# Volume 68, № 1, 2016

### Rings whose nonsingular modules have projective covers

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 3-13

We determine rings $R$ with the property that all (finitely generated) nonsingular right $R$-modules have projective covers. These are just the rings with $t$-supplemented (finitely generated) free right modules. Hence, they are called right (finitely) $\Sigma -t$-supplemented. It is also shown that a ring $R$ for which every cyclic nonsingular right $R$-module has a projective cover is exactly a right $t$-supplemented ring. It is proved that, for a continuous ring $R$, the property of right $\Sigma -t$-supplementedness is equivalent to the semisimplicity of $R/Z_2(R_R)$, while the property of being right finitely $\Sigma -t$-supplemented is equivalent to the right self-injectivity of $R/Z_2(R_R)$. Moreover, for a von Neumann regular ring $R/Z_2(R_R)$, the properties of being right $\Sigma -t$-supplemented, right finitely \Sigma -t-supplemented, and right t-supplemented are equivalent to the semisimplicity, right self-injectivity, and right continuity of $R/Z_2(R_R)$, respectively.

### On the solution of the problem of stochastic stability of the integral manifold by the Lyapunov’s second method

Tleubergenov M. I., Vasilina G. K.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 14-27

By using the method of Lyapunov functions, we establish sufficient conditions of stability and asymptotic stability in probability for the integral manifold of the Itˆo differential equations in the presence of random perturbations from the class of processes with independent increments. Theorems on the stochastic stability of the analytically given integral manifold of differential equations are proved in the first approximation and under the permanent action of small (in the mean) random perturbations.

### Functions with nondegenerate critical points on the boundary of the surface

Hladysh B. I., Prishlyak A. O.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 28-37

We prove an analog of the Morse theorem in the case where the critical point belongs to the boundary of an $n$-dimensional manifold and find the least number of critical points for the Morse functions defined on the surfaces whose critical points coincide with the critical points of their restriction to boundary.

### Degenerate Backlund transformation

Gor'kavyi V. A., Nevmerzhitskaya E. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 38-51

A concept of degenerate B¨acklund transformation is introduced for two-dimensional surfaces in many-dimensional Euclidean spaces. It is shown that if a surface in $R^n, n \geq 4$, admits a degenerate B¨acklund transformation, then this surface is pseudospherical, i.e., its Gauss curvature is constant and negative. The complete classification of pseudospherical surfaces in $R^n, n \geq 4$ that admit degenerate Bianchi transformations is obtained. Moreover, we also obtain a complete classification of pseudospherical surfaces in $R^n, n \geq 4$, admitting degenerate Backlund transformations into straight lines.

### Finite groups with given systems of $K-\mathfrak{U}$-subnormal subgroups

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 52-63

A subgroup $H$ of a finite group $G$ is called $\mathfrak{U}$-subnormal in Kegel’s sense or $K-\mathfrak{U}$-subnormal in $G$ if there exists a chain of subgroups $H = H_0 \leq H_1 \leq . . . \leq H_t = G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})H_i$ is supersoluble for any $i = 1, . . . , t$. We describe finite groups for which every 2-maximal or every 3-maximal subgroup is $K-\mathfrak{U}$-subnormal.

### Hahn-Jordan decomposition as an equilibrium state of the conflict system

Koshmanenko V. D., Petrenko S. M.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 64-77

The notion of conflict system is introduced in terms of couples of probability measures. We construct several models of conflict systems and show that every trajectory with initial state given by a couple of measures $\mu, \nu$ converges to an equilibrium state specified by the normalized components $\mu_+, \nu_+$ of the classical Hahn – Jordan decomposition of the signed measure $\omega = \mu - \nu$.

### Wiman-type inequality for functions analytic in a polydisc

Kurylyak A. O., Shapovalovs’ka L. O., Skaskiv O. B.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 78-86

We prove an analog of Wiman-type inequality for analytic functions in a polydisc $\mathbb{D}^p = \{z \in \mathbb{C}^p : |z_j| < 1,\; j \in \{ 1, . . . ,p\} \} , p \in N.$ The obtained inequality is sharp.

### Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 87-105

We discuss the Norlund means of quadratic partial sums of the Walsh – Kaczmarz – Fourier series of a function in $L_p$. We investigate the rate of approximation by this means, in particular, in $\text{Lip}(\alpha , p)$, where $\alpha > 0$ and $1 \leq p \leq \infty$. For $p = \infty$, by $L_p$, we mean $C$, i.e., the collection of continuous functions. Our main theorem states that the approximation behavior of this two-dimensional Walsh – Kaczmarz –Norlund means is as good as the approximation behavior of the one-dimensional Walsh– and Walsh – Kaczmarz –Norlund means. Earlier results for one-dimensional N¨orlund means of the Walsh – Fourier series was given by M´oricz and Siddiqi [J. Approxim. Theory. – 1992. – 70, № 3. – P. 375 – 389] and Fridli, Manchanda and Siddiqi [Acta Sci. Math. (Szeged). – 2008. – 74. – P. 593 – 608], for one-dimensional Walsh – Kaczmarz –N¨orlund means by the author [Georg. Math. J. –2011. – 18. – P. 147 – 162] and for two-dimensional trigonometric system by M´oricz and Rhoades [J. Approxim. Theory. – 1987. – 50. – P. 341 – 358].

### Normality of the Orlicz - Sobolev classes

Ryazanov V. I., Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 106-116

We establish a series of new criteria of equicontinuity and, hence, normality of the mappings of Orlicz – Sobolev classes in terms of inner dilatations.

### On one class of quaternionic mappings

Kuz’menko T. S., Shpakovskii V. S.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 117-130

We consider a new class of quaternionic mappings associated with spatial partial differential equations. We obtain a description of all mappings from this class by using four analytic functions of complex variable.

### The $D&P$ Shapley value: a weighted extension

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 131-141

First, we propose a weighted extension of the D&P Shapley value and then study several equivalences among the potentializability and some properties. On the basis of these equivalences and consistency, two axiomatizations are also proposed.

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk A. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144