### On the Convergence of Fourier Series with Orthogonal Polynomials inside and on the Closure of a Region

Abdullayev F. G., Küçükaslan M.

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1299-1312

We study the rate of convergence of Fourier series of orthogonal polynomials over an area inside and on the closure of regions of the complex plane.

### Cumulant Representation of Solutions of the BBGKY Hierarchy of Equations

Gerasimenko V. I., Ryabukha T. V.

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1313-1328

We construct a cumulant representation of solutions of the Cauchy problem for the BBGKY hierarchy of equations and for the dual hierarchy of equations. We define the notion of dual nonequilibrium cluster expansion. We investigate the convergence of the constructed cluster expansions in the corresponding functional spaces.

### Invariant Geometric Objects of the Canonical Almost-Geodesic Mapping π_{2} (*e* = 0)

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1329-1335

For the canonical almost-geodesic mapping π_{2} (*e* = 0), we prove an analog of the Beltrami theorem in the theory of geodesic mappings. We introduce canonical π_{2}-flat spaces and obtain metrics for them in a special coordinate system.

### On One Method for the Introduction of Local Coordinates in a Neighborhood of an Invariant Toroidal Set

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1336-1347

We consider one method for the introduction of local coordinates in a neighborhood of an *m*-dimensional invariant torus of a dynamical system of differential equations in the Euclidean space **R** ^{ n } in dimensions satisfying the inequalities *m* + 1 < *n* ≤ 2*m*.

### Strengthening of the Kolmogorov Comparison Theorem and Kolmogorov Inequality and Their Applications

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1348-1356

We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions *x* ∈ *L* _{∞} ^{ x } (** r**), namely, $$\left\| {x^{(k)} } \right\|_{L_\infty (R)} \leqslant \frac{{\left\| {\phi _{r - k} } \right\|_\infty }}{{\left\| {\phi _r } \right\|_\infty ^{1 - k/r} }}M(x)^{1 - k/r} \left\| {x^{(r)} } \right\|_{L_\infty (R)}^{k/r} ,$$ where $$M(x): = \frac{1}{2}\mathop {\sup }\limits_{\alpha ,\beta } \left\{ {\left| {x(\beta ) - x(\alpha )} \right|:x'(t) \ne 0{\text{ }}\forall t \in (\alpha ,\beta )} \right\}{\text{,}}$$

*k*,

*r*∈

**,**

*N**k*<

*r*, and ϕ

_{ r }is a perfect Euler spline of order

*r*. Using this inequality, we strengthen the Bernstein inequality for trigonometric polynomials and the Tikhomirov inequality for splines. Some other applications of this inequality are also given.

### Galerkin Method for First-Order Hyperbolic Systems with Two Independent Variables

Lavrenyuk S. P., Oliskevych M. O.

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1356-1371

We investigate a mixed problem for a weakly nonlinear first-order hyperbolic system with two independent variables in bounded and unbounded domains. Assuming that the nonlinearities are monotonic, we obtain conditions for the existence and uniqueness of a generalized solution; these conditions do not depend on the behavior of a solution as *x* → +∞.

### Stationary Mode and Binomial Moments for Networks of the Type $[SM|GI|∞]^r$

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1371-1380

By using methods of the Markov renewal theory, we find conditions for the existence of a stationary mode for multichannel networks with semi-Markov input stream. As a tool of stationary distribution analysis, we introduce multivariate binomial moments and investigate their asymptotic properties. For a multichannel queuing system with periodic input stream, we construct the generating function of the stationary distribution in explicit form in terms of the parameters of the system under consideration.

### $P$-Faithful Partially Ordered Sets

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1381-1395

We prove a theorem that describes $P$-faithful partially ordered sets.

### On One Class of Almost Bounded Perturbations of Smooth Restrictions of a Closed Operator

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1396-1402

We investigate one class of perturbations of a closed densely-defined operator in a Hilbert space. These perturbations change the domain of definition of the operator. We prove that the perturbed operator *S* is closed and densely defined. We construct the adjoint operator *S**.

### On Periodic Solutions of Degenerate Singularly Perturbed Linear Systems with Multiple Elementary Divisor

Akymenko A. M., Yakovets V. P.

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1403-1415

We establish sufficient conditions for the existence and uniqueness of a periodic solution of a system of linear differential equations with a small parameter and a degenerate matrix of coefficients of derivatives in the case of a multiple spectrum of a boundary matrix pencil. We construct asymptotics of this solution.

### Conditional Symmetry and Exact Solutions of a Multidimensional Diffusion Equation

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1416-1420

We investigate the conditional symmetry of a multidimensional nonlinear reaction–diffusion equation by its reduction to a radial equation. We construct exact solutions of this equation and infinite families of exact solutions for the corresponding one-dimensional diffusion equation.

### Singularities of Solutions of Singular Integral Equations

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1429-1436

From the point of view of singularities of differential mappings, we determine the dependence between the dimension of a family of singular equations and the types of singularities of their solutions.

### The characteristics of points of strong summability of the Fourier - Laplace series for functions from the class $L(S^m)$ in the case of critical indicator

↓ Abstract

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1437-1439

We announce the result that enables one to determine fairly constructive characteristics of a set of points of full measure on a sphere $L(S^m)$ at which the strong means converge to a given function $f(\cdot)$.

### International Scientific Conference on the Theory of Evolution Equations (Fifth Bogolyubov Readings)

Konet I. M., Perestyuk N. A., Samoilenko A. M., Teplinsky Yu. V.

Ukr. Mat. Zh. - 2002νmber=3. - 54, № 10. - pp. 1440