# Volume 51, № 1, 1999

### 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.

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

Buldygin V. V., Mel'nik V. M., Shportyuk V. G.

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.

### 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 $f$.

### 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.

### 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.

### 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.

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

Mokhon'ko A. Z., Mokhon'ko V. D.

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 $Δ < ∞$.

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

Samoilenko A. M., Samoilenko V. G., Sidorenko Yu. M.

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-dimensional*k*-constrained KP-hierarchy, briefly called the 2*d k-c*-hierarchy). As examples of (2+1)-dimensional nonlinear models belonging to the 2*d 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 2*d k-c* KP-hierarchy is proposed.

### 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.

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

Okhrimenko S. A., Tamrazov P. M.

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).

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

Babenko V. F., Uedraogo Zh. B.

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.

### 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.

### 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.

### 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.

### 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.

### 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.