In the symmetric case $ \left( {Q = \bar{Q}} \right) $
we describe self-adjoint and maximal dissipative/accumulative extensions of the minimal operator and its generalized resolvents. In the general (nonself-adjoint) case, we establish the conditions of convergence for the resolvents of the analyzed operators in norm. The case where m = 2 and $ Q \in {L_2}\left( {\left[ {a,\,b} \right];\mathbb{C}} \right) $
was studied earlier.
19 citations
TL;DR: In this article, the authors investigated the problem of optimal control in which the state of the controlled system is described by impulsive differential equations with nonlocal boundary conditions, which is a natural generalization of the Cauchy problem.
Abstract: We investigate the problem of optimal control in which the state of the controlled system is described by impulsive differential equations with nonlocal boundary conditions, which is a natural generalization of the Cauchy problem. Using the principle of contracting mappings, we prove the existence and uniqueness of a solution of a nonlocal boundary-value problem with impulse perturbations and fixed admissible controls. Under certain conditions for the initial data of the problem, we calculate the gradient of a functional and obtain necessary optimality conditions.
17 citations
TL;DR: In this paper, the generalized Littlewood theorem about a contour integral involving the logarithm of an analytic function is used to establish an infinite number of integral equalities involving integrals of the Riemann zeta-function.
Abstract: By using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytic function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard-Saias-Yor equality, and an equality established by one of the authors, are certain special cases of our general approach.
17 citations
TL;DR: In this article, a solution of a reflection problem on a half-line similar to the Skorokhod reflection problem but with possible jump-like exit from zero was obtained.
Abstract: For a solution of a reflection problem on a half-line similar to the Skorokhod reflection problem but with possible jump-like exit from zero, we obtain an explicit formula and study its properties. We also construct a Wiener process on a half-line with Wentzell boundary condition as a strong solution of a certain stochastic differential equation.
15 citations
TL;DR: In this article, conditions for the existence of solutions of nonlinear differential equations in the space of functions bounded on the axis were obtained by using a local linear approximation of these equations. But these conditions are not applicable to nonlinear problems.
Abstract: We obtain conditions for the existence of solutions of nonlinear differential equations in the space of functions bounded on the axis by using a local linear approximation of these equations.
14 citations
TL;DR: In this paper, it was shown that the indicated mapping cannot grow faster than an integral of a special type that corresponds to the distortion of the capacity under this mapping, which is an analog of the well-known Ikoma growth estimate for quasiconformal mappings of the unit ball into itself and of the classic Schwartz lemma for analytic functions.
Abstract: We obtain results on the local behavior of open discrete mappings $ f:D \to {\mathbb{R}^n} $
, n ≥ 2, that satisfy certain conditions related to the distortion of capacities of condensers. It is shown that, in an infinitesimal neighborhood of zero, the indicated mapping cannot grow faster than an integral of a special type that corresponds to the distortion of the capacity under this mapping, which is an analog of the well-known Ikoma growth estimate proved for quasiconformal mappings of the unit ball into itself and of the classic Schwartz lemma for analytic functions. For mappings of the indicated type, we also obtain an analog of the well-known Liouville theorem for analytic functions.
14 citations
TL;DR: In this article, the Kolmogorov-Nikolskii problem for biharmonic Poisson integrals on the classes of (ψ, β)-differentiable periodic functions of low smoothness in the uniform metric was solved.
Abstract: We solve the Kolmogorov–Nikol’skii problem for biharmonic Poisson integrals on the classes of (ψ, β)-differentiable periodic functions of low smoothness in the uniform metric.
14 citations
TL;DR: In this article, the authors established a series of criteria for the existence of regular solutions of the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains and formulated corresponding criteria for pseudoregular and multivalued solutions in the case of finitely connected domains.
Abstract: We establish a series of criteria for the existence of regular solutions of the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains. We also formulate the corresponding criteria for the existence of pseudoregular and multivalued solutions of the Dirichlet problem in the case of finitely connected domains.
TL;DR: In this article, a one-parameter family of positive polynomial operators of one and two variables that approximate the Urysohn operator was studied, and Bernstein-type polynomials were obtained.
