# Volume 52, № 7, 2000

### Information Complexity of Multidimensional Fredholm Integral Equations with Harmonic Coefficients

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 867-874

For the class of multidimensional Fredholm integral equations with free terms and kernels periodic and harmonic in each variable, we determine the exact order of the minimum radius of information in the logarithmic scale.

### A Remark on the Completeness of Systems of Exponentials with Weight in $L^2(ℝ)$

Shapovalovskii A. V., Vynnyts’kyi B. V.

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 875-880

We establish new conditions for the completeness of systems of exponentials with weight in *L* ^{2}(ℝ), which complement and generalize the results obtained earlier by the authors.

### New Generalizations of the Scorza-Dragoni Theorem

Gaidukevich O. L., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 881-888

We consider Carathéodory functions *f* : *T* × *X* → *Y*, where *T* is a topological space with regular σ-finite measure, the spaces *X* and *Y* are metrizable and separable, and *X* is locally compact. We show that every function of this sort possesses the Scorza-Dragoni property. A similar result is also established in the case where the space *T* is locally compact and *X* = ℝ^{∞}.

### Determination of the Spectral Index of Ergodicity of a Birth-and-Death Process

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 889-897

We obtain a new explicit relation for the calculation of the spectral index of ergodicity of a birth-and-death process with continuous time. The calculation of the index is reduced to the solution of an optimization problem of nonlinear programming that contains the infinitesimal matrix of the process. As an example, we use the proposed method for finding the exact values of the indices of exponential ergodicity for certain Markov queuing systems.

### Estimates of the Best $M$-Term Trigonometric Approximations of the Classes $L_{β, p}^{ψ}$ of Periodic Functions of Many Variables in the Space $L_q$

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 898-907

We obtain order estimates for the best trigonometric approximations of the classes $L_{β, p}^{ψ}$ of periodic functions of many variables in the space$L_q$ for $1 < p < q ≤ 2$ and $1 < q ≤ p < ∞$.

### Description of Convex Curves

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 908-922

We present a description of convex curves, which enables one to reduce the problem of approximation of a convex curve by piecewise circular lines in the Hausdorff metric to the problem of approximation of 2π-periodic functions by trigonometric splines in the uniform metric. We describe certain properties of convex curves.

### Time-Irreversibility and Existence and Uniqueness of Solutions of Problems in Linear Viscoelasticity

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 923-930

We study a problem of linear viscoelasticity for the case where the relation between the Cauchy stress and strain tensors is described by a linear integral relation. Theorems on the existence and uniqueness of a solution of the problem are proved.

### Smooth Solution of the Dirichlet Problem for a Quasilinear Hyperbolic Equation of the Second Order

Khoma N. H., Khoma S. G., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 931-935

On the basis of the exact solution of the linear Dirichlet problem \(u_{tt} - u_{xx} = f\left( {x,t} \right)\) , \(u\left( {0,t} \right) = u\left( {\pi ,t} \right) = 0,{\text{ }}u\left( {x,0} \right) = u\left( {x,2\pi } \right) = 0,\) \(0 \leqslant x \leqslant \pi ,{\text{ }}0 \leqslant t \leqslant 2\pi ,\) we obtain conditions for the solvability of the corresponding Dirichlet problem for the quasilinear equation *u* _{tt} − *u* _{xx} = *f*(*x*, *t*, *u*, *u* _{t}).

### General Solution of Systems of Nonlinear Difference Equations with Continuous Argument

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 936-953

We investigate the structure of the general solution of a system of nonlinear difference equations with continuous argument in the neighborhood of an equilibrium state.

### General Theorems on the Existence and Uniqueness of Solutions of Impulsive Differential Equations

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 954-964

We study the Cauchy problem for impulsive differential equations in the general case.

### On Periodic Locally Solvable Groups Decomposable into the Product of Two Locally Nilpotent Subgroups

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 965-970

We establish new results concerning various properties of a periodic locally solvable group *G* = *A* *B* with locally nilpotent subgroups *A* and *B* one of which is hyper-Abelian.

### On the Approximation of Functions of the Hölder Class by Biharmonic Poisson Integrals

Kharkevych Yu. I., Zhyhallo K. M.

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 971-974

We determine the exact value of the upper bound of the deviation of biharmonic Poisson integrals from functions of the Hölder class.

### Attractors of Differential Inclusions and Their Approximation

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 975-979

We investigate the properties of solutions of differential inclusions in a Banach space. We prove a theorem on the existence of a global attractor for a multivalued semidynamical system generated by these solutions and a theorem on the approximation of an attractor in the Hausdorff metric.

### Perturbation of a Two-Point Problem

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 980-984

We investigate the problem of the effect of integral terms in boundary conditions on the well-posedness of nonlocal boundary-value problems for partial differential equations.

### On the Denseness of Subspaces of Analytic Vectors of a Closed Linear Operator in a Banach Space

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 985-989

We establish conditions for the behavior of the resolvent and the location of the spectrum of a linear closed operator *A* densely defined in a Banach space *E* under which its Gevrey spaces *G* _{(β)}(*A*), β < 1, are dense in *E*.

### On the Boundedness of Singular Integral Operators in Symmetric Spaces

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 988-993

We consider the integral convolution operators \(T_\varepsilon f\left( x \right) = \int\limits_{|x - y| > \varepsilon } {k\left( {x - y} \right)f\left( y \right)dy}\) defined on spaces of functions of several real variables. For the kernels *k*(*x*) satisfying the Hörmander condition, we establish necessary and sufficient conditions under which the operators {*T* _{ε}} are uniformly bounded from Lorentz spaces into Marcinkiewicz spaces.

### Approximation of Periodic Functions of High Smoothness by Interpolation Trigonometric Polynomials in the Metric of $L_1$

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 994-998

We establish an asymptotically exact estimate for the error of approximation of ℝ^{2}-periodic functions of high smoothness by interpolation trigonometric polynomials in the metric of *L* _{1}.

### Estimation of an Unknown Parameter in the Cauchy Problem for a First-Order Partial Differential Equation under Small Gaussian Perturbations

Bondarev B. V., Dzundza A. I., Simogin A. A.

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 999-1006

On the basis of observation of a realization of a solution of the Cauchy problem, we establish a maximum-likelihood estimate for an unknown parameter. We construct an exponential inequality for the probabilities of large deviations of the estimate from the real value of the parameter.

### International Conference on Geometry "in the Large"

Diskant V. I., Gor'kavyi V. A., Milka A. D.

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 1007-1008