# Volume 50, № 7, 1998

### On the structure of solutions of a system of singularly perturbed differential equations with nondiagonalizable limit operator

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 867–876

We construct uniform asymptotics of a solution of a heterogeneous system of singularly perturbed differential equations in the case of nondiagonalizable limit operator. We consider the case where the spectrum of the limit operator contains an unstable element at the point *x* = 0.

### A problem with formal initial conditions for differential equations with constant pseudodifferential coefficients

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 877–888

We establish conditions for the unique existence of a solution of a problem with formal initial conditions. We investigate the problem of its solvability in the case where a solution is not unique.

### On the regularity of the growth of the modulus and argument of an entire function in the metric of $L^p [0, 2π]$

Kalinets R. Z., Koval’chuk Yu. A.

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 889-896

Under sufficiently general assumptions, we describe sets of entire functions $f$, sets of growing functions $λ$, and sets of complex-valued functions $H$ from $L^p [0, 2π]$, $p ∈ [1, + ∞]$, for which $$\left\{ {\frac{1}{{2\pi }}\int\limits_0^{2\pi } {|\log f(re^{i\theta } ) - \lambda (r)H(\theta )|^p } d\theta } \right\}^{1/p} = o(\lambda (r)),r \to \infty.$$

### Boundary-value problems with random initial conditions and functional series from sub_{φ} (Ω). II

Koval’chuk Yu. A., Kozachenko Yu. V.

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 897–906

We study conditions for convergence and the rate of convergence of random functional series from the space sub_{φ}(Ω) in various norms. The results obtained are applied to the investigation of a boundary-value problem for a hyperbolic equation with random initial conditions.

### Permutations and piecewise-constant approximation of continuous functions of *n* variables

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 907–918

We consider the problem of approximation of a continuous function *f* given on an *n*-dirnensional cube by step functions in the metrics of *C* and *L* _{p}. We obtain exact error estimates in terms of the modulus of continuity of the function *f* or a special permutation of it.

### Pseudoparabolic variational inequalities without initial conditions

Lavrenyuk S. P., Ptashnik M. B.

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 919–929

We consider a pseudoparabolic variational inequality in a cylindrical domain semibounded in a variable *t*. Under certain conditions imposed on the coefficients of the inequality, we prove theorems on the unique existence of a solution for a class of functions with exponential growth as *t* → ∞.

### Distribution of the spectrum and representation of solutions of degenerate dynamical systems

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 930–936

We propose algebraic methods for the investigation of the spectrum and structure of solutions of degenerate dynamical systems. These methods are based on the construction and solution of new classes of matrix equations. We prove theorems on the inertia of solutions of the matrix equations, which generalize the well-known properties of the Lyapunov equation.

### Asymptotic normality and efficiency of a weighted correlogram

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 937–947

For a process *X*(*t*)=Σ _{ j=1} ^{ M } *g* _{ j }(*t*)ξ_{ j }(), where g_{j}(*t*) are nonrandom given functions, \((\xi _j (t),j = \overline {1,M} )\) is a stationary vector-valued Gaussian process, Eξ_{k}(*t*) = 0, and Eξ_{k}(0) Eξ_{l}(τ) = *r* _{kl}(τ), we construct an estimate \(\hat r_{kl} (\tau ,T)\) for the functions *r* _{kl}(τ) on the basis of observations *X(t), t* ∈ [0, *T*]. We establish conditions for the asymptotic normality of \(\sqrt T (\hat r_{kl} (\tau ,T) - r_{kl} (\tau ))\) as *T* → ∞. We consider the problem of the optimal choice of parameters of the estimate \(\hat r_{kl} \) depending on observations.

### Construction of a separately continuous function with given oscillation

Maslyuchenko O. V., Maslyuchenko V. K.

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 948–959

We investigate the problem of construction of a separately continuous function *f* whose oscillation is equal to a given nonnegative function *g*. We show that, in the case of a metrizable Baire product, the problem under consideration is solvable if and only if *g* is upper semicontinuous and its support can be covered by countably many sets, which are locally contained in products of sets of the first category.

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

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

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 960–979

We analyze results concerning the application of the numerical-analytic method suggested by Samoilenko to delay differential equations, differential equations with “maxima,” functional-differential, operator-differential, and integro-differential equations.

### Minimum-Area ellipse containing a finite set of points. I

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 980–988

From the geometric point of view, we consider the problem of construction of a minimum-area ellipse containing a given convex polygon. For an arbitrary triangle, we obtain an equation for the boundary of the minimum-area ellipse in explicit form. For a quadrangle, the problem of construction of a minimumarea ellipse is connected with the solution of a cubic equation. For an arbitrary polygon, we prove that if the boundary of the minimum-area ellipse has exactly three common points with the polygon, then this ellipse is the minimum-area ellipse for the triangle obtained.

### On *I*-Rigid and *q*-Rigid Rings

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 989–993

We describe *q*-rigid rings that have no simple noncommutative homomorphic images, and *I*-rigid rings with periodic and mixed additive groups.

### On the convergence of difference schemes for the diffusion equation of fractional order

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 994–996

For the diffusion equation of fractional order, we construct an approximation difference scheme of order 0(*h* ^{2} + τ). We present an algorithm for the solution of boundary-value problems for a generalized transfer equation of fractional order.

### Filtration and prediction of random solutions of a system of linear differential equations with coefficients depending on a finite-valued Markov process

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 997–1000

We obtain an equation of optimal filtration for processes of Markov random evolution, which is a solution of systems of linear differential equations with Markov switchings.

### Rate of convergence of the Taylor series for some classes of analytic functions

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 1001–1003

We study the rate of convergence of the Taylor series for functions from the classes *A* ^{Ψ} *H* _{p}, p = 1, ∞, in the uniform and integral metrics.

### On one generalization of the Langevin equation with determinate modulus of velocity

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 1004–1006

For a special class of systems of Itô stochastic equations with random coefficients, we establish conditions under which the modulus of the vector of state is not a random variable. We also consider possible ways of generalization of this problem.

### The International Conference “Nonlinear Partial Differential Equations”

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 1007–1008