# Volume 61, № 1, 2009

### Integral group ring of Rudvalis simple group

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 3-13

Using the Luthar–Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group *Ru *. As a consequence, for this group we confirm the Kimmerle conjecture on prime graphs.

### Algebraic-geometric operators and Galois differential theory

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 14-27

We show that, by using the Galois differential theory, one can substantially improve the description of algebraic-geometric operators. In particular, we give a complete description of all elementary algebraic-geometric operators, present simple relations for the construction of all second-order operators of this type, and give a criterion for testing the algebraic-geometric properties of a linear differential operator with meromorphic coefficients.

### Inverse problem for the strongly degenerate heat equation in a domain with free boundary

Hryntsiv N. M., Ivanchov N. I.

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 28-43

In a domain with free boundary, we establish conditions for the existence and uniqueness of a solution of the inverse problem of finding the time-dependent coefficient of heat conductivity. We study the case of strong degeneration where the unknown coefficient tends to zero as $t → +0$ as a power function $t^{β}$, where $β ≥ 1$.

### On one class of modules over integer group rings of locally solvable groups

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 44-51

We study a $Z G$-module $A$ in the case where the group $G$ is locally solvable and satisfies the condition min–naz and its cocentralizer in A is not an Artinian $Z$-module. We prove that the group G is solvable under the conditions indicated above. The structure of the group $G$ is studied in detail in the case where this group is not a Chernikov group.

### Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 52-60

We describe the structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental.

### On one extremal problem of Pompeiu sets

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 61-72

We determine upper bounds for the least radius of a ball in which a given set is a Pompeiu set (the set considered is a half right circular cone). The obtained estimates significantly improve known results.

### Approximation of conjugate differentiable functions by their Abel–Poisson integrals

Kharkevych Yu. I., Zhyhallo K. M.

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 73-82

We obtain the exact values of upper bounds of approximations of classes of periodic conjugate differentiable functions by their Abel–Poisson integrals in uniform and integral metrics.

### Full measure of a set of singular continuous measures

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 83-91

On the space of structurally similar measures, we construct a nontrivial measure **m** such that the subclass of all purely singular continuous measures is a set of full **m**-measure.

### Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 92-98

We investigate exact constants in Jackson-type inequalities in the space $L_2$ for the approximation of functions on an axis by the subspace of entire functions of exponential type.

### Exact constants in Jackson-type inequalities for $L_2$-approximation on an axis

Doronin V. G., Ligun A. A., Serdyuk A. S., Shydlich A. L.

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 92-98

We investigate exact constants in Jackson-type inequalities in the space $L_2$ for the approximation of functions on an axis by the subspace of entire functions of exponential type.

### On Γ-convergence of integral functionals defined on various weighted Sobolev spaces

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 99-115

We consider weighted Sobolev spaces correlated with a sequence of $n$-dimensional domains. We prove a theorem on the choice of a subsequence $Γ$-convergent to an integral functional defined on a “limit” weighted Sobolev space from a sequence of integral functionals defined on the spaces indicated.

### Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for Q-mappings

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 116-126

We prove that an open discrete *Q*-mapping \( f:D \to \overline {{\mathbb{R}^n}} \) has a continuous extension to an isolated boundary point if the function *Q*(*x*) has finite mean oscillation or logarithmic singularities of order at most *n* – 1 at this point. Moreover, the extended mapping is open and discrete and is a *Q*-mapping. As a corollary, we obtain an analog of the well-known Sokhotskii–Weierstrass theorem on *Q*-mappings. In particular, we prove that an open discrete *Q*-mapping takes any value infinitely many times in the neighborhood of an essential singularity, except, possibly, for a certain set of capacity zero.

### Almost critical branching processes and limit theorems

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 127-133

We study almost critical branching processes with infinitely increasing immigration and prove functional limit theorems for these processes.

### Classification of topologically conjugate affine mappings

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 134-139

We consider affine mappings from $ℝ^n$ into $ℝ^n, n ≥ 1$. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding linear mapping. We give a classification, up to topological conjugacy, for affine mappings from $ℝ$ into $ℝ$ and also for affine mappings from $ℝ^n$ into $ℝ^n, n > 1$, having at least one fixed point and the nonperiodic linear part.