# Volume 67, № 4, 2015

### Integral Functionals of the Gasser–Muller Regression Function

Arabidze D., Babilua P., Nadaraya E., Sokhadze G. A.

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 435-446

For integral functionals of the Gasser–Muller regression function and its derivatives, we consider the plug-in estimator. The consistency and asymptotic normality of the estimator are shown.

### Infinite Groups with Complemented Non-Abelian Subgroups

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 447-455

We obtain a description of locally finite *A* -groups with complemented non-Abelian subgroups.

### The Stone–Čech Compactification of Groupoids

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 456–466

Let $G$ be a discrete groupoid. Consider the Stone–Čech compactification $βG$ of $G$ . We extend the operation on the set of composable elements $G^{(2)}$ of $G$ to the operation * on a subset $(βG)^{(2)}$ of $βG×βG$ such that the triple $(βG, (βG)^{(2)}$, *) is a compact right topological semigroupoid.

### On the Theory of Prime Ends for Space Mappings

Kovtonyuk D. A., Ryazanov V. I.

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 467-479

We present a canonical representation of prime ends in regular domains and, on this basis, study the boundary behavior of the so-called lower *Q*-homeomorphisms obtained as a natural generalization of quasiconformal mappings. We establish a series of effective conditions imposed on a function *Q*(*x*) for the homeomorphic extension of given mappings with respect to prime ends in domains with regular boundaries. The developed theory is applicable, in particular, to mappings of the Orlicz–Sobolev classes and also to finitely bi-Lipschitz mappings, which can be regarded as a significant generalization of the well-known classes of isometric and quasiisometric mappings.

### On the Norm of Decomposable Subgroups in Locally Finite Groups

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 480-488

We study the relationships between the norm of decomposable subgroups and the norm of Abelian noncyclic subgroups in the class of locally finite groups. We also describe some properties of periodic locally nilpotent groups in which the norm of decomposable subgroups is a non-Dedekind norm.

### On The Boundary Behavior of Regular Solutions of the Degenerate Beltrami Equations

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 489-498

We study the boundary behavior of regular solutions to the degenerate Beltrami equations with constraints of the integral type imposed on the coefficient.

### On the Limit Behavior of a Sequence of Markov Processes Perturbed in a Neighborhood of the Singular Point

Pilipenko A. Yu., Prikhod’ko Yu. E.

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 499-516

We study the limit behavior of a sequence of Markov processes whose distributions outside any neighborhood of a “singular” point are attracted to a certain probability law. In any neighborhood of this point, the limit behavior can be irregular. As an example of application of the general result, we consider a symmetric random walk with unit jumps perturbed in the neighborhood of the origin. The invariance principle is established for the standard time and space scaling. The limit process is a skew Brownian motion.

### Differential Equations with Bistable Nonlinearity

Nizhnik L. P., Samoilenko A. M.

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 517-554

We study bounded solutions of differential equations with bistable nonlinearity by numerical and analytic methods. A simple mechanical model of circular pendulum with magnetic suspension in the upper equilibrium position is regarded as a bistable dynamical system simulating a supersensitive seismograph. We consider autonomous differential equations of the second and fourth orders with discontinuous piecewise linear and cubic nonlinearities. Bounded solutions with finitely many zeros, including solitonlike solutions with two zeros and kinklike solutions with several zeros are studied in detail. It is shown that, to within the sign and translation, the bounded solutions of the analyzed equations are uniquely determined by the integer numbers \( n=\left[\frac{d}{l}\right] \) where *d* is the distance between the roots of these solutions and *l* is a constant characterizing the intensity of nonlinearity. The existence of bounded chaotic solutions is established and the exact value of space entropy is found for periodic solutions.

### Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex

Qi Feng, Xi Bo-Yan, Zhang Tian-Yu

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 555-567

We establish some new Hermite–Hadamard-type inequalities for functions whose first derivatives are of convexity and apply these inequalities to construct inequalities for special means.

### The Energy of a Domain on the Surface

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 568-573

We compute the energy of a unit normal vector field on a Riemannian surface *M.* It is shown that the energy of the unit normal vector field is independent of the choice of an orthogonal basis in the tangent space. We also define the energy of the surface. Moreover, we compute the energy of spheres, domains on a right circular cylinder and torus, and of the general surfaces of revolution.

### Skewed-Gentle Algebras are Nodal

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 574-576

We prove that any gentle or skewed-gentle algebra is a nodal algebra of type $A$.