Volume 51, № 8, 1999

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

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

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.

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.

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.

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

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.

A new criterion of constancy of complex functions is proved.

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

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 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.

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

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.

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

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 $n \geqslant \sqrt {k - 1}$.

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.

By using the transformationsSA(x)R(x), whereS andR(x) are invertible matrices, we reduce a polynomial matrixA(x) whose elementary divisors are pairwise relatively prime to a direct sum of irreducible triangular summands with invariant factors on the principal diagonals.

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 + ∞, andF(s) =a ₀ + ∑ _n=1 ^∞ a _n e ^sλn ,s=σ+it, is the Dirichlet entire series.