2018
Том 70
№ 8

# Volume 51, № 1, 1999

Article (Ukrainian)

### On differentiability of mappings of finite-dimensional domains into Banach spaces

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 3–11

The well-known Stepanov criterion of the differentiability (approximate differentiability) of real functions is generalized to mappings of subsets of $R^n$ n into Banach spaces satisfying the Rieffel sharpness condition, in particular, reflexive Banach spaces. For Banach spaces that do not satisfy the Rieffel sharpness condition, this criterion is not true.

Article (Ukrainian)

### On the Levy-Baxter theorems for shot-noise fields. II

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 12–31

We establish sufficient conditions under which shot-noise fields with a response function of a certain form possess the Levy-Baxter property on an increasing parametric set.

Article (Ukrainian)

### Asymptotics of the logarithmic derivative of an entire function of zero order

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 32–40

We find asymptotic formulas for the logarithmic derivative of a zero-order entire functionf whose zeros have an angular density with respect to the comparison function $v(r) = r^{\lambda(r)}$, where $λ(r)$ is the zero proximate order of the counting function $n(r)$ of zeros of $f$.

Article (Russian)

### Decomposability of topological groups

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 41–47

We prove that every countable Abelian group with finitely many second-order elements can be decomposed into countably many subsets that are dense in any nondiscrete group topology.

Article (Ukrainian)

### Basic boundary-value problems for one equation with fractional derivatives

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 48–59

We prove some properties of solutions of an equation $\cfrac{\partial^{2\alpha}u}{\partial x_1^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_2^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_3^{2\alpha}} = 0, \quad \alpha \in \left( \cfrac 12\, ; 1 \right ]$, in a domain $\Omega \subset R^3$ which are similar to the properties of harmonic functions. By using the potential method, we investigate principal boundary-value problems for this equation.

Article (Ukrainian)

### Local properties of gaussian random fields on compact symmetric spaces and theorems of the Jackson-Bernstein type

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 60–68

We consider local properties of sample functions of Gaussian isotropic random fields on the compact Riemann symmetric spaces $\mathcal{M}$ of rank one. We give conditions under which the sample functions of a field almost surely possess logarithmic and power modulus of continuity. As a corollary, we prove the Bernshtein-type theorem for optimal approximations of functions of this sort by harmonic polynomials in the metric of space $L_2(\mathcal{M})$. We use the Jackson-Bernshtein-type theorems to obtain sufficient conditions of almost surely belonging of the sample functions of a field to classes of functions associated with Riesz and Cesaro means.

Article (Russian)

### On the order of growth of solutions of algebraic differential equations

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 69–77

Assume that $f$ is an integer transcendental solution of the differential equation $P_n(z,f,f′)=P_{n−1}(z,f,f′,...,f(p))),$ $P_n, P_{n−1}$ are polynomials in all the variables, the order of $P_n$ with respect to $f$ and $f′$ is equal to $n$, and the order of $P_{n−1}$ with respect to $f, f′, ... f(p)$ is at most $n−1$. We prove that the order $ρ$ of growth of $f$ satisfies the relation $12 ≤ ρ < ∞$. We also prove that if $ρ = 1/2$, then, for some real $η$, in the domain $\{z: η < \arg z < η+2π\} E∗$, where $E∗$ is some set of disks with the finite sum of radii, the estimate $\ln f(z) = z^{1/2}(β+o(1)),\; β ∈ C$, is true (here, $z=\re i^{φ}, r ≥ r(φ) ≥ 0$, and if $z = \text{re } i^{φ}, r ≥ r(φ) ≥ 0$ and, on a ray $\{z: \arg z=η\}$, the relation $\ln |f(\text{re } i^{η})| = o(r^{1/2}), \; r → +∞,\; r > 0, r \bar \in \Delta$, holds, where $Δ$ is some set on the semiaxis $r > 0$ with mes $Δ < ∞$.

Article (Ukrainian)

### Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 78–97

We present a spatially two-dimensional generalization of the hierarchy of Kadomtsev-Petviashvili equations under nonlocal constraints (the so-called 2-dimensionalk-constrained KP-hierarchy, briefly called the 2d k-c-hierarchy). As examples of (2+1)-dimensional nonlinear models belonging to the 2d k-c KP-hierarchy, both generalizations of already known systems and new nonlinear systems are presented. A method for the construction of exact solutions of equations belonging to the 2d k-c KP-hierarchy is proposed.

Article (Russian)

### Necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 98–109

We obtain necessary and sufficient conditions for the oscillation of solutions of nonlinear second-order differential equations with pulse influence in a Banach space.

Article (Russian)

### Pairwise products of moduli of families of curves on a Riemannian Möbius strip

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 110–116

We investigate pairwise products of moduli of families of curves on a Riemannian Möbius strip and obtain estimates for these products. As one of the factors, we consider the modulus of a family of arcs from a broad class of families of this sort (for each of these families, we determine the modulus and extremal metric).

Brief Communications (Russian)

### On exact constants in inequalities for norms of derivatives on a finite segment

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 117–119

We prove that, in an additive inequality for norms of intermediate derivatives of functions defined on a finite segment and equal to zero at a given system of points, the least possible value of a constant coefficient of the norm of a function coincides with the exact constant in the corresponding Markov-Nikol'skii inequality for algebraic polynomials that are also equal to zero at this system of points.

Brief Communications (Ukrainian)

### Condition number of the matrix of transition to the normal Jordan form

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 120–122

We establish necessary and sufficient conditions for the well-conditioned reduction of a matrix to the Jordan normal form.

Article (Russian)

### Visiting measures and an ergodic theorem for a sequence of iterations with random perturbations

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 123–127

By using local visiting measures, we describe the limit behavior of a sequence of iterations with random unequally distributed perturbations. As a corollary, we obtain a version of the local ergodic theorem.

Brief Communications (Ukrainian)

### On types of distributions of sums of one class of random power series with independent identically distributed coefficients

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 128–132

By using the method of characteristic functions, we obtain sufficient conditions for the singularity of a random variable. $$ξ = \sum_{k=1}^{∞} 2^{−k}ξ_k,$$ where $ξ_k$ are independent identically distributed random variables taking values $x_0, x_1$, and $x_2$ $(x_0 < x_1 < x_2)$ with probabilities $p_0, p_1$ and $p_2$, respectively, such that $p_i ≥ 0,\; p_0 + p_1 + p_2 = 1$ and $2(x_1 − x_0)/(x_2−x_0)$ is a rational number.

Brief Communications (Russian)

### Linear singularly perturbed problems with pulse influence

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 133–139

We establish the closeness of solutions of a linear singularly perturbed problem with asymptotically large pulse influence and the corresponding degenerate problem.

Brief Communications (Russian)

### Group analysis of boundary-value problems of mathematical physics

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 140–144

We obtain conditions for invariance and invariant solvability of boundary-value problems of mathematical physics.