# Volume 59, № 10, 2007

### On Kolmogorov-type inequalities for functions defined on a semiaxis

Babenko V. F., Skorokhodov D. S.

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1299–1312

Necessary and sufficient conditions for the existence of a function from the class *S*^{ -}
with prescribed values of integral norms of three successive derivatives (generally speaking, of a fractional order) are obtained.

### Application of a separating transformation to estimates of inner radii of open sets

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1313–1321

We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets.

### Integro-differential systems with fuzzy noise

Plotnikov A. V., Vasilkovskaya V. S.

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1322–1330

For a controlled integro-differential equation with fuzzy noise, we introduce the notions of a fuzzy bundle of trajectories and a fuzzy reachability set and prove some properties of fuzzy bundles.

### On some groups all subgroups of which are nearly pronormal

Kurdachenko L. A., Russo A., Vincenzi G.

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1331–1338

A subgroup $H$ of a group $G$ is said to be nearly pronormal in $G$ if, for each subgroup $L$ of the group $G$ including H, the normalizer $N_L ( H)$ is contranormal in $L$. We prove that if $G$ is a (generalized) soluble group in which every subgroup is nearly pronormal, then all subgroups of $G$ are pronormal.

### Behavior of classical risk processes after ruin and a multivariate ruin function

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1339–1352

We establish relations for the distribution of functionals associated with the behavior of a classical risk process after ruin and a multivariate ruin function.

### Characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1353–1362

We give a characterization of the semilattice of idempotents of a finite-rank permutable inverse semigroup with zero.

### Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1363–1375

We establish asymptotic representations for unbounded solutions of nonlinear nonautonomous differential equations of the third order that are close, in a certain sense, to equations of the Emden-Fowler type.

### Generalized hybrid Mehler-Fock-type integral transformation of the first kind and its applications

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1376–1390

We introduce a generalized hybrid integral transformation of the Mehler-Fock type on a segment [0; *R*] with *n* conjugate points. We consider examples of application of this transformation to the solution of typical singular boundary-value problems for linear partial differential equations of the second order in piecewise-homogeneous media.

### On representations of a general solution in the theory of micropolar thermoelasticity without energy dissipation

Giordano P., Svanadze M., Zampoli V.

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1391–1398

In the present paper, the linear theory of micropolar thermoelasticity without energy dissipation is considered. This work is organized as follows: Section 2 is devoted to basic equations for micropolar thermoelastic materials, supposed to be isotropic and homogeneous, and to assumptions on constitutive constants. In Section 3, some theorems related to representations of a general solution are studied.

### On some new criteria for infinite differentiability of periodic functions

Serdyuk A. S., Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1399–1409

The set $\mathcal{D}^{\infty}$ of infinitely differentiable periodic functions is studied in terms of generalized $\overline{\psi}$-derivatives defined by a pair $\overline{\psi} = (\psi_1, \psi_2)$ of
sequences $\psi_1$ and $\psi_2$ .
It is shown that every function $f$ from the set $\mathcal{D}^{\infty}$ has at least one derivative whose parameters $\psi_1$ and $\psi_2$ decrease faster than any power function, and, at the same time, for an arbitrary
function $f \in \mathcal{D}^{\infty}$ different from a trigonometric polynomial, there exists a pair $\psi$ whose parameters $\psi_1$ and $\psi_2$ have the same rate of decrease and for which the $\overline{\psi}$-derivative no longer exists.

### On one criterion for analyticity of functions

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1410–1418

We prove a generalization of the well-known Dzyadyk theorem that gives an interesting geometric criterion for the analyticity of functions.

### Weighted sharp boundedness for multilinear commutators

Hong Xu, Liu Lanzhe, Zeng Jiasheng

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1419–1431

In this paper, the sharp estimates for some multilinear commutators related to certain sublinear integral operators are obtained. The operators include the Littlewood - Paley operator and the Marcinkiewicz operator. As application, we obtain the weighted $L^p (p > 1)$ inequalities and $L \log L$-type estimate for the multilinear commutators.

### Satisfiability of bright formulas

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1432–1435

We investigate one solvable subclass of quantified formulas in pure predicate calculus and obtain a necessary and sufficient condition for the satisfiability of formulas from this subclass.

### Weak bases of vector measures

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1436–1440

We solve the problem of representation of measures with values in a Banach space as the limits of weakly convergent sequences of vector measures whose basis is a given nonnegative measure.