Showing papers in "Ukrainian Mathematical Journal in 1998"
TL;DR: In contrast to the Lasota-Wazewska model, in this paper, the authors consider the delay differential equations, which were proposed by Mackey and Glass as a model of blood cell production, and establish the existence of the number δj(n, τ) > 0 such that the equilibrium of the equation under consideration is a global attractor for all δ e (0, δ j] independently of β0 and θ.
Abstract: We consider the delay differential equations
$$P'(t) = \frac{{\beta _0 \theta ^n [P(t - \tau )]^j }}{{\theta ^n + [P(t - \tau )]^n }} - \delta P(t),{\rm{ }}j = 0,1,$$
which were proposed by Mackey and Glass as a model of blood cell production. We suggest new conditions sufficient for the positive equilibrium of the equation considered to be a global attractor. In contrast to the Lasota-Wazewska model, we establish the existence of the number δj = δj(n, τ) > 0 such that the equilibrium of the equation under consideration is a global attractor for all δ e (0, δj] independently of β0 and θ.
41 citations
TL;DR: In this paper, conditions for convergence and the rate of convergence of random functional series from the space subφ (Ω) in various norms were studied for a boundary-value problem for a hyperbolic equation with random initial conditions.
Abstract: We study conditions for convergence and the rate of convergence of random functional series from the space subφ (Ω) in various norms. The results are applied to the investigation of a boundary-value problem for a hyperbolic equation with random initial conditions.
29 citations
TL;DR: By using the theory of generalized inverse operators, the authors established a criterion of the solvability of the Lyapunov-type matrix equations AX - XB = D and X - AXB =D and investigated the structure of the set of their solutions.
Abstract: By using the theory of generalized inverse operators, we establish a criterion of the solvability of the Lyapunov-type matrix equations AX - XB = D and X - AXB = D and investigate the structure of the set of their solutions.
24 citations
TL;DR: By using a singular perturbation problem, this article obtained sufficient conditions for the stability of a dynamical system with rapid Markov switchings under the condition of exponential stability of the averaged diffusion process.
Abstract: By using a solution of a singular perturbation problem, we obtain sufficient conditions for the stability of a dynamical system with rapid Markov switchings under the condition of exponential stability of the averaged diffusion process.
20 citations
TL;DR: In this article, the authors investigated the convergence of Fourier series on the classes of convolutions of functions from N with kernels with slowly decreasing coefficients and obtained asymptotic equalities for the upper bounds of deviations of the Fourier sums on the sets which are solutions of the Kolmogorov-Nikol'skii problem.
Abstract: We investigate the rate of convergence of Fourier series on the classes \(L^{\bar \psi } \)N in the uniform and integral metrics The results obtained are extended to the case where the classes \(L^{\bar \psi } \)N are the classes of convolutions of functions from N with kernels with slowly decreasing coefficients In particular, we obtain asymptotic equalities for the upper bounds of deviations of the Fourier sums on the sets \(L^{\bar \psi } \)N which are solutions of the Kolmogorov-Nikol’skii problem In addition, we establish an analog of the well-known Lebesgue inequality
17 citations
TL;DR: In this article, the application of the numerical-analytic method proposed by A.M. Samoilenko in 1965 to multipoint boundary value problems is analyzed, and the results show that the numerical analysis can be used to solve the problem.
Abstract: We analyze the application of the numerical-analytic method proposed by A.M. Samoilenko in 1965 to multipoint boundary-value problems.
16 citations
TL;DR: In this article, the authors considered the Stefan problem for a parabolic equation with a small parameter as the coefficient of the derivative with respect to time, and proved the classical solvability of the Hele-Shaw problem with free boundary in the small domain in the time domain.
Abstract: We consider the Stefan problem for a parabolic equation with a small parameter as the coefficient of the derivative with respect to time. We justify the limit transition as the small parameter tends to zero, which enables us to prove the classical solvability of the Hele-Shaw problem with free boundary in the small with respect to time.
14 citations
TL;DR: In this paper, the moduli and extremal metrics of families of curves on certain nonorientable or twisted Riemannian manifolds are investigated. And the authors also introduce and investigate weighted l-moduli for families of curve and the corresponding extremal metric.
Abstract: We present results on the moduli and extremal metrics of families of curves on certain nonorientable or twisted Riemannian manifolds. We also introduce and investigate weighted l-moduli of families of curves and the corresponding extremal metrics.
