# Volume 54, № 3, 2002

### Dmytro Ivanovych Martynyuk (On the 60th Anniversary of His Birth)

Danilov V. Ya., Mitropolskiy Yu. A., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 291-292

### On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 293-303

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form ω(δ_{1}, ..., δ_{ n }) = ω_{1}(δ_{1}) + ... + ω_{ n }(δ_{ n }), where ω_{ i }(δ_{ i }) are ordinary moduli of continuity that depend on one variable. In the case where ω_{ i }(δ_{ i }) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric *L* _{2} for ω(δ_{1}, ..., δ_{ n }) = ω_{1}(δ_{1}) + ... + ω_{ n }(δ_{ n }); (ii) in the integral metric *L* _{p} (*p* ≥ 1) for ω(δ_{1}, ..., δ_{ n }) = *c* _{1}δ_{1} + ... + *c* _{n}δ_{ n }; (iii) in the integral metric *L* _{(2, ..., 2, 2r)} (*r* = 2, 3, ...) for ω(δ_{1}, ..., δ_{ n }) = ω_{1}(δ_{1}) + ... + ω_{ n − 1}(δ_{ n − 1}) + *c* _{n}δ_{ n }.

### Distribution of Overjump Functionals of a Semicontinuous Homogeneous Process with Independent Increments

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 303-321

We establish relations for the distributions of functionals associated with an overjump of a process ξ(*t*) with continuously distributed jumps of arbitrary sign across a fixed level *x* > 0 (including the zero level *x* = 0 and infinitely remote level *x* → ∞). We improve these relations in the case where the distributions of maxima and minima of ξ(*t*) may have an atom at zero. The distributions of absolute extrema of semicontinuous processes are defined in terms of these atomic probabilities and the cumulants of the corresponding monotone processes.

### Filtration and Finite-Dimensional Characterization of Logarithmically Convex Measures

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 323-331

We study the classes *C*(α, β) and *C* _{H}(α, β) of logarithmically convex measures that are a natural generalization of the notion of Boltzmann measure to an infinite-dimensional case. We prove a theorem on the characterization of these classes in terms of finite-dimensional projections of measures and describe some applications to the theory of random series.

### Recovery of a Function from Information on Its Values at the Nodes of a Triangular Grid Based on Data Completion

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 332-341

We consider a method for binary completion of two-dimensional data. On the basis of information about a surface given by a triangular grid, we construct a continuous polygonal surface based on a denser grid (than the one given). We determine the error and norm of this method and study its properties.

### On Noncyclic Norm of Infinite Locally Finite Groups

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 342-349

We study relationships between the properties of a group and its noncyclic norm. We obtain a description of infinite locally finite groups whose noncyclic norms are non-Dedekind.

### Existence Theorems for Equations with Noncoercive Discontinuous Operators

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 349-364

In a Hilbert space, we consider equations with a coercive operator equal to the sum of a linear Fredholm operator of index zero and a compact operator (generally speaking, discontinuous). By using regularization and the theory of topological degree, we establish the existence of solutions that are continuity points of the operator of the equation. We apply general results to the proof of the existence of semiregular solutions of resonance elliptic boundary-value problems with discontinuous nonlinearities.

### Fading Markov Random Evolution

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 364-372

We introduce the notion of fading Markov random evolution and study the properties and characteristics of this process.

### Minimal Hereditary ω-Local Non-ℌ-Formations

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 373-380

We describe minimal hereditary ω-local non-ℌ-formations, where ℌ is a formation of the classical type.

### Best $M$-Term Trigonometric Approximations of the Classes $B_{p,θ}^Ω$ of Functions of Many Variables

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 381-394

We obtain exact order estimates for the best $M$-term trigonometric approximations of the classes $B_{p,θ}^Ω$ of functions of many variables in the space $L_{q, 1} < p < q < ∞, q > 2$.

### On Stability in Time of Space Asymptotics of Solutions of Evolution Equations

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 395-401

We obtain solutions of the heat-conduction equation on a semi-axis that preserve in time the asymptotic representation of the function that determines a solution at initial time. This property is preserved in the presence of a complex-valued power-decreasing potential. We present an estimate for the rate of “destruction” of the structure of a solution.

### On Global Solutions of Systems of Nonlinear Functional Differential Equations with Deviating Argument Dependent on Unknown Functions

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 402-407

For a system of nonlinear functional differential equations with nonlinear deviations of an argument, we obtain sufficient conditions for the existence of a continuously differentiable solution bounded for *t* ∈ *R*.

### On Some Properties of the Behavior of Linear Extensions of Dynamical Systems on a Torus under Perturbation of Phase Variables

Samoilenko A. M., Stepanenko N. V.

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 408-412

We investigate classes of linear extensions of dynamical systems on a torus for which the Lyapunov functions exist for an arbitrary flow on the torus. Linear extensions for which the Lyapunov functions exist only with varying coefficients are considered separately. We investigate the problem of preservation of regularity under perturbation of phase variables.

### Stochastic Stability of Processes Determined by Poisson Differential Equations with Delay

Svishchuk A. V., Svishchuk M. Ya.

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 413-418

We prove an existence theorem and establish the property of stochastic stability for processes determined by the Poisson stochastic differential equations with delay.

### On Reducibility of Systems of Linear Differential Equations with Quasiperiodic Skew-Adjoint Matrices

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 419-424

We prove that there exists an open set of irreducible systems in the space of systems of linear differential equations with quasiperiodic skew-adjoint matrices and fixed frequency module.

### Topological Limit of Trajectories of Intervals of Simplest One-Dimensional Dynamical Systems

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 425-430

We consider dynamical systems generated by continuous maps of an interval into itself. We investigate the asymptotic behavior of the trajectories of subsets of the interval. In particular, we prove that if the ω-limit set of an arbitrary trajectory is a fixed point, then the topological limit of the trajectory of any subinterval exists.