2017
Том 69
№ 7

All Issues

Volume 63, № 8, 2011

Article (Ukrainian)

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

Buryachenko K. O.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1011-1020

We consider a Cauchy-type boundary-value problem of, a problem with three boundary conditions, and the Dirichlet problem for a general fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain $\Omega \subset R^2$ with smooth boundary. Using the method of the Green formula, the theory of expansion of differential operators, and the theory of $L$-traces (i.e., traces associated with a differential operation $L$), we obtain necessary and sufficient (for elliptic operators) conditions for the solvability of each of the problems under consideration in the space $H^m(\Omega),\;\; m \geq 4$.

Article (Ukrainian)

Sojourn time of almost semicontinuous integral-valued processes in a fixed state

Gusak D. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1021-1029

Let $\xi(t)$ be an almost lower semicontinuous integer-valued process with the moment generating function of the negative part of jumps $\xi_k : \textbf{E}[z^{\xi_k} / \xi_k < 0] = \frac{1 − b}{z − b},\quad 0 ≤ b < 1.$ For the moment generating function of the sojourn time of $\xi(t)$ in a fixed state, we obtain relations in terms of the roots $z_s < 1 < \widehat{z}_s$ of the Lundberg equation. By passing to the limit $(s → 0)$ in the obtained relations, we determine the distributions of $l_r(\infty)$.

Article (Ukrainian)

On the asymptotic distribution of the Koenker?Bassett estimator for a parameter of the nonlinear model of regression with strongly dependent noise

Ivanov O. V., Savych I. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1030-1052

We prove that, under certain regularity conditions, the asymptotic distribution of the Koenker - Bassett estimator coincides with the asymptotic distribution of the integral of the indicator process generated by a random noise weighted by the gradient of the regression function.

Article (Ukrainian)

Cauchy problem for a differential equation in the Banach space with generalized strongly positive operator coefficient

Chaikovs'kyi A. V., Il'chenko Yu. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1053-1070

The concept of strongly positive operator is generalized, and properties of the operators introduced are analyzed. The solutions of the Cauchy problem for a linear inhomogeneous differential equation with generalized strongly positive operator coefficient are found.

Article (English)

Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion

Yin J. L.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1071-1077

The orbital stability problem of the smooth solitary waves in the generalized Korteweg - de Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for an arbitrary speed of wave propagation.

Article (Russian)

On the boundary behavior of solutions of the Beltrami equations

Kovtonyuk D. A., Petkov I. V., Ryazanov V. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1078-1091

We show that every homeomorphic solution of the Beltrami equation $\overline{\partial} f = \mu \partial f$ in the Sobolev class $W^{1, 1}_{\text{loc}}$ is a so-called lower $Q$-homeomorphism with $Q(z) = K_{\mu}(z)$, where $K_{\mu}$ is a dilatation quotient of this equation. On this basis, we develop the theory of the boundary behavior and the removability of singularities of these solutions.

Article (English)

Weyl's theorem for algebrascally $wF(p, r, q)$ operators with $p, q > 0$ and $q \geq 1$

Rashid M. H. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1092-1102

If $T$ or $T*$ is an algebraically $wF(p, r, q)$ operator with $p, r > 0$ and $q ≥ 1$ acting on an infinite-dimensional separable Hilbert space, then we prove that the Weyl theorem holds for $f(T)$, for every $f \in \text{Hol}(\sigma(T))$, where $ \text{Hol}(\sigma(T))$ denotes the set of all analytic functions in an open neighborhood of $\sigma(T)$. Moreover, if $T^*$ is a $wF(p, r, q)$ operator with $p, r > 0$ and $q ≥ 1$, then the $a$-Weyl theorem holds for $f(T)$. Also, if $T$ or $T^*$ is an algebraically $wF(p, r, q)$ operators with $p, r > 0$ and $q ≥ 1$, then we establish spectral mapping theorems for the Weyl spectrum and essential approximate point spectrum of T for every $f \in \text{Hol}(\sigma(T))$, respectively. Finally, we examine the stability of the Weyl theorem and $a$-Weyl theorem under commutative perturbation by finite-rank operators.

Article (Ukrainian)

On asymptotic equivalence of solutions of stochastic and ordinary equations

Novak I. H., Samoilenko A. M., Stanzhitskii A. N.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1103-1127

For a weakly nonlinear stochastic system, we construct a system of ordinary differential equations the behavior of solutions of which at infinity is similar to the behavior of solutions of the original stochastic system.

Article (Russian)

On the openness and discreteness of mappings with unbounded characteristic of quasiconformality

Sevost'yanov E. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1128-1134

The paper is devoted to the investigation of the topological properties of space mappings. It is shown that sense-preserving mappings $f : D \rightarrow \overline{\mathbb{R}^n}$ in a domain $D \subset \mathbb{R}^n$, n ≥ 2, which are more general than mappings with bounded distortion, are open and discrete if a function Q corresponding to the control of the distortion of families of curves under these mappings has slow growth in the domain f(D), e.g., if Q has finite mean oscillation at an arbitrary point $y0 \in f(D)$.

Anniversaries (Ukrainian)

Volodymyr Leonidovych Makarov (on his 70th birthday)

Korolyuk V. S., Lukovsky I. O., Samoilenko A. M.

Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1135-1136

Anniversaries (Ukrainian)

Yurii Serhiiovych Osypov (on his 75th birthday)

Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1137-1139

Brief Communications (English)

Strongly radical supplemented modules

Büyükaşık Е., Türkmen E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1140-1146

Zoschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a no-local Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.

Brief Communications (Ukrainian)

Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Prokip V. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1147-1152

Polynomial matrices $A(x)$ and $B(x)$ of size $n \times n$ over a field $\mathbb{F}$ are called semiscalar equivalent if there exist a nonsingular $n \times n$ matrix $P$ over $\mathbb{F}$ and an invertible $n \times n$ matrix $Q(x)$ over $\mathbb{F}[x]$ such that $A(x) = PB(x)Q(x)$. We give a canonical form with respect to the semiscalar equivalence for a matrix pencil $A(x) = A_0x - A_1$, where $A_0$ and $A_1$ are $n \times n$ matrices over $\mathbb{F}$, and $A_0$ is nonsingular.