Abstract: We study a one-parameter family of positive polynomial operators of one and two variables that approximate the Urysohn operator. In the case of two variables, the integration domain is an “isosceles right triangle.” As a special case, Bernstein-type polynomials are obtained. The Stancu asymptotic formulas for remainders are refined.
TL;DR: In this paper, the problems of continuous and homeomorphic extensions to the boundary for ring Q-homeomorphisms between domains on the Riemannian manifolds were studied and conditions for the function Q(x) and the boundaries of the domains under which every ring Q -homeomorphism admits a continuous or homeomorphic extension.
Abstract: We study the problems of continuous and homeomorphic extensions to the boundary for so-called ring Q-homeomorphisms between domains on the Riemannian manifolds and establish conditions for the function Q(x) and the boundaries of the domains under which every ring Q-homeomorphism admits a continuous or homeomorphic extension to the boundary. This theory can be applied, in particular, to the Sobolev classes.
TL;DR: In this article, necessary and sufficient conditions for the sum of subspaces of a Hilbert space H to be a subspace were given, and various properties of the n-tuples of subspace with closed sum were presented.
Abstract: We give necessary and sufficient conditions for the sum of subspaces H
1,…, H
n
, n ≥ 2, of a Hilbert space H to be a subspace and present various properties of the n-tuples of subspaces with closed sum.
TL;DR: In this paper, it was shown that an open discrete mapping with quasiconformality characteristic Q(x) can be extended to the boundary of a domain D by continuity.
Abstract: We study the problem of extension of mappings $ f:D\to \overline{{{{\mathbb{R}}^n}}} $
, n ≥ 2, to the boundary of a domain D. Under certain conditions imposed on a measurable function Q(x), Q:D → [0, ∞], and the boundaries of the domains D and D′ = f (D), we show that an open discrete mapping $ f:D\to \overline{{{{\mathbb{R}}^n}}} $
, n ≥ 2, with quasiconformality characteristic Q(x) can be extended to the boundary ∂D by continuity. The obtained statements extend the corresponding Srebro’s result to mappings with bounded distortion.
TL;DR: In this article, the authors extend the common fixed-point theorem established by Zhang and Song in 2009 to generalized (ψ, φ) -weak contractions, and give an example that illustrates the main result.
Abstract: We extend the common fixed-point theorem established by Zhang and Song in 2009 to generalized (ψ, φ)
f,g
-weak contractions. Moreover, we give an example that illustrates the main result. Finally, some common fixed-point results are obtained for mappings satisfying a contraction condition of the integral type in complete metric spaces.
TL;DR: In this paper, a version of the Laplace operator for functions on a Hilbert space with measure is proposed, and the Dirichlet problem for the Poisson equation is investigated in terms of this operator.
Abstract: We propose a version of the Laplace operator for functions on a Hilbert space with measure. In terms of this operator, we investigate the Dirichlet problem for the Poisson equation.
TL;DR: In this paper, the p-nilpotency and supersolubility of a finite group G under the assumption that every subgroup of a Sylow p-subgroup of G with given order is SΦ-supplemented in G is characterized.
Abstract: We call H an SΦ-supplemented subgroup of a finite group G if there exists a subnormal subgroup T of G such that G = HT and H ∩ T ≤ Φ(H), where Φ(H) is the Frattini subgroup of H. In this paper, we characterize the p-nilpotency and supersolubility of a finite group G under the assumption that every subgroup of a Sylow p-subgroup of G with given order is SΦ-supplemented in G: Some results about formations are also obtained.
TL;DR: For an arbitrary countable infinite amenable group G, the ideal of sets having μ-measure zero for every Banach measure μ on G is an Fσδ subset of {0; 1} groups.
Abstract: Answering a question posed by Banakh and Lyaskovska, we prove that, for an arbitrary countable infinite amenable group G, the ideal of sets having μ-measure zero for every Banach measure μ on G is an Fσδ subset of {0; 1}
G
.
TL;DR: In this paper, it was shown that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect.
Abstract: Zoschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.
