# Volume 62, № 5, 2010

### Some remarks concerning Riemannian extensions

Aslanci S., Kazimova S., Salimov A. A.

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 579–590

We study some properties of Riemannian extensions in cotangent bundles with the help of adapted frames.

### Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 591–603

We consider the homogeneous Dirichlet problem in the unit disk $K ⊂ R^2$ for a general typeless differential equation of any even order $2m,\; m ≥ 2$, with constant complex coefficients whose characteristic equation has multiple roots $± i$. For each value of multiplicity of the roots $i$ and $–i$, we either formulate criteria of the nontrivial solvability of the problem or prove that the analyzed problem possesses solely the trivial solution. A similar result generalizes the well-known Bitsadze examples to the case of typeless equations of any even order.

### On sufficient conditions for the existence of bounded solutions of inhomogeneous linear extensions of dynamical systems

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 604–611

We study sufficient conditions for the existence of bounded solutions of linear extensions of the dynamical systems.

### Semiperfect ipri-rings and right Bézout rings

Dokuchaev M. A., Gubareni N. M., Kirichenko V. V.

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 612–624

We present a survey of some results on ipri-rings and right Bézout rings. All these rings are generalizations of principal ideal rings. From the general point of view, decomposition theorems are proved for semiperfect ipri-rings and right Bézout rings.

### Linear system of differential equations with turning point

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 625–642

A system of linear differential equations with small parameter as a coefficient of a part of derivatives is reduced to the canonical form and the properties of the transformation matrix are investigated.

### Problem of large deviations for Markov random evolutions with independent increments in the scheme of asymptotically small diffusion

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 643–650

The problem of large deviations for random evolutions with independent increments is solved in the scheme of asymptotically small diffusion by passing to the limit in the nonlinear (exponential) generator of semigroups by using the solution of the problem of singular perturbation for a reducibly invertible operator.

### Kernel of a map of a shift along the orbits of continuous flows

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 651–659

Let $F: M × R → M$ be a continuous flow on a topological manifold $M$. For every subset $V ⊂ M$, we denote by $P(V)$ the set of all continuous functions $ξ: V → R$ such that $F(x,ξ(x)) = x$ for all $x ∈ V$. These functions vanish at nonperiodic points of the flow, while their values at periodic points are integer multiples of the corresponding periods (in general, not minimal). In this paper, the structure of $P(V)$ is described for an arbitrary connected open subset $V ⊂ M$.

### Summation of *p*-Faber series by the Abel–poisson method in the integral metric

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 660–673

We establish conditions on the boundary \( \Gamma \) of a bounded simply connected domain \( \Omega \subset \mathbb{C} \) under which the *p*-Faber series of an arbitrary function from the Smirnov space \( {E_p}\left( \Omega \right),1 \leqslant p < \infty \), can be summed by the Abel–Poisson method on the boundary of the domain up to the limit values of the function itself in the metric of the space \( {L_p}\left( \Gamma \right) \).

### Convergence of a semi-Markov process and an accompanying

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 674–681

We propose an approach to the proof of the weak convergence of a semi-Markov process to a Markov process under certain conditions imposed on local characteristics of the semi-Markov process.

### Boundary behavior of ring *Q*-homeomorphisms in metric spaces

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 682–689

We investigate the problem of extension of so-called ring *Q*-homeomorphisms between domains in metric spaces with measures to the boundary. We establish conditions for the function *Q*(*x*) and the boundary of the domain under which any ring *Q*-homeomorphism admits a continuous or a homeomorphic extension to the boundary. The results are applicable, in particular, to Riemannian manifolds, Löwner spaces, and Carnot and Heisenberg groups.

### Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 690–697

We use the scheme of the classic least-squares method for the construction of an approximate pseudosolution of a linear ill-posed boundary-value problem with pulse action for a system of ordinary differential equations in the critical case. The pseudosolution obtained is represented in the form of partial sums of a generalized Fourier series.

### Polynomial extensions of generalized quasi-Baer rings

Ghalandarzadeh S., Javadi H. S., Khoramdel M.

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 698–701

In this paper, we consider the behavior of polynomial rings over generalized quasi-Baer rings and show that the generalized quasi-Baer condition on a ring *R* is preserved by many polynomial extensions.

### On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 702–707

We study nonuniform ergodic averages of the Kozlov – Treshchev type for operator semigroups and obtain estimates for the corresponding maximal functions.

### Properties of reciprocal derivatives

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 708–713

New properties of reciprocal derivatives are established.

### On the convergence of solutions of certain inhomogeneous fourth-order differential equations

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 714–721

The main purpose of this paper is to give sufficient conditions for the convergence of solutions of a certain class of fourth-order nonlinear differential equations using Lyapunov’s second method. Nonlinear functions involved are not necessarily differentiable, but a certain incrementary ratio for a function *h* lies in a closed subinterval of the Routh–Hurwitz interval.