# Volume 56, № 5, 2004

### Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Babenko V. F., Korneichuk N. P., Pichugov S. A.

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 579-594

Let $γ = (γ_1 ,..., γ_d )$ be a vector with positive components and let $D^γ$ be the corresponding mixed derivative (of order $γ_j$ with respect to the $j$ th variable). In the case where $d > 1$ and $0 < k < r$ are arbitrary, we prove that $$\sup_{x \in L^{r\gamma}_{\infty}(T^d)D^{r\gamma}x\neq0} \frac{||D^{k\gamma}x||_{L_{\infty}(T^d)}}{||x||^{1-k/r}||D^{r\gamma}||^{k/r}_{L_{\infty}(T^d)}} = \infty$$ and $$||D^{k\gamma}x||_{L_{\infty}(T^d)} \leq K||x||^{1 - k/r}_{L_{\infty}(T^d)}||D^{r\gamma}x||_{L_{\infty}(T^d)}^{k/r} \left(1 + \ln^{+}\frac{||D^{r\gamma}x||_{L_{\infty}(T^d)}}{||x||_{L_{\infty} (T^d)}}\right)^{\beta}$$ for all $x \in L^{r\gamma}_{\infty}(T^d)$ Moreover, if \(\bar \beta \) is the least possible value of the exponent β in this inequality, then $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$

### Jackson-type inequalities and exact values of widths of classes of functions in the spaces $S^p , 1 ≤ p < ∞$

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 595–605

In the spaces $S^p , 1 ≤ p < ∞$, introduced by Stepanets, we obtain exact Jackson-type inequalities and compute the exact values of widths of classes of functions determined by averaged moduli of continuity of order $m$.

### On configurations of subspaces of a Hilbert space with fixed angles between them

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 606–615

We investigate the set of irreducible configurations of subspaces of a Hilbert space for which the angle between every two subspaces is fixed. This is the problem of *-representations of certain algebras generated by idempotents and depending on parameters (on the set of angles). We separate the class of problems of finite and tame representation type. For these problems, we indicate conditions on angles under which the configurations of subspaces exist and describe all irreducible representations.

### On random measures on spaces of trajectories and strong and weak solutions of stochastic equations

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 625–633

We investigate stationary random measures on spaces of sequences or functions. A new definition of a strong solution of a stochastic equation is proposed. We prove that the existence of a weak solution in the ordinary sense is equivalent to the existence of a strong measure-valued solution.

### On algebras of the Temperley-Lieb type associated with algebras generated by generators with given spectrum

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 634–641

We introduce and study algebras of the Temperley-Lieb type associated with algebras generated by linearly connected generators with given spectrum. We study their representations and the sets of parameters for which representations of these algebras exist.

### On the solution of a one-dimensional stochastic differential equation with singular drift coefficient

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 642–655

We determine generalized diffusion coefficients and describe the structure of local times for a process defined as a solution of a one-dimensional stochastic differential equation with singular drift coefficient.

### Properties of solutions of the cauchy problem for essentially infinite-dimensional evolution equations

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 656–662

We investigate properties of solutions of the Cauchy problem for evolution equations with essentially infinite-dimensional elliptic operators.

### Approximation of $\bar {\omega}$ -integrals of continuous functions defined on the real axis by Fourier operators

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 663-676

We obtain asymptotic formulas for the deviations of Fourier operators on the classes of continuous functions $C^{ψ}_{∞}$ and $\hat{C}^{\bar{\psi} } H_{\omega}$ in the uniform metric. We also establish asymptotic laws of decrease of functionals characterizing the problem of the simultaneous approximation of $\bar{\psi}$-integrals of continuous functions by Fourier operators in the uniform metric.

### Entire solutions of the euler—poisson equations

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 677–686

All entire solutions of Euler—Poisson equations are presented.

### Second-order moment equations for a system of differential equations with random right-hand side

Dzhalladova I. A., Valeyev K. G.

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 687-691

We present a method for the derivation of second-order moment equations for solutions of a system of nonlinear equations that depends on a finite-valued semi-Markov or Markov process. For systems of linear differential equations with random coefficients, the case where the inhomogeneous part contains white noise is considered.

### Rarefaction of moving diffusion particles

Gasanenko V. A., Roitman A. B.

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 691-694

We investigate a flow of particles moving along a tube together with gas. The dynamics of particles is determined by a stochastic differential equation with different initial states. The walls of the tube absorb particles. We prove that if the incoming flow of particles is determined by a random Poisson measure, then the number of remained particles is characterized by the Poisson distribution. The parameter of this distribution is constructed by using a solution of the corresponding parabolic boundary-value problem.

### Solution of a nonlinear singular integral equation with quadratic nonlinearity

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 695-704

Using methods of the theory of boundary-value problems for analytic functions, we prove a theorem on the existence of solutions of the equation $$u^2 \left( t \right) + \left( {\frac{1}{\pi }\int\limits_{ - \infty }^\infty {\frac{{u\left( \tau \right)}}{{\tau - t}}d\tau } } \right)^2 = A^2 \left( t \right)$$ and determine the general form of a solution by using zeros of an entire function $A^2 (z)$ of exponential type.

### Integral conditions for the invertibility of Markov chains on a half-line with general measure of irreducibility

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 705-712

We present conditions for the invertibility of Markov chains with values from ℝ+ and general measure of irreducibility. The results are obtained by the classical method of test functions combined with the method of perturbation of partial potentials.

### Continuous procedure of stochastic approximation in a semi-Markov medium

Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 713–720

Using the Lyapunov function for an averaged system, we establish conditions for the convergence of the procedure of stochastic approximation $$du(t)=a(t)[C(u(t),x(t))dt+σ(u(t))dw(t)]$$ in a random semi-Markov medium described by an ergodic semi-Markov process $x(t)$.