# Volume 63, № 10, 2011

### Boundary behavior of ring *Q*-homeomorphisms on Riemannian manifolds

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1299-1313

We study the problems of a continuous and homeomorphic extension of so-called ring $Q$-homeomorphisms between domains on Riemannian manifolds to the boundary. We establish conditions for a function $Q(x)$ and the boundaries of domains under which every ring $Q$-homeomorphism admits a continuous or homeomorphic extension to the boundary. This theory can be applied, in particular, to Sobolev classes.

### On one generalization of modular subgroups

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1314-1325

We study the influence of generalized modular subgroups on the structure of finite groups.

### On some qualitative properties of monotone linear extensions of dynamical systems

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1326-1335

We study monotone linear extensions of dynamical systems. The problem of existence of invariant manifolds and exponential separation is investigated for linear extensions on vector bundles that preserve the order structure. We also study the relationship between the monotonicity of linear extensions and the existence of bounded solutions of inhomogeneous linear extensions (weak regularity, quasiregularity).

### A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1336-1351

We study strong limit theorems for a bivariate function sequence of an nonhomogeneous Markov chain indexed by a generalized Bethe tree on a generalized random selection system by constructing a nonnegative martingale. As corollaries, we generalize results of Yang and Ye and obtain some limit theorems for frequencies of states, ordered couples of states, and the conditional expectation of a bivariate function on Cayley tree.

### Symmetry analysis and exact solutions of one class of (1 + 3)-dimensional boundary-value problems of the Stefan type

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1352-1359

We present the group classification of one class of (1 + 3)-dimensional nonlinear boundary-value problems of the Stefan type that model the processes of melting and evaporation of metals. The results obtained are used for the construction of the exact solution of one boundary-value problem from the class under study.

### On the unconditional almost-everywhere convergence of general orthogonal series

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1360-1367

The Orlicz and Tandori theorems on the unconditional almost-everywhere convergence, with respect to Lebesgue measure, of real orthogonal series defined on the interval (0; 1) are extended to general complex orthogonal series defined on an arbitrary measure space.

### Analogs of the Ikoma?Schwartz lemma and Liouville theorem for mappings with unbounded characteristic

Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1368-1380

In the present paper, we obtain results on the local behavior of open discrete mappings $f:\;D \rightarrow \mathbb{R}^n, \quad n \geq 2,$, that satisfy certain conditions related to the distortion of capacities of condensers. It is shown that, in an infinitesimal neighborhood of zero, the indicated mapping cannot grow faster than an integral of a special type that corresponds to the distortion of the capacity under this mapping, which is an analog of the well-known growth estimate of Ikoma proved for quasiconformal mappings of the unit ball into itself and of the classical Schwartz lemma for analytic functions. For mappings of the indicated type, we also obtain an analogue of the well-known Liouville theorem for analytic functions.

### On closeness of the sum of *n* subspaces of a Hilbert space

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1381-1425

We give necessary and sufficient conditions for the sum of subspaces $H_1,..., H_n,, \quad n \geq 2,$ of a Hilbert space $H$ to be a subspace and present various properties of $n$-tuples of subspaces with closed sum.

### Some results on MP-injectivity and MGP-injectivity of rings and modules

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1426-1433

We study MP-injective rings and MGP-injective rings satisfying some additional conditions. Using the concepts of MP-injectivity and MGP-injectivity of rings and modules, we present some new characterizations of QF-rings, semisimple Artinian rings, strongly regular rings, and simple Artinian rings.

### Exact Jackson - Stechkin-type inequalities for 2π -periodic functions in *L *_{2 } and widths of some classes of functions

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1434-1440

We consider the problem of finding exact inequalities for the best approximations of periodical differentiable
functions by trigonometric polynomials and the *m *-order moduli of continuity in the space *L *_{2 } and present
their applications. For some classes of functions defined by the indicated moduli of continuity, we calculate
the exact values of n-widths in the space *L *_{2 }.