Volume 61, № 1, 2009

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.

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.

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 $β ≥ 1$.

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.

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

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.

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.

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.

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.

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.

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.

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.

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

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.