# Volume 50, № 8, 1998

### Comparison of approximation properties of generalized polynomials and splines

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1011–1020

We establish that, for *p* ∈ [2, ∞), *q* = 1 or *p* = ∞, *q ∈* [ 1, 2], the classes *W* _{p} ^{r} of functions of many variables defined by restrictions on the *L* _{p}-norms of mixed derivatives of order *r* = (*r* _{1}, *r* _{2}, ..., *r* _{m}) are better approximated in the *L* _{q}-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide.

### Criterion of the solvability of matrix equations of the Lyapunov type

Boichuk О. A., Krivosheya S. A.

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1021–1026

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.

### Isometry of functional spaces with different number of variables

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1027–1045

We construct spaces of real functions of *n + k* variables that are isometric to spaces of real functions given on an *n*-dimensional Euclidean space. We present certain properties and examples of delta-like kernels used for the construction of isometric spaces of functions with different number of variables. We prove certain assertions that enable one to construct delta-like kernels with many variables by using delta-like kernels with smaller number of variables.

### Generalized horseshoes and indecomposability for one-dimensional continua

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1046–1054

We consider dynamical systems given by a sequence of continuous maps of graphs. We obtain results that generalize the known results concerning the existence of indecomposable subcontinua in terms of the corresponding maps of one-dimensional continua.

### $q$-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1055-1063

We obtain algebraic relations (identities) for $q$-numbers that do not contain $q^{α}$-factors. We derive a formula that expresses any $q$-number $[x]$ in terms of the $q$-number [2]. We establish the relationship between the $q$-numbers $[n]$ and the Fibonacci numbers, Chebyshev polynomials, and other special functions. The sums of combinations of $q$-numbers, in particular, the sums of their powers, are calculated. Linear and bilinear generating functions are found for “natural” $q$-numbers.

### Spectral theory of some matrix differential operators of mixed order

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1064–1072

We develop spectral and scattering theory for one class of self-adjoint matrix operators of mixed order.

### Criteria of the mean-square asymptotic stability of solutions of systems of linear stochastic difference equations with continuous time and delay

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1073–1081

We obtain spectral and algebraic coefficient criteria and sufficient conditions for the mean-square asymptotic stability of solutions of systems of linear stochastic difference equations with continuous time and delay. We consider the case of a rational correlation between delays and a “white-noise”-type stochastic perturbation of coefficients. We use the method of Lyapunov functions. Most results are presented in terms of the Sylvester and Lyapunov matrix algebraic equations.

### On finite convolutions of singular distributions and a “singular analog” of the Jessen-Wintner theorem

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1082–1088

We study the fractal properties of a convolution of two Cantor distributions. By using the method of characteristic functions, we establish sufficient conditions for the singularity of the convolution of an arbitrary finite number of distributions of random variables with independent *s*-adic digits. We disprove the hypothesis on the validity of a “singular analog” of the Jessen-Wintner theorem for anomalously fractal distributions.

### Trigonometric widths of the classes $B_{p,θ}^ r$ of functions of many variables in the space $L_q$

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1089-1097

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 $1 ≤ p ≤ 2 < q < p/(p - 1)$.

### Minimum-Area ellipse containing a finite set of points. II

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1098–1105

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.

### On new exact solutions of a nonlinear diffusion system that describes the growth of protein crystals

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1106–1120

By using the method of additional generating conditions, we construct multiparameter families of exact solutions of a nonlinear diffusion system that describes the growth of protein crystals. We demonstrate the efficiency of the application of the solutions obtained to the solution of the corresponding nonlinear problem with moving boundary.

### On one class of solutions of a countable quasilinear system of differential equations with slowly varying parameters

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1121–1128

For a countable quasilinear differential system whose coefficients are represented as Fourier series with slowly varying coefficients and frequency, we present conditions under which solutions of this system have analogous structure.

### Estimates of the heat kernel on a manifold of nonpositive curvature

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1129–1136

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.

### Investigation of a system of linear differential equations with random coefficients

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1137–1143

We investigate a system of linear differential equations with random coefficients that depend on a periodic Markov process.

### On topological spaces with π_{2} = 0

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1144–1146

We consider one construction over topological spaces and study its influence on the group π_{2}.

### On the renewal of a solution of the Dirichlet boundary-value problem for a biharmonic equation

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1147–1151

We study the problem of renewal of a solution of the Dirichlet boundary-value problem for a biharmonic equation on the basis of the known information about the boundary function. The obtained estimates of renewal error are unimprovable in certain cases.

### Jubilees Iosyp Zakharovych Shtokalo (on the 100th anniversary of his birth)

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1152–1153