13 citations
TL;DR: In this article, one class of Gaussian random processes that are not semimartingales but their increments can play the role of a random measure is considered and for an extended stochastic integral with respect to the processes considered, the Ito formula is obtained.
Abstract: We consider one class of Gaussian random processes that are not semimartingales but their increments can play the role of a random measure. For an extended stochastic integral with respect to the processes considered, we obtain the Ito formula.
13 citations
TL;DR: In this article, the authors established sufficient conditions for the existence of generalized Besicovitch almost (quasi)periodic solutions and classical quasiperiodic solutions of natural Lagrangian systems with force functions convex on a compact set.
Abstract: By using the variational method, we establish sufficient conditions for the existence of generalized Besicovitch almost (quasi)periodic solutions and classical quasiperiodic solutions of natural Lagrangian systems with force functions convex on a compact set.
12 citations
TL;DR: In this article, the authors study the existence of periodic and almost periodic solutions of the scalar equation x′ (t) = − δx(t) + pmaxu∈[t − h, t]x(u) + f(t), where δ, p ∈ R, with a periodic (almost periodic) perturbation f( t).
Abstract: We study the problem of existence of periodic and almost periodic solutions of the scalar equation x′ (t) = − δx(t) + pmaxu∈[t − h, t]x(u) + f(t) where δ, p ∈ R, with a periodic (almost periodic) perturbation f(t). For these solutions, we establish conditions of global exponential stability and prove uniqueness theorems.
TL;DR: In this article, the authors developed spectral and scattering theory for one class of self-adjoint matrix operators of mixed order, i.e., matrix operators that can be expressed as
Abstract: We develop spectral and scattering theory for one class of self-adjoint matrix operators of mixed order.
TL;DR: In this paper, it was shown that every non-discrete topological Abelian group can be decomposed into countably many dense subsets and that every submaximal left-topological ABG is σdiscrete.
Abstract: We prove that there exist ZFC models in which every nondiscrete topological Abelian group can be decomposed into countably many dense subsets. This statement is an answer to the question raised by Comfort and van Mill. We also prove that every submaximal left-topological Abelian group is σ-discrete.
TL;DR: Under sufficiently general assumptions, the authors describes sets of entire functions f, sets of growing functions λ, and sets of complex-valued functions H from L ∈ [1, + ∞], for which
Abstract: Under sufficiently general assumptions, we describe sets of entire functions f, sets of growing functions λ, and sets of complex-valued functions H from L
p
[0, 2π], p ∈ [1, + ∞], for which
$$\left\{ {\frac{1}{{2\pi }}\int\limits_0^{2\pi } {|\log f(re^{i\theta } ) - \lambda (r)H(\theta )|^p } d\theta } \right\}^{1/p} = o(\lambda (r)),r \to \infty $$
TL;DR: On a Riemannian manifold of nonpositive curvature, this article obtained dimension-independent estimates for the fundamental solution of a parabolic equation and for the logarithmic derivative of this solution.
Abstract: On a Riemannian manifold of nonpositive curvature, we obtain dimension-independent estimates for the fundamental solution of a parabolic equation and for the logarithmic derivative of this solution.
TL;DR: In this article, the properties of functions with lacunary Fourier series were studied in a many-dimensional space, depending only on the values of these functions in a neighborhood of a certain point.
Abstract: In a many-dimensional space, we study some properties of functions with lacunary Fourier series depending only on the values of these functions in a neighborhood of a certain point.
TL;DR: In this article, it was shown that the boundary of the minimum-area ellipse has exactly four or five common points with the polygon, and that the ellipses formed by these common points are the minimum area ellipsse for the quadrangles and pentagons formed by the common points.
Abstract: We continue the investigation of the problem of construction of a minimum-area ellipse for a given convex polygon (this problem is solved for a rectangle and a trapezoid). For an arbitrary polygon, we prove that, in the case where the boundary of the minimum-area ellipse has exactly four or five common points with the polygon, this ellipse is the minimum-area ellipse for the quadrangles and pentagons formed by these common points.
TL;DR: Two-dimensional quadratic systems are considered as a Lienard equation with certain special nonlinearities in this paper, and theorems on the existence or absence of cycles are given.
Abstract: Two-dimensional quadratic systems are considered as a Lienard equation with certain special nonlinearities. Theorems on the existence or absence of cycles are given.