TL;DR: This article established asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the sets of (ψ, β)-differentiable functions generated by sequences ψ(k) that satisfy the d'Alembert conditions.
Abstract: We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the sets $ C_{\beta}^{\psi }{L_p} $
of (ψ, β)-differentiable functions generated by sequences ψ(k) that satisfy the d’Alembert conditions. We find asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials on the classes $ C_{{\beta, p}}^{\psi } $
, 1 ≤ p ≤ ∞.
TL;DR: In this article, the value sharing problem, versions of the Hayman conjecture, and the uniqueness problem for p-adic meromorphic functions, as well as their difference operators and difference polynomials, are discussed.
Abstract: We discuss the value-sharing problem, versions of the Hayman conjecture, and the uniqueness problem for p-adic meromorphic functions, as well as their difference operators and difference polynomials.
TL;DR: In this paper, a criterion for the improved regular growth of functions of positive order with zeros on a finite system of half-lines in terms of their Fourier coefficients was established.
Abstract: We establish a criterion for the improved regular growth of entire functions of positive order with zeros on a finite system of half-lines in terms of their Fourier coefficients.
TL;DR: For example, National Natural Science Foundation of China [61170324] as mentioned in this paper, National Defense Basic Scientific Research Program [B1420110155] and Natural Sciences Foundation of Fujian Province, China [2010J01012]
Abstract: National Natural Science Foundation of China [61170324]; Natural Science Foundation of Fujian Province of China [2010J01012]; National Defense Basic Scientific Research Program of China [B1420110155]
TL;DR: In this paper, new oscillation criteria were established for the nonlinear damped differential equation (DDE), and the results obtained extend and improve some existing results, such as the following:
Abstract: Some new oscillation criteria are established for the nonlinear damped differential equation
$${\left( {r\left( t \right){k_1}\left( {x,x'} \right)} \right)^\prime } + p\left( t \right){k_2}\left( {x,x'} \right)x' + q\left( t \right)f\left( {x\left( t \right)} \right) = 0,\;t \ge {t_0}.$$
The results obtained extend and improve some existing results.
TL;DR: In this article, it is proved that an Arratia flow x(u,t) is a Markov process whose phase space is a certain subset K of the Skorokhod space.
Abstract: We study an Arratia flow x(u,t) It is proved that x(∙,t) is a Markov process whose phase space is a certain subset K of the Skorokhod space. We introduce the notion of total local time at zero for an Arratia flow. We prove that it is an additive, nonnegative, continuous functional of the flow and calculate its characteristic.
TL;DR: In this article, the influence of generalized modular subgroups on the structure of finite groups was studied and it was shown that generalized modular subsets have a strong influence on finite groups.
Abstract: We study the influence of generalized modular subgroups on the structure of finite groups.
TL;DR: In this paper, the numerical solutions of nonlinear Volterra integral equations by the block-by-block method were investigated, and a convergence theorem was proved showing that the method has at least sixth order of convergence.
Abstract: We investigate the numerical solutions of nonlinear Volterra integral equations by the block-by-block method especially useful for the solution of integral equations on large-size intervals. A convergence theorem is proved showing that the method has at least sixth order of convergence. Finally, the performance of the method is illustrated by some numerical examples.
TL;DR: In this paper, the authors describe all pairs of linear continuous operators that act in spaces of functions analytic in domains and satisfy a relation that is an operator analog of the Rubel equation.
Abstract: We describe all pairs of linear continuous operators that act in spaces of functions analytic in domains and satisfy a relation that is an operator analog of the Rubel equation
TL;DR: In this paper, the direct Jackson theorem for approximation by trigonometric polynomials is proved only if the lower dilation index of the function ψ is not equal to zero.
Abstract: In the spaces L
ψ (T
m
) of periodic functions with metric
$$ \uprho {\left( {f,0} \right)_\uppsi } = \int\limits_{{{\text{T}}^m}} {\uppsi \left( {\left| {f(x)} \right|} \right)dx}, $$
where ψ is a function of the type of modulus of continuity, we study the direct Jackson theorem in the case of approximation by trigonometric polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function ψ is not equal to zero.