Том 51, № 1, 1999

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.

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

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

We prove that every countable Abelian group with a finite number of second order elements can be decomposed into countable number of subsets which are dense in any nondiscrete group topology.

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.

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.

Assume that $f$ is an integer transcendental solution of the differential equation $P_n(z,f,f') = P_{n-1}(z, f,f',... ,f^{(p)})),\quad P_n, P_{n-1}$ 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 $\rho$ of growth of $f$ satisfies the relation $\cfrac12 \leq \rho

The spatially two-dimensional generalization of hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints or the so-called 2-dimensional $k$-constrained $KP$-hierarchy is given (its abrigded notation is $2 d k - c$-hierarchy). As examples of $(2+ l)$-dimensional nonlinear models belonging to the $2 d k-с\; KP$-hierarchy, both generalizations of already known systems and new nonlinear systems are presented. A method for constructing exact solutions of equations belonging to the $2 d k-c \; KP$-hierarchy is proposed.

We obtain necessary and sufficient conditions of oscillation of solutions of second order nonlinear differentiel equations with puise influence in the Banach space.

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).

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

Necessary and sufficient conditions of well conditioned reduction of a matrix to the Jordan normal form are established.

By using local visitation 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.

By using the method of characteristic functions, we obtain sufficient conditions of the singularity of a random variable $$\xi = \sum_{k=1}^{\infty}2^{-k}\xi_k,$$ where $\xi_k,$ are independent equally distributed random variables taking the values $x_0, x_1$ and $x_2$ $(x_0

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

Invariance conditions and conditions of invariant solvability are obtained for boundary-value problems in mathematical physics.