TL;DR: In this article, the authors investigate the problem of parametric excitation of oscillations in systems of parabolic and hyperbolic equations with small coefficient of diffusion, and establish the existence of an arbitrary fixed number of stable periodic solutions for a proper choice of the parameters of equations.
Abstract: We investigate the problem of parametric excitation of oscillations in systems of parabolic and hyperbolic equations with small coefficient of diffusion. We establish the phenomenon of parametric bufferness, i.e., the existence of an arbitrary fixed number of stable periodic solutions for a proper choice of the parameters of equations.
TL;DR: For one class of degenerate parabolic equations of the Kolmogorov type, the authors proved uniqueness theorems for the solutions of the Cauchy problem for the classes of functions with bounded growth and for nonnegative functions.
Abstract: For one class of degenerate parabolic equations of the Kolmogorov type, we establish the property of normality, the convolution formula, the property of positivity, and a lower bound for the fundamental solution. We also prove uniqueness theorems for the solutions of the Cauchy problem for the classes of functions with bounded growth and for the class of nonnegative functions.
TL;DR: For the perturbed nonlinear Klein-Gordon equation, this paper constructed an asymptotic solution by using Ateb-functions for both autonomous and non-autonomous cases.
Abstract: For the perturbed nonlinear Klein-Gordon equation, we construct an asymptotic solution by using Ateb-functions. We consider autonomous and nonautonomous cases.
TL;DR: For quasidifferential equations in semilinear metric spaces, the authors considered the existence, uniqueness, and continuity of solutions and the problem of justification of the averaging method.
Abstract: For quasidifferential equations in semilinear metric spaces, we consider the problem of existence, uniqueness, and continuity of solutions and the problem of justification of the averaging method.
TL;DR: In this article, the maximum of the difference of two renewal processes with discrete time is shown to be semicontinuous in discrete topology. But the distribution of the maximum is not known.
Abstract: We find the distribution of the maximum of the difference of two renewal processes with discrete time that is semicontinuous in discrete topology.
TL;DR: In this paper, the frequency locking of an asymptotically orbitally stable rotating wave solution of an autonomous S1-equivariant differential equation under the forcing of a rotating wave was described.
Abstract: We describe the frequency locking of an asymptotically orbitally stable rotating wave solution of an autonomous S1-equivariant differential equation under the forcing of a rotating wave.
TL;DR: In this article, the existence and uniqueness of invariant toroidal manifolds of countable systems of linear equations given on an infinite-dimensional torus was studied, where the uniqueness of the manifolds was also studied.
Abstract: We study the problem of existence and uniqueness of invariant toroidal manifolds of countable systems of linear equations given on an infinite-dimensional torus.
TL;DR: For the trigonometric widths of the Besov classes of periodic functions of many variables in the space L ≥ q for 1 ≤ p ≤ 2 < q < p/(p - 1) as mentioned in this paper.
Abstract: We obtain estimates exact in order for the trigonometric widths of the Besov classes B
p,θ
r
of periodic functions of many variables in the space L
q for 1 ≤ p ≤ 2 < q < p/(p - 1).
TL;DR: In this paper, the covariant derivatives of Jacobi fields along a geodesic on a Riemannian manifold of negative curvature were estimated for the case of a Gaussian manifold.
Abstract: We obtain estimates of the covariant derivatives of Jacobi fields along a geodesic on a Riemannian manifold of negative curvature.
TL;DR: For the diffusion equation of fractional order, the authors constructed an approximation difference scheme of order 0(h2 + τ) for boundary-value problems and presented an algorithm for the solution of boundary value problems.
Abstract: For the diffusion equation of fractional order, we construct an approximation difference scheme of order 0(h2 + τ). We present an algorithm for the solution of boundary-value problems for a generalized transfer equation of fractional order.
TL;DR: For a system of nonlinear difference equations, this paper established conditions for the existence and uniqueness of a solution bounded on the entire real axis and studied its properties. But the uniqueness of the solution was not considered in this paper.
Abstract: For a system of nonlinear difference equations, we establish conditions for the existence and uniqueness of a solution bounded on the entire real axis and study its properties.
TL;DR: For a second-order quasilinear differential system whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, the authors proved that there exists a particular solution with a similar structure in the case of purely imaginary roots of the characteristic equation for the matrix of coefficients of the linear part.
Abstract: For a second-order quasilinear differential system whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we prove that, under certain conditions, there exists a particular solution with a similar structure in the case of purely imaginary roots of the characteristic equation for the matrix of coefficients of the linear part.