# Volume 62, № 7, 2010

### Distribution of eigenvalues and trace formula for the Sturm–Liouville operator equation

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 867–877

We study the asymptotic distribution of eigenvalues of the problem generated by the Sturm–Liouville operator equation. A formula for the regularized trace of the corresponding operator is obtained.

### $A$-deformations of a surface with stationary lengths of $LGT$-lines

Bezkorovaina L. L., Vashpanova T. Yu.

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 878–884

We investigate infinitesimal areal deformations ($A$-deformations) of the first order under which the lengths of $LGT$-lines of a surface are preserved in the $E_3$ -space. We prove that any regular surface of the class $C^4$ of nonzero Gaussian curvature without umbilical points admits nontrivial $A$-deformations with stationary lengths of LGT-lines.

### Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function

Bodnar O. V., Zabolotskii N. V.

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 885–893

For entire functions whose zero counting functions are slowly increasing, we establish criteria for the regular growth of their logarithms of moduli and arguments in the metric of $L^p [0, 2π]$.

### Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 894–912

We consider the Neumann initial boundary-value problem for the equation $$u_t=\text{div}(u^{m−1}|Du|^{λ−1}Du)−u^p$$ in domains with noncompact boundary and with initial Dirac delta function. In the case of slow diffusion $(m + λ − 2 > 0)$ and critical absorption exponent $(p = m + λ − 1 +\frac{λ + 1}{N})$, we prove that the singularity at the point $(0, 0)$ is removable.

### Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 913–922

We supplement recent results on a class of Bernstein–Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein–Durrmeyer operators.

### Algebraic dependences of meromorphic mappings in several complex variables

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 923–936

We give some theorems on algebraic dependence of meromorphic mappings in several complex variables into complex projective spaces.

### On a mapping of a projective space into a sphere

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 937–944

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for $n = 2, 3$. For $n ≥ 4$, we prove that this estimate does not exceed 4. Several open questions are formulated.

### Estimate for Euclidean parameters of a mixture of two symmetric distributions

Maiboroda R. E., Suhakova O. V.

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 945–953

A sample from a mixture of two symmetric distributions is observed. The considered distributions differ only by a shift. Estimates are constructed by the method of estimating equations for parameters of mean locations and concentrations (mixing probabilities) of both components. We obtain conditions for the asymptotic normality of these estimates. The greatest lower bounds for the coefficients of dispersion of the estimates are determined.

### Period functions for $\mathcal{C}^0$- and $\mathcal{C}^1$-flows

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 954–967

Let $F:\; M×R→M$ be a continuous flow on a manifold $M$, let $V ⊂ M$ be an open subset, and let $ξ:\; V→R$ be a continuous function. We say that $ξ$ is a period function if $F(x, ξ(x)) = x$ for all $x ∈ V$. Recently, for any open connected subset $V ⊂ M$; the author has described the structure of the set $P(V)$ of all period functions on $V$. Assume that $F$ is topologically conjugate to some $\mathcal{C}^1$-flow. It is shown in this paper that, in this case, the period functions of $F$ satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.

### Approximation by de la Vallée-Poussin operators on the classes of functions locally summable on the real axis

Chaichenko S. O., Rukasov V. I.

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 968–978

For the least upper bounds of deviations of the de la Vallée-Poussin operators on the classes $\widehat{L}^{\psi}_{\beta}$ of rapidly vanishing functions $ψ$ in the metric of the spaces $\widehat{L}_p,\; 1 ≤ p ≤ ∞$, we establish upper estimates that are exact on some subsets of functions from $\widehat{L}_p$.

### Linear approximation methods and the best approximations of the Poisson integrals of functions from the classes $H_{ω_p}$ in the metrics of the spaces $L_p$

Serdyuk A. S., Sokolenko I. V.

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 979–996

We obtain upper estimates for the least upper bounds of approximations of the classes of Poisson integrals of functions from $H_{ω_p}$ for $1 ≤ p < ∞$ by a certain linear method $U_n^{*}$ in the metric of the space $L_p$. It is shown that the obtained estimates are asymptotically exact for $р = 1$: In addition, we determine the asymptotic equalities for the best approximations of the classes of Poisson integrals of functions from $H_{ω_1}$ in the metric of the space $L_1$ and show that, for these classes, the method $U_n^{*}$ is the best polynomial approximation method in a sense of strong asymptotic behavior.

### Automorphisms of a finitary factor power of an infinite symmetric group

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 997–1001

We consider a semigroup $FP^{+}_{\text{fin}}(\mathfrak{S}_{\text{fin}}(\mathbb{N}))$ defined as a finitary factor power of a finitary symmetric group of countable order. It is proved that all automorphisms of $FP^{+}_{\text{fin}}(\mathfrak{S}_{\text{fin}}(\mathbb{N}))$ are induced by permutations from $\mathfrak{S}_{\text{fin}}(\mathbb{N})$.

### On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities

Azhymbaev D. T., Tleubergenov M. I.

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 1002–1008

We construct the Lagrange equation, Hamilton equation, and Birkhoff equation on the basis of given properties of motion under random perturbations. It is assumed that random perturbation forces belong to the class of Wiener processes and that given properties of motion are independent of velocities. The obtained results are illustrated by an example of motion of an Earth satellite under the action of gravitational and aerodynamic forces.