# Volume 68, № 5, 2016

### On the optimal reconstruction of the convolution of $n$ functions according to the linear information

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 579-585

We determine the optimal linear information and the optimal method of its application to the recovery of convolution of $n$ functions on some convex and centrally symmetric classes of $2\pi$ -periodic functions.

### Verification of the hypotheses on the equality of densities of distributions

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

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 586-600

We construct new criteria for the verification of the hypotheses that $p \geq 2$ independent samplings have identical densities of distributions (homogeneity hypothesis) or identically defined densities of distributions (compatibility hypothesis). We determine the ultimate powers of the constructed criteria for some local “close” alternatives.

### $L_p$ -dual mixed affine surface areas

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 601-609

Lutwak proposed the notion of $L_p$-affine surface area according to the Lp-mixed volume. Recently,Wang and He introduced the concept of Lp-dual affine surface area combing with the $L_p$-dual mixed volume. In the article, we give the concept of $L_p$-dual mixed affine surface areas associated with the $L_p$-dual mixed quermassintegrals. Further, some inequalities for the $L_p$-dual mixed affine surface areas are obtained.

### Classification of finite nilsemigroups for which the inverse monoid of local automorphisms is permutable semigroup

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 610-624

A semigroup $S$ is called permutable if $\rho \circ \sigma = \sigma \circ \rho$ for any pair of congruences $\rho$, $\sigma$ on $S$. A local automorphism of the semigroup $S$ is defined as an isomorphism between two subsemigroups of this semigroup. The set of all local automorphisms of a semigroup $S$ with respect to an ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. In the proposed paper, we present a classification of all finite nilsemigroups for which the inverse monoid of local automorphisms is permutable. Полугруппа $S$ называется перестановочной, если для любой пары конгруэнций $\rho$, $\sigma$ на $S$ имеет место равенство $\rho \circ \sigma = \sigma \circ \rho$.

### Topological stability of the averagings of functions

Maksimenko S. I., Marunkevych O. V.

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 625-633

We present sufficient conditions for the topological stability of the averagings of piecewise smooth functions $f : R \rightarrow R$ with finitely many extrema with respect to discrete measures with finite supports.

### Kolmogorov widths of the anisotropic Besov classes of periodic functions of many variables

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 634-643

We establish exact-order estimates for the Kolmogorov widths of the anisotropic Besov classes of periodic functions of many variables in the spaces $L_q,\; 1 \leq q \leq \infty$.

### On the classification of functions integrable on a segment

Motornaya O. V., Motornyi V. P., Sedunova V. V.

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 642-656

The problem of classification of functions integrable on a segment is considered. Estimates for the integral moduli of continuity of functions from generalized Potapov’s classes are obtained.

### On the completely integrable calogero-type discretizations of Lax-integrable nonlinear dynamical systems and related coadjoint Markov-type orbits

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 657-664

The Calogero-type matrix discretization scheme is applied to THE construction of Lax-type integrable discretizations of one sufficiently wide class of nonlinear integrable dynamical systems on functional manifolds. Their Lie-algebraic structure and complete integrability related to the coadjoint orbits on the Markov coalgebras is discussed. It is shown that the set of conservation laws and the associated Poisson structure can be obtained as a byproduct of the proposed approach. Based on the quasirepresentation property of Lie algebras, the limiting procedure of finding nonlinear dynamical systems on the corresponding functional spaces is demonstrated.

### Boundary-value problem with mixed conditions for linear typeless partial differential equations

Ptashnik B. I., Repetylo S. M.

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 665-682

In the domain obtained as the Cartesian product of a segment $0 \leq t \leq T$ by a $p$-dimensional torus in variables $x_1, ..., x_p$, $p \geq 1$, we study the problem with mixed boundary conditions in the variable $t$ for general (no restrictions are imposed on the type) linear partial differential equations of high order with constant coefficients isotropic with respect to the order of differentiation for all independent variables. We establish conditions for the unique solvability of the problem in various functional spaces and construct its solution in the form of a series with respect to systems of orthogonal functions of the variables $x_1, ..., x_p$.

### On the removability of isolated singularities of Orlicz – Sobolev classes with branching

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 683-693

The local behavior of closed-open discrete mappings of the Orlicz – Sobolev classes in $R^n,\; n \geq 3$, is investigated. It is proved that the indicated mappings have continuous extensions to an isolated boundary point $x_0$ of the domain $D \setminus \{ x0\}$, whenever the $n - 1$ degree of its inner dilatation has FMO (finite mean oscillation) at this point and, in addition, the limit sets of $f$ at $x_0$ and $\partial D$ are disjoint. Another sufficient condition for the possibility of continuous extension can be formulated as a condition of divergence of a certain integral.

### Triangular models of commutative systems of linear operators close to unitary operators

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 694-711

Triangular models are constructed for commutative systems of linear bounded operators close to unitary operators. The construction of these models is based on the continuation of basic relations for the characteristic function along the general chain of invariant subspaces.

### A modified Newton method for a quadratic vector equation arising in markovian binary trees

Deng Liang-Jian, He Jun, Huang Ting-Zhu

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 712-720

We prove the existence of solution for the quadratic vector equation arising in Markovian binary trees. A modified Newton method for finding the minimal solution of the equation is presented. The monotone convergence of the modified Newton method is proved. Numerical experiments show the efficiency of our method.