# Volume 48, № 6, 1996

### Maximal sectorial extensions and closed forms associated with them

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 723-738

We describe all closed sesquilinear forms associated with *m*-sectorial extensions of a densely defined sectorial operator with vertex at the origin.

### Some geometric-differential models in the class of formal operator power series

Baranovich A. M., Daletskii Yu. L.

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 739-746

We consider an example of a formal construction of local differential geometry in which smooth functions regarded as morphisms are replaced by formal operator power series.

### On parameter dependence of bounded invariant manifolds of autonomous systems of differential equations

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 747-752

We consider bounded invariant manifolds of autonomous systems of differential equations and study the problem of their continuity and continuous differentiability with respect to a parameter.

### Quasiwidths and optimization of methods of mixed approximation of multidimensional singular integrals with kernels of hilbert type

Shabozov M. Sh., Vakarchuk S. B.

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 753-770

We consider the problem of application of mixed methods to the construction of algorithms, optimal in accuracy, for the calculation of multidimensional singular integrals with Hilbert-type kernels. We propose a method for the optimization of cubature formulas for singular integrals with Hilbert-type kernels based on the theory of quasiwidths.

### On manifolds of eigenfunctions and potentials generated by a family of periodic boundary-value problems

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 771-781

We consider a family of boundary-value problems with some potential as a parameter. We study the manifold of normalized eigenfunctions with even number of zeros in a period, and the manifold of potentials associated with double eigenvalues. In particular, we prove that the manifold of normalized eigenfunctions is a trivial fiber space over a unit circle and that the manifold of potentials with double eigenvalues is a homotopically trivial manifold trivially imbedded into the space of potentials.

### On groups close to metacyclic groups

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 782-790

We study groups whose structure is similar to the structure of metacyclic groups. These groups play an important role in the investigation of groups with normal subgroups.

### One version of the Linearized theory of nonstationary boundary-value problems with free boundary

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 791-804

We analyze the principle of linearization and linear boundary-value problems obtained by using this principle in the nonlinear theory of motion for a bounded volume of liquid with free surface subjected to the action of a nonstationary oscillating load. We formulate and study the problem of vibrocapillary equilibrium state, spectral problems in the theory of linear waves, and problems of stability of equilibrium states, including the problem of bifurcation of equilibrium states.

### Weak convergence of the extreme values of independent random variables in banach spaces with unconditional bases

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 805-812

We generalize well-known results concerning the weak convergence of maxima of real independent random variables to the case of random variables taking values in the Banach spaces with unconditional bases.

### Distribution of eigenvalues of the Sturm-Liouville problem with slowly increasing potential

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 813-825

We establish an asymptotic representation of the function \(\tilde n(R) = \int\limits_0^R {\frac{{n(r) - n(0)}}{r}dr, R \in \Re } \subseteq [0, \infty ), R \to \infty ,\) where *n*(*r*) is the number of eigenvalues of the Sturm-Liouville problem on [0,∞) in (λ:¦λ¦≤*r*) (counting multiplicities). This result is obtained under assumption that *q*(*x*) slowly (not faster than In *x*) increases to infinity as *x*→∞ and satisfies additional requirements on some intervals \([x_ - (R), x_ + (R)],R \in \Re \) .

### The second Lyapunov method for the investigation of stability of differential equations with Pulse perturbations and Markov coefficients

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 826-833

The stability of solutions of differential equations with pulse perturbations and Markov coefficients is studied by the second Lyapunov method.

### Differential Contour-Solid problem of Analytic Functions

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 834-842

The paper gives a survey of results completely solving the differential contour-solid problem of analytic functions in an open subset *G* of the complex plane, which was discussed as an open problem at the informal seminar held in 1994 in Zurich by participants of the International Congress of Mathematicians. This problem has a long prehistory and includes questions (unsolved at that time) concerning conditions of validity of differential contour-solid statements on the continuous extendability of a derivative to boundary points and on the differentiability of an analytic function at boundary points of the set *G*. In June, 1995, the author established that these statements are always true for arbitrary open sets G and any boundary points. These and more general theorems are given in this paper. We also present some other results, among which contour-solid theorems and a representation formula for the generalized solution of the Dirichlet problem for the derivative of a function should be mentioned.

### Generalized (CO)Homology length of a Manifold and functions with Singular Submanifolds

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 843-846

We introduce a topological invariant of a manifold. In terms of this invariant, we obtain an estimate for the generalized Lyustemik-Shnirel’man category of the manifold considered and an estimate for the minimal number of singular submanifolds of a function on this manifold.

### On the order of convergence of a semigroup to the identity operator

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 847-851

We describe classes of vectors *f* from a Hilbert space **H** for which the quantity ‖*T*(*t*)*f−f*‖, where *T*(*t*)=*e* ^{−tA }, *t*≥0, and *A* is a self-adjoint nonnegative operator in **H**, has a certain order of convergence to zero as *t*→+0.

### On the Hayman-Wu theorem for quasilines

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 852-856

For a function ω, we establish a condition sufficient for the sum ∑_{i}, ω(diam φ(*L* _{ i })) to be finite for any quasiconformal curve *L* _{ i }, simply connected domain Ω, and a function φ which conformally and univalently maps this domain onto the unit disk. Here, *L* _{ i }denote the components of Ω∩*L*.

### Uniform integrabblity and the lebesgue theorem on convergence in $L_0$-valued measures

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 857-860

We study integrals $∫fdμ$ of real functions over $L_0$-valued measures. We give a definition of convergence of real functions in quasimeasure and, as a special case, in $L_0$-measure. For these types of convergence, we establish conditions of convergence in probability for integrals over $L_0$-valued measures, which are analogous to the conditions of uniform integrability and to the Lebesgue theorem.

### Asymptotics of solutions of à system of differential equations with “turning points”

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 861-862

We construct an algorithm for determination of the principal term of the asymptotics of a solution of a system of differential equations with slow time and “turning points.”

### School-Seminar “Nonlinear boundary-value problems in Mathematical Physics and their applications”

Berezovsky A. A., Lenyuk M. P., Samoilenko A. M.

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 863-865