# Volume 68, № 3, 2016

### Local times of self-intersection

Dorogovtsev A. A., Izyumtseva O. L.

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 290-340

This survey article is devoted to the local times of self-intersection as the most important geometric characteristics of random processes. The trajectories of random processes are, as a rule, very nonsmooth curves. This is why to characterize the geometric shape of the trajectory it is impossible to use the methods of differential geometry. Instead of this, one can consider the local times of self-intersection showing how much time the process stays in “small” vicinities of its self-crossing points. In our paper, we try to describe the contemporary state of the theory of local times of self-intersection for Gaussian and related processes. Different approaches to the definition, investigation, and application of the local times of self-intersection are considered.

### Robust feedback synthesis for the canonical system

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 341-356

The paper deals with the problems of global and local robust feedback synthesis of bounded control for a system with unknown bounded perturbation. Our approach is based on the method of controllability function proposed by V. I. Korobov. The ranges of perturbations are found from the condition that the total derivative of the controllability function caused by the perturbed system must be negative. We determine the largest segment of variation of the perturbation and construct a positional control that steers an arbitrary initial point to the origin within a finite period of time. The length of this period is estimated both from below and from above. A two-dimensional system is considered as an example.

### Conditions for a submanifold $F^n$ from $E^{n + p}$ to lie in $E^{2n − 1}$

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 357-362

We consider a submanifold $F^n$ with $n$ principal directions in the space $E^{n + p}$, where $p \geq n_1$.

### Boundary-value problems for the Helmholts equation in domains of the complex plane

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 363-376

By using the conformal mappings of plane with elliptic hole and a plane with cross-shaped hole into the outside of the circle, we construct systems of functions playing the role of bases in the spaces of the functions analytic in these domains. The Faber polynomials are biorthogonal with the basis functions. We construct the solutions of the Helmholtz equation in the plane with holes whose boundary values coincide with the boundary values of analytic functions represented in the form of series in these bases.

### Semigroups of endotopisms of efficient and connected relations

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 377-387

For binary relations satisfying the conditions of efficiency and connectedness, it is shown that the semigroup of all endotopisms of any effective and connected binary relation characterizes this relation to within an isotopism or an antiisotopism.

### Best trigonometric and bilinear approximations for the classes of $(ψ, β)$ -differentiable periodic functions

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 386-399

We establish the exact-order estimates for the best $m$-term trigonometric approximations of the classes $L^{\psi}_{\beta, 1}$ in the space $L_q,\; 2 < q < \infty$. We also determine the exact orders of the best bilinear approximations for the classes of functions of two variables generated by functions of a single variable from the class$L^{\psi}_{\beta, p}$ by the shifts of the argument in the space $L_{q_1,q_2},\; 1 \leq q_1,\; q_2 \leq \infty$.

### A note on similarity to contraction for stable $2 \times 2$ companion matrices

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 400-407

We consider companion matrices of size $2 \times 2$ with general complex spectra satisfying a root condition with respect to the closed complex unit circle or the closed left complex half plane. For both cases, smooth and naturally conditioned basis transformations are constructed such that the resulting, transformed matrix is contractive or dissipative, respectively, with respect to the $\ell_2$-norm.

### Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary value problems on time scales

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 408-422

In the paper, the four functionals fixed-point theorem is used to study the existence of positive solutions for nonlinear thirdorder $m$-point impulsive boundary-value problems on time scales. As an application, we give an example demonstrating our results.

### Existence and nonexistence of solutions of reaction-diffusion equation with Robin boundary condition

Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 423-432

We investigate the long-time behavior of the reaction-diffusion equation, which has a nonlinearity of polynomial growth of any order, with Robin boundary condition. Sufficient conditions are obtained for the solutions of the problem to be bounded or approaching infinity at a finite time.