# Volume 51, № 7, 1999

### On the almost-everywhere convergence of the Riesz means of double orthogonal series

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 867–880

We establish coefficient conditions of the classical type that guarantee the almost-everywhere summability of double orthogonal series by the Riesz methods of nonnegative order. We also prove certain equiconvergence theorems.

### Optimization of approximate integration of monotone functions of two variables

Babenko V. F., Borodachov S.V.

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 881–889

We solve the problem of optimization of approximate integration of functions of two variables defined on a rectangle and monotonic in each variable by using the quadrature formulas with nodes at points of a rectangle net.

### On Hölder continuity of solutions of doubly nonlinear parabolic equations with weight

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 890–903

We prove the Hölder regularity of bounded weak solutions of doubly nonlinear degenerate parabolic equations with measurable coefficients.

### On zeros of functions of given proximate formal order analytic in a half-plane

Sharan V.L., Vynnyts’kyi B. V.

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 904–909

We describe sequences of zeros of functions*f*≢0 that are analytic in the half-plane ℂ_{+}={*z*:Re*z*> and satisfy the condition $$\forall \varepsilon > 0\exists c_1 \in (0; + \infty )\forall z \in \mathbb{C}_{\text{ + }} :\left| {f(z)} \right| \leqslant c_1 \exp \left( {(\sigma + \varepsilon )\left| z \right|\eta (\left| z \right|)} \right)$$ where 0≤σ<+∞ and η is a positive function continuously differentiable on [0; +∞) and such that*x*η′(*x*)/η(*x*)→0 as*x*→+∞.

### Impulsive boundary-value problems for weakly nonlinear systems with control

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 910–917

For weakly nonlinear impulsive differential systems with control, we obtain necessary and sufficient conditions for the existence of control and the corresponding solutions of differential systems with general boundary conditions.

### Distribution of the supremum of random processes from quasi-Banach $K_{σ}$-spaces

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 918-930

We study random processes from quasi-Banach $K_{σ}$-spaces of random variables whose domain of definition is not necessarily a compact set. We establish conditions under which the supremum of a properly normalized process belongs to the same space as the process itself. We also obtain estimates for the norm of this supremum.

### Gaussian and non-Gaussian limit distributions of estimates of the regression coefficients of a long-memory time series

Leonenko N. N., Sharapov M. M.

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 931–939

We obtain Gaussian and non-Gaussian distributions of estimates of regression coefficients of a long-memory time series.

### Approximation of fractional-order integrals by algebraic polynomials. II

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 940–951

We investigate the approximation of functions that are fractional-order integrals of bounded functions by algebraic polynomials.

### Asymmetric approximations in the space $L_{p(t)}$

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 952-959

We introduce the notion of $(α,β)$-norm in the space $L_{p(t)}$ of functions $x(t)$ for which $$\int\limits_0^1 {\left| {x(t)} \right|^{p(t)}< \infty }$$ where $p(t)$ is a positive measurable function. We establish a criterion for the element of the best $(α,β)$-approximation in the space $L_{p(t)}$. We obtain inequalities of the type of duality relations.

### The theory of the numerical-analytic method: Achievements and new trends of development. VI

Ronto M. I., Samoilenko A. M., Trofimchuk S. I.

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 960–971

We analyze the application of the numerical-analytic method proposed by A.M. Samoilenko in 1965 to multipoint boundary-value problems.

### Stability of semi-Markov risk processes in schemes of averaging and diffusion approximation

Goncharova S. Ya., Svishchuk A. V.

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 972–979

We investigate the asymptotic stability of semi-Markov risk processes with probability one in schemes of averaging and diffusion approximation.

### Generalization of the Lidskii theorem on the localization of the spectrum of a product of Hermitian operators

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 980–988

We consider products*C=AB* of Hermitian operators in an*n*-dimensional unitary space. Two equivalent localization theorems are proved in the case where one of the factors*A* and*B* is positive definite.

### Functional polystability of some nonautonomous quasilinear differential systems

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 989–995

For quasilinear differential systems with a boundary matrix of coefficients of the system of the first approximation, we obtain sufficient conditions of functional polystability, which generalizes the notion of exponential polystability.

### One version of the projection-iterative method based on the method of chords

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 996–1000

We consider the problem of application of one version of the projection-iterative method to nonlinear integral equations. Sufficient conditions for the convergence of this method are established.

### On minimal prime ideals of commutative Bezout rings

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 1001–1005

We study the spectrum of minimal prime ideals of commutative Bezout rings. We apply the results obtained to the problem of diagonal reduction of matrices over rings of this sort.

### An attractor of a semiflow generated by a system of phase-field equations without the uniqueness of a solution

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 1006–1009

We prove the existence of a global compact attractor for a multivalued semiflow generated by a system of phase-field equations with conditions on nonlinearity that do not guarantee the uniqueness of a solution.