# Volume 62, № 6, 2010

### Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 723 – 732

We show that there exists a countable set of eigenfunctions of the Tricomi spectral problem for multidimensional mixed hyperbolic–parabolic equations.

### Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 733–753

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space $C[0, T]$. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding Itô stochastic equation.

### Conditions of regularity of a general differential boundary-value problem for improperly elliptic equations

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 754–761

The Fredholm property and well-posedness of a general differential boundary-value problem for a general improperly elliptic equation are analyzed in a two-dimensional bounded domain with smooth boundary.

### General algorithm of computation of $c$-table and detection of valleys

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 762–772

We present a review of all interesting results concerning the c-table obtained by the authors for the last two decades. These results are not widely known because they were presented in publications of limited circulation. We discuss different computational aspects of software producing the $c$-tables in the presence of blocs and their evolution following the evolution of the computer environment: effects of the use of 32-bit arithmetic .≈8 digits), 64-bit arithmetic (double precision, ≈16 digits), and Bailey’s Fortran multiprecision package .32 or 64 digits), competition between the ascending and descending algorithms, relationship between the complexity of computation and precision, overflow and underflow problems, competition between different formulas allowing one to overcome the blocs in the $c$-table, practical simple criterion of detecting numerical zeros in the c-table allowing to identify the blocs, and automatic detection of valleys.

### Quasiperiodic solutions of degenerate linear systems of second-order ordinary differential equations

Aliluiko A. M., Er’omenko V. O.

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 773–783

We establish sufficient conditions for the existence of quasiperiodic solutions of a system of ordinary second-order differential equations with degenerate symmetric matrix of the second derivatives for an arbitrary quasiperiodic inhomogeneity.

### Exponential stability of a program manifold of indirect control systems

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 784–790

We establish sufficient conditions for the exponential stability of a program manifold of indirect control systems and conditions for the fast operation of a regulator, overcontrol, and monotone damping of a transient process in the neighborhood of the program manifold.

### Points of joint continuity and large oscillations

Maslyuchenko V. K., Nesterenko V. V.

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 791–800

For topological spaces $X$ and $Y$ and a metric space $Z$, we introduce a new class $N(X × Y,Z)$ of mappings $f:\; X × Y → Z$ containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping $f$ from this class and any countable-type set $B$ in $Y$, the set $C_B (f)$ of all points $x$ from $X$ such that $f$ is jointly continuous at any point of the set $\{x\} × B$ is residual in $X$: We also prove that if $X$ is a Baire space, $Y$ is a metrizable compact set, $Z$ is a metric space, and $f ∈ N(X×Y,Z)$, then, for any $ε > 0$, the projection of the set $D^{ε} (f)$ of all points $p ∈ X × Y$ at which the oscillation $ω_f (p) ≥ ε$ onto $X$ is a closed set nowhere dense in $X$.

### On solutions of one class of second-order operator differential equations in the class of holomorphic vector functions

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 801–813

We establish sufficient conditions for the completeness of a part of root vectors of one class of the second-order operator bundles corresponding to the characteristic numbers from a certain sector and prove the theorem on completeness of a system of elementary holomorphic solutions of the corresponding second-order homogeneous operator differential equations. We also indicate the conditions of correct and unique solvability of a boundary-value problem for the analyzed equation with linear operator in the boundary condition and estimate the norm of the operator of the intermediate derivative in the perturbed part of the equation.

### On the mean convergence of Fourier–Jacobi series

Goncharov S. V., Motornyi V. P., Nitiema P. K.

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 814–828

The convergence of Fourier–Jacobi series in the spaces $L_{p,A,B}$ is studied in the case where the Lebesgue constants are unbounded.

### On the order of growth of ring $Q$-homeomorphisms at infinity

Salimov R. R., Smolovaya E. S.

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 829 – 836

For ring homeomorphisms $f : ℝn → ℝn$ , we establish the order of growth at infinity.

### Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 837–846

Let $E$ be a finite-dimensional Banach space, let $C^0(R; E)$ be a Banach space of functions continuous and bounded on $R$ and taking values in $E$; let $K:\;C^0(R ,E) → C^0(R, E)$ be a $c$-continuous bounded mapping, let $A:\;E → E$ be a linear continuous mapping, and let $h ∈ C^0(R, E)$. We establish conditions for the existence of bounded solutions of the nonlinear equation $$\frac{dx(t)}{dt} + (Kx)(t)Ax(t) = h(t),\;t ∈ R.$$

### Unitarization of Schur representations of a partially ordered set associated with $\widetilde{E}_7$

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 847–853

We prove that every Schur representation of a partially ordered set associated with graph $\widetilde{E}_7$ can be unitarized with some character.

### Singularities of the structure of two-sided ideals of a domain of elementary divisors

Bilyavs’ka S. I., Zabavskii B. V.

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 854 – 856

We prove that, in a domain of elementary divisors, the intersection of all nontrivial two-sided ideals is equal to zero. We also show that a Bézout domain with finitely many two-sided ideals is a domain of elementary divisors if and only if it is a 2-simple Bézout domain.

### On a spherical code in the space of spherical harmonics

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 857 – 859

We propose a new method for the construction of new “nice” configurations of vectors on the unit sphere $S^d$ with the use of spaces of spherical harmonics.

### Kostin-type criterion for abstract linear differential equations of arbitrary order in banach spaces

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 860 – 864

We consider linear differential equations with operator coefficients in a Banach space. We construct necessary and sufficient conditions for the well-posedness of the Cauchy problem for these equations of arbitrary order that are analogous to the Kostin conditions for incomplete equations of the second order.

### Letter to the Editor

Khachatryan A. Kh., Khachatryan Kh. A.

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 865