# Volume 51, № 8, 1999

### On the 90th Anniversary of the Birth of Academician N.N. Bogolyubov

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1012–1013

### On the role of N.N. Bogolyubov in the development of the theory of nonlinear oscillations

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1014–1035

A review of N. N. Bogolyubov's works on investigations in the theory of nonlinear oscillations is presented.

### Tauberian theorems with remainder for (*H, p, α, β*)-and (*C, p, α, β*)-methods of summation of functions of two variables

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1036–1044

We consider a general method of obtaining Tauberian theorems with remainder for Hölder- and Cesarotype methods of summation.

### Singularly perturbed normal operators

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1045–1053

We present a generalization of definition'of-singularly perturbed operators to the case of normal operators. To do this, we use an idea of normal expansions of a prenormal operator and prove the relation for resolvents of normal expansions similar to the M. Krein relation for resolvents of self-adjoint expansions. In addition, we establish one-to-one correspondence between the set of singular perturbations of rank one and the set of perturbed (of rank one) operators.

### Limit theorems for conditional distributions with regard for large deviations

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1054–1064

Possible limit laws are studied for the multivariate conditional distribution of a subset of components of the sum of independent identically distributed random vectors under the condition that other components belong to the domain of large deviations. It is assumed that the considered distribution is absolutely continuous and belongs to the domain of attraction of the normal law but possesses “heavy tails.” The approach suggested is based on the local theorem for large deviations.

### Scattering problem for a multidimensional system of first-order partial differential equations

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1065–1076

We construct transformation operators, which enables us to study a scattering problem and investigate the properties of a scattering operator for a multidimensional system of first-order partial differential equations.

### Asymptotic integration of a singularly perturbed quasilinear cauchy problem in a banach space

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1077–1086

We investigate the asymptotic behavior of a solution of a singularly perturbed quasilinear abstract differential equation in a Banach space.

### Integrals of certain random functions with respect to general random measures

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1087–1095

For random functions that are sums of random functional series, we determine an integral over a general random measure and prove limit theorems for this integral. We consider the solution of an integral equation with respect to an unknown random measure.

### On one criterion of constancy of a complex function

Safonov V. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1096–1104

A new criterion of constancy of complex functions is proved.

### On the asymptotics of integral manifolds of singularly perturbed systems with delay

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1105–1111

We investigate the problem of constructing asymptotic decompositions of integral manifolds of slow variables for linear and nonlinear singularly perturbed systems with delay.

### On the asymptotics of the maximal eigenvalue for a family of branching processes

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1112–1117

We establish an asymptotic representation of the maximal eigenvalue ρ_{ε} for a family of branching processes with arbitrary number of types of particles which are close to critical ones.

### On estimates of minima of criterion functions in optimization on combinations

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1118–1121

On the basis of the approach proposed, we obtain new estimates of extremal values of strongly convex differentiable functions and strengthened estimates of minima on a set of combinations with repetions.

### On the differential properties of continuous functions

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1122–1125

We introduce and investigate some new differential properties of continuous functions by means of the geometrical properties of their derivatives.

### Asymptotic solution of the cauchy problem for a singularly perturbed linear system

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1126–1128

We construct the asymptotics of the solution of the Cauchy problem for a degenerate singularly perturbed linear system in the case of multiple spectrum of the principal operator.

### Classification of*m*-functions on surfaces

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1129–1135

We establish a necessary and sufficient condition of conjugacy of*m*=functions on surfaces.

### Singly generatedC *-algebras

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1136-1141

We consider a $C*$-algebra $A$ generated by $k$ self-adjoint elements. We prove that, for $n \geqslant \sqrt {k - 1}$ , the algebra $M_n (A)$ is singly generated, i.e., generated by one non-self-adjoint element. We present an example of algebraA for which the property that $M_n (A)$ is singly generated implies the relation $n \geqslant \sqrt {k - 1}$.

### An exact solution of boundary-value problem

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1142-1143

We establish conditions under which the problem $u_{tt} — u_{xx} = f(x, t),\; u(x, 0) = u(x, \pi) = 0$ possesses the classical solution.

### Decomposition of polynomial matrices into a direct sum of triangular summands

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1144–1148

By using the transformations*SA(x)R(x)*, where*S* and*R(x)* are invertible matrices, we reduce a polynomial matrix*A(x)* whose elementary divisors are pairwise relatively prime to a direct sum of irreducible triangular summands with invariant factors on the principal diagonals.

### On the growth of an entire dirichlet series

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1149–1153

We establish the relation between the increase of the quantityM(σ,*F*) = ∣*a* _{0}∣ + ∑ _{ n=1} ^{∞} ∣*a* _{ n }∣ exp (σλ_{ n }) and the behavior of sequences (|*a* _{ n }|) and (λ_{ n }), where (λ_{ n }) is a sequence of nonnegative numbers increasing to + ∞, and*F*(*s*) =*a* _{0} + ∑ _{ n=1} ^{∞} *a* _{ n } *e* ^{ sλn },*s*=σ+*it*, is the Dirichlet entire series.