# Volume 65, № 11, 2013

### On Infinite Groups with Complemented Non-Abelian Subgroups

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1443–1455

We present the description of locally finite groups containing at least one non-Abelian Sylow subgroup in which all non-Abelian subgroups are complemented.

### Schreier Graphs for a Self-Similar Action of the Heisenberg Group

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1456–1462

We construct a faithful self-similar action of the discrete Heisenberg group with the following properties: This action is self-replicating, finite-state, level-transitive, and noncontracting. Moreover, there exist orbital Schreier graphs of action on the boundary of the tree with different degrees of growth.

### Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1463–1475

By using the coincidence degree theory, we study the existence of solutions for a coupled system of fractional differential equations with periodic boundary conditions. A new result on the existence of solutions of the indicated fractional boundary-value problem is obtained.

### Almost MGP-Injective Rings

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1476–1481

A ring *R* is called right almost MGP-injective (or AMGP-injective) if, for any 0 ≠ *a* ∈ *R*, there exists an element *b* ∈ *R* such that *ab* = *ba* ≠ 0 and any right *R*-monomorphism from *abR* to *R* can be extended to an endomorphism of *R*. In the paper, several properties of these rings are establshed and some interesting results are obtained. By using the concept of right AMGP-injective rings, we present some new characterizations of QF-rings, semisimple Artinian rings, and simple Artinian rings.

### On Preservation of the Order of Flattening by an Induced Diffeomorphism

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1482–1497

We consider the structure of a smooth curve from the viewpoint of the concept of flattening and establish conditions under which an *r*-geodesic curve of the base manifold is the projection of the *r*-geodesic curve in a tangent bundle of the second order. The necessary and sufficient condition under which a 2-geodesic diffeomorphism of affine-connected spaces induces a 2-geodesic diffeomorphism of tangent bundles of the second order is established.

### On Preservation of the Invariant torus for Multifrequency Systems

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1498–1505

We establish new conditions for the preservation of an asymptotically stable invariant toroidal manifold of the linear extension of a dynamical system on a torus under small perturbations in a set of nonwandering points. The proposed approach is applied to the investigation of the existence and stability of the invariant tori of linear extensions of the dynamical systems with simple structures of limit sets and recurrent trajectories.

### Codecomposition of a Transformation Semigroup

Hosseini A., Sabbaghan M., Shirazi F. Ayatollah Zadeh

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1506–1514

The present paper deals with the concept of “codecomposition” of a transformation semigroup interacting with the phase semigroup. In this way, we distinguish new classes of transformation semigroups with meaningful relations, e.g., we show the class of all distal transformation semigroups *⊂,* the class of all transformation semigroups decomposable into distal semigroups *⊂,* and the class of all transformation semigroups (here, *⊂* is strict inclusion).

### Two-Phase Solitonlike Solutions of the Cauchy Problem for a Singularly Perturbed Korteweg-De-Vries Equation with Variable Coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1515–1530

We describe a set of initial conditions for which the Cauchy problem for a singularly perturbed Korteweg–de-Vries equation with variable coefficients has an asymptotic two-phase solitonlike solution. The notion of the manifold of initial data of the Cauchy problem for which this solution exists is proposed.

### Determination of the Lowest Coefficient for a One-Dimensional Parabolic Equation in a Domain with Free Boundary

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1531–1549

We establish conditions for the unique solvability of the inverse problem of finding the lower coefficient with two unknown time-dependent parameters in a one-dimensional parabolic equation with integral overdetermination conditions in a domain with free boundary.

### On the 100th birthday of O. Yu. Ishlinskii

Lukovsky I. O., Samoilenko A. M., Storozhenko V. A.

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1550-1554

### On Finite Groups with Permutable Generalized Subnormal Subgroups

Semenchuk V. N., Velesnitskii V. F.

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1555–1559

We study the Kegel–Shemetkov problem of finding the classes of finite groups \( {\mathfrak F} \) such that, in any finite group, the product of permutable \( {\mathfrak F} \) -subnormal subgroups is a \( {\mathfrak F} \) -subnormal subgroup.

### ON *M*-Projectively Flat LP-Sasakian Manifolds

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1560–1566

In the present paper, we study the nature of LP-Sasakian manifolds admitting the *M*-projective curvature tensor. It is examined whether this manifold satisfies the condition *W*(*X, Y* )*.R* = 0*.* Moreover, it is proved that, in the *M*-projectively flat LP-Sasakian manifolds, the conditions *R*(*X, Y* )*.R* = 0 and *R*(*X, Y* )*.S* = 0 are satisfied. In the last part of the paper, an *M*-projectively flat space-time is introduced, and some properties of this space are obtained.

### ON the Openness of Functors of *k*-Nonexpanding and Weakly Additive Functionals

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1567–1574

We study the property of openness of the functors for *k* -nonexpanding and weakly additive functionals. In particular, it is shown that these functors preserve the openness of mappings between finite compact sets but they are not open.

### Asymptotic Behavior of a Counting Process in the Maximum scheme

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1575–1579

We determine the exact asymptotic behavior of the logarithm of a counting process in the maximum scheme.

### Some Properties of Multivalent Functions Associated with a Certain Operator

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1580–1584

We obtain some subordination and superordination results involving a new operator. By means of the new introduced operator \( \mathrm{C}_{p,n}^{\lambda }(a,c)f(z), \) for some multivalent functions in the open unit disc, we establish the differential sandwich theorem.