# Volume 53, № 3, 2001

### Naum Il'ich Akhiezer (on his 100-th birthday)

Khruslov E. Ya., Marchenko V. O., Mitropolskiy Yu. A., Pogorelov A. V., Samoilenko A. M., Skrypnik I. V.

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 291-293

### Oleksii Mykolajovych Boholyubov (on His 90th Birthday)

Koshlyakov V. N., Mitropolskiy Yu. A., Samoilenko A. M., Urbansky V. M.

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 294-295

### On the Manifolds of Eigenvectors of Linear and Quasilinear Finite-Dimensional Self-Adjoint Operators. II

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 296-301

We investigate the manifold of normalized eigenvectors of self-adjoint operators. We present the homotopic classification of typical quasilinear eigenvector problems based on the properties of this manifold.

### Qualitative Analysis of an Implicit Singular Cauchy Problem

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 302-310

We consider a singular Cauchy problem for a first-order ordinary differential equation unsolved with respect to the derivative of the unknown function. We prove the existence of continuously differentiable solutions with required asymptotic properties.

### Exterior Tensor Product of Perverse Sheaves

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 311-322

Under certain assumptions, we prove that the Deligne tensor product of the categories of constructible perverse sheaves on pseudomanifolds *X* and *Y* is the category of constructible perverse sheaves on *X*×*Y*. The functor of the exterior Deligne tensor product is identified with the exterior geometric tensor product.

### Stability of Linear Positive Systems

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 323-330

We establish criteria of asymptotic stability for positive differential systems in the form of conditions of monotone invertibility of linear operators. The structure of monotone and monotonically invertible operators in the space of matrices is investigated.

### Approximation of Certain Classes of Singular Integrals by Algebraic Polynomials

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 331-345

We study the problem of pointwise approximation by algebraic polynomials for classes of functions that are singular integrals of bounded functions. We obtain asymptotically exact estimates of approximations.

### Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. II

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 346-355

In normed spaces of functions analytic in the Jordan domain Ω⊂ℂ, we establish exact order estimates for the Kolmogorov widths of classes of functions that can be represented in Ω by Cauchy-type integrals along Γ = ∂Ω with densities *f*(·) such that \(f \circ \Psi \in L_{\beta ,p}^\Psi (T)\) . Here, Ψ is a conformal mapping of \(C\backslash \overline \Omega \) onto {*w*: |*w*| > 1}, and \(L_{\beta ,p}^\Psi (T)\) is a certain subset of infinitely differentiable functions on *T* = {*w*: |*w*| = 1}.

### On the Exponential Dichotomy on $\mathbb{R}$ of Linear Differential Equations in $\mathbb{R}^n$

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 356-371

New results of the investigation of exponential dichotomy on the entire axis of linear differential equations in $\mathbb{R}^n$ are presented.

### Existence and Extendability of Solutions of the Equation $g(t, x) = 0$

Kaplun Yu. I., Samoilenko V. G.

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 372-382

We consider the problem of extendability and existence of solutions of the equation *g*(*t*, *x*) = 0 on the maximum interval of their definition.

### Characterization of Groups with a Layer-Finite Periodic Part

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 383-391

We prove a theorem that characterizes groups with a layer-finite periodic part in the class of the Shunkov groups with solvable finite subgroups.

### Approximation Characteristics of Spaces $S_\phi ^p$

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 392-416

We introduce linear vector spaces $S_\phi ^p$ and study their approximation properties. By using the general results obtained, we establish statements concerning the approximation of classes of periodic functions of many variables by trigonometric polynomials.

### On Multipliers in Hardy Spaces

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 417-421

We investigate the accuracy of certain sufficient conditions for multipliers of power series in Hardy spaces.

### On $(n, m)$-Convex Sets

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 422-427

We investigate the class of generalized convex sets on Grassmann manifolds, which includes known generalizations of convex sets for Euclidean spaces. We extend duality theorems (of polarity type) to a broad class of subsets of the Euclidean space. We establish that the invariance of a mapping on generalized convex sets is equivalent to its affinity.

### Law of the Iterated Logarithm for Unstable Gaussian Autoregressive Models

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 428-432

We investigate the asymptotic properties of one-dimensional Gaussian autoregressive processes of the second order. We prove the law of the iterated logarithm in the case of an unstable autoregressive model.