# Volume 67, № 8, 2015

### Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. II

Bodenchuk V. V., Serdyuk A. S.

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1011-1018

It is shown that the lower bounds of the Kolmogorov widths $d_{2n}$ in the space $C$ established in the first part of our work for the function classes that can be represented in the form of convolutions of the kernels $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),\kern1em h>0,\kern1em \beta \in \mathbb{R},}$$ with functions $φ ⊥ 1$ from the unit ball in the space $L_{∞}$ coincide (for all $n ≥ nh$) with the best uniform approximations of these classes by trigonometric polynomials whose order does not exceed $n − 1$. As a result, we obtain the exact values of widths for the indicated classes of convolutions. Moreover, for all $n ≥ nh$, we determine the exact values of the Kolmogorov widths $d_{2n-1}$ in the space $L_1$ of classes of the convolutions of functions $φ ⊥ 1$ from the unit ball in the space $L_1$ with the kernel $H_{h,β}$.

### Potential Method in the Limit Problems for the Processes with Independent Increments

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1019-1029

We propose a new approach to the application of the Korolyuk potential method for the investigation of limit functionals for processes with independent increments. The formulas for the joint distribution of functionals related to crossing a level by the process are obtained and their asymptotic analysis is performed. The possibility of crossing a level by the process in a continuous way is also investigated.

### An Example of Neutrally Nonwandering Points for the Inner Mappings that are Not Neutrally Recurrent

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1030-1033

In the previous papers, the author offered a new theory of topological invariants for the dynamical systems formed by noninvertible inner mappings. These invariants are constructed by using the analogy between the trajectories of homeomorphisms and directions in the set of points with common iteration. In particular, we introduce the sets of neutrally recurrent and neutrally nonwandering points. We also present an example of the so-called “neutrally nonwandering but not neutrally recurrent” points, which shows that these sets do not coincide.

### On the Moment-Generating Functions of Extrema and Their Complements for Almost Semicontinuous Integer-Valued Poisson Processes on Markov Chains

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1034-1049

For an integer-valued compound Poisson process with geometrically distributed jumps of a certain sign [these processes are called almost upper (lower) semicontinuous] defined on a finite regular Markov chain, we establish relations (without projections) for the moment-generating functions of extrema and their complements. Unlike the relations obtained earlier in terms of projections, the proposed new relations for the moment-generating functions are determined by the inversion of the perturbed matrix cumulant function. These matrix relations are expressed via the moment-generating functions for the distributions of the corresponding jumps.

### On the Whittle Estimator of the Parameter of Spectral Density of Random Noise in the Nonlinear Regression Model

Ivanov O. V., Prykhod’ko V. V.

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1050-1067

We consider a nonlinear regression model with continuous time and establish the consistency and asymptotic normality of the Whittle minimum contrast estimator for the parameter of spectral density of stationary Gaussian noise.

### Problem of Optimal Control for Parabolic-Hyperbolic Equations with Nonlocal Point Boundary Conditions and Semidefinite Quality Criterion

Kapustyan V. E., Pyshnograev I. A.

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1068-1081

We consider a problem of optimal control for parabolic-hyperbolic equations with nonlocal boundary conditions and semidefinite quality criterion. The optimality conditions are constructed by reducing the problem to a sequence of one-dimensional problems, the optimal control is obtained in a closed form, and its convergence is proved. The form of the quality criterion is substantiated.

### A New Sufficient Condition for Belonging to the Algebra of Absolutely Convergent Fourier Integrals and Its Application to the Problems of Summability of Double Fourier Series

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1082-1096

We establish a general sufficient condition for the possibility of representation of functions $$f\left( \max \left\{\left|{x}_1\right|,\left|{x}_2\right|\right\}\right)$$ in the form of absolutely convergent double Fourier integrals and study the possibility of its application to various problems of summability of double Fourier series, in particular, by using the Marcinkiewicz–Riesz method.

### $p$-Regularity Theory. Tangent Cone Description in the Singular Case

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1097-1106

We present a new proof of the theorem which is one of the main results of the $p$-regularity theory. This gives us a detailed description of the structure of the zero set of a singular nonlinear mapping. We say that $F : X → Y$ is singular at some point $x_0$, where $X$ and $Y$ are Banach spaces, if Im $F′(x_0) ≠ Y$. Otherwise, the mapping $F$ is said to be regular.

### Certain Regularity of the Entropy Solutions for Nonlinear Parabolic Equations with Irregular Data

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1107-1121

We introduce new sets of functions different both from the space introduced in [Ph. Bénilan, L. Boccardo, T. Gallou?t, R. Gariepy, M. Pierre, and J. L. Vazquez, *Ann. Scuola Norm. Super. Pisa*, 22, No. 2, 241–273 (1995)] and from the Rakotoson *T* -set introduced in [J. M. Rakotoson, *Different. Integr. Equat.*, 6, No. 1, 27–36 (1993); *J. Different. Equat.*, 111, No. 2, 458–471 (1994)]. In the new framework of sets, we present some summability results for the entropy solutions of nonlinear parabolic equations.

### Existence of the Category $DTC_2 (K)$ Equivalent to the Given Category $KAC_2$

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1122–1133

For a given category $KAC_2$ , the present paper deals with the existence problem for the category $DTC_2 (K)$, which is equivalent to $KAC_2$ , where $DTC_2 (K)$ is the category whose objects are simple closed $K$-curves with even number $l$ of elements in $Z^n,\; l ≠ 6$, and morphisms are (digitally) $K$-continuous maps, and $KAC_2$ is a category whose objects are simple closed $A$-curves and morphisms are $A$-maps. To address this issue, the paper starts from the category denoted by $KAC_1$ whose objects are connected $nD$ Khalimsky topological subspaces with Khalimsky adjacency and morphisms are $A$-maps in [S. E. Han and A. Sostak, Comput. Appl. Math., 32, 521–536 (2013)]. Based on this approach, in $KAC_1$ the paper proposes the notions of $A$-homotopy and $A$-homotopy equivalence and classifies the spaces in $KAC_1$ or $KAC_2$ in terms of the $A$-homotopy equivalence. Finally, the paper proves that, for Sa given category $KAC_2$, there is $DTC_2 (K)$, which is equivalent to $KAC_2$.

### On Bijective Continuous Images of Absolute Null Sets

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1134-1138

The images of absolute null sets (spaces) under bijective continuous mappings are studied. It is shown that, in general, these images do not possess regularity properties from the viewpoint of topological measure theory.

### Approximating Characteristics of the Classes $L_{β,p}^{ψ}$ of Periodic Functions in the Space $L_q$

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1139-1150

We obtain the exact-order estimates of the best $m$-term and orthogonal trigonometric approximations and establish the order of trigonometric widths for the classes $L_{β,p}^{ψ}$ in the space $L_q$ for some relations between the parameters $p$ and $q$.

### Volodymyr Semenovych Korolyuk (on his 90th birthday)

Bratiichuk N. S., Gusak D. V., Kovalenko I. N., Lukovsky I. O., Makarov V. L., Samoilenko A. M., Samoilenko I. V.

Ukr. Mat. Zh. - 2015. - 67, № 8. - pp. 1151-1152