# Volume 59, № 9, 2007

### Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series

Baranovskyi O. M., Pratsiovytyi M. V., Torbin H. M.

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1155–1168

We study topological and metric properties of the set $$C\left[\overline{O}^1, \{V_n\}\right] = \left\{x:\; x= ∑_n \frac{(−1)^{n−1}}{g_1(g_1 + g_2)…(g_1 + g_2 + … + g_n)},\quad g_k ∈ V_k ⊂ \mathbb{N}\right\}$$ with certain conditions on the sequence of sets $\{V_n\}$. In particular, we establish conditions under which the Lebesgue measure of this set is (a) zero and (b) positive. We compare the results obtained with the corresponding results for continued fractions and discuss their possible applications to probability theory.

### Rate of convergence to ergodic distribution for queue length in systems of the $M^{θ}/G/1/N$

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1169–1178

For finite capacity queueing systems of the type *M *^{θ}/*G*/1, the convenient formulas for the ergodic distribution of a queue length are obtained.
An estimate of a rate of convergence of the distribution of queue length in the trasient regime to the ergodic distribution is obtained and computational algorithms for finding the convergence rate are presented.

### General Kloosterman sums over ring of Gaussian integers

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1179-1200

The general Kloosterman sum $K(m, n; k; q)$ over $\mathbb{Z}$ was studied by $S$. Kanemitsu, Y. Tanigawa, Yi. Yuan, Zhang Wenpeng in their research of problem of D. H. Lehmer. In this paper, we obtain the similar estimations of $K(\alpha, \beta; k; \gamma)$ over $\mathbb{Z}[i]$. We also consider the sum $\widetilde{K}(\alpha, \beta; h, q; k)$ which has not an analogue in the ring $\mathbb{Z}$ but it can be used for the inversigation of the second moment of the Hecke zeta-fonction of field $\mathbb{Q}(i)$.

### Approximation of ( ψ, β )-differentiable functions defined on the real axis by Weierstrass operators

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1201–1220

Asymptotic equalities are obtained for upper bounds of approximations by the Weierstrass operators on the functional classes $\widehat{C}^{\psi}_{\beta, \infty}$ and $\widehat{L}^{\psi}_{\beta, 1}$ in metrics of the spaces $\widehat{C}$ and $\widehat{L}_1$, respectively.

### On moduli of smoothness and Fourier multipliers in $L_p, 0 < *p* < 1$

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1221–1238

We obtain the theorem on the relationship between a modulus of smoothness and the best approximation in *L _{p }* , 0 <

*p*< 1, and theorems on the extension of functions with the preservation of the modulus of smoothness in

*L*, 0 <

_{p }*p*< 1.

In addition, we present a complete description of multipliers of periodic functions in the spaces

*L*, 0 <

_{p }*p*< 1.

### Invariants of knots, surfaces in **R** _{3}, and foliations

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1239–1252

We give a survey of some known results related to combinatorial and geometric properties of finite-order invariants of knots in a three-dimensional space. We study the relationship between Vassiliev invariants and some classical numerical invariants of knots and point out the role of surfaces in the investigation of these invariants. We also consider combinatorial and geometric properties of essential tori in standard position in closed braid complements by using the braid foliation technique developed by Birman, Menasco, and other authors. We study the reductions of link diagrams in the context of finding the braid index of links.

### Approximation of holomorphic functions by Taylor-Abel-Poisson means

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1253–1260

We investigate approximations of functions $f$ holomorphic in the unit disk by means $A_{\rho, r}(f)$ for $\rho \rightarrow 1_-$.
In terms of an error of the approximation by these means, the constructive characteristic of classes of holomorphic functions $H_p^r \text{\;Lip\,}\alpha$ is given.
The problem of the saturation of $A_{\rho, r}(f)$ in the Hardy space $H_p$ is solved.

### On Schur classes for modules over group rings

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1261–1268

We consider the problem of the coupling between a factor-module $A / C_A(G)$ and a submodule $A(\omega RG)$, where $G$ is a group, $R$ is a ring, and $A$ is an $RG$-module. It is possible to consider $C_A (G)$ as an analog of the center of the group and the submodule $A(\omega RG)$ as an analog of the derived subgroup of the group.

### Expansion of weighted pseudoinverse matrices with singular weights into matrix power products and iteration methods

Deineka V. S., Galba E. F., Sergienko I. V.

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1269–1289

We obtain expansions of weighted pseudoinverse matrices with singular weights into matrix power products with negative exponents and arbitrary positive parameters. We show that the rate of convergence of these expansions depends on a parameter. On the basis of the proposed expansions, we construct and investigate iteration methods with quadratic rate of convergence for the calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions. Iteration methods for the calculation of weighted normal pseudosolutions are adapted to the solution of least-squares problems with constraints.

### Stability of a dynamical system with semi-Markov switchings under conditions of diffusion approximation

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1290–1296

We obtain sufficient conditions for the stability of a dynamical system in a semi-Markov medium under the conditions of diffusion approximation by using asymptotic properties of the compensation operator for a semi-Markov process and properties of the Lyapunov function for an averaged system.