# Volume 50, № 10, 1998

### Estimates of generalized solutions of the Dirichlet problem for quasilinear elliptic equations of the second order in a domain with conical boundary point

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1299–1309

We obtain *a priori* estimates for generalized second derivatives (in the Sobolev weighted norm) of solutions of the Dirichlet problem for the elliptic equation $$\frac{d}{{dx_i }}a_i (x,u,u_x ) + a(x,u,u_x ) = 0,x \in G,$$ in the neighborhood of a conical boundary point of the domain *G*. We give an example that demonstrates that the estimates obtained are almost exact.

### Spherical means on Euclidean spaces

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1310–1315

We give a description of certain classes of functions with zero spherical means.

### Explicit realization of irreducible representations of classical compact lie groups in the spaces of sections of line bundles

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1316–1323

We use the Borel-Weil scheme for the construction of irreducible representations of compact Lie groups in the spaces of holomorphic sections of line bundles over homogeneous manifolds. We find the explicit form of the space of sections and construct an invariant scalar product. We show that the space of holomorphic sections locally satisfies the Zhelobenko indicator system.

### Diametral theory of algebraic surfaces and geometric theory of invariants of groups generated by reflections. III

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1324–1340

We present the geometric theory of invariants of wild groups of skew symmetries in a real Euclidean space.

### Localization of the spectrum and representation of solutions of linear dynamical systems

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1341–1351

We develop a general method for the localization of eigenvalues of matrix polynomials and functions based on the solution of matrix equations. For a broad class of equations, we formulate theorems that generalize the known properties of the Lyapunov equation. A new method for the representation of solutions of linear differential and difference systems is proposed.

### On sufficient conditions for the technical stability of controlled processes with lumped parameters

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1352–1358

On the basis of the theory of differential inequalities and Lagrange multipliers, we develop a method for the investigation of conditions for the technical stability of continuously controlled processes with lumped parameters.

### Asymptotic properties of the norm of the extremum of a sequence of normal random functions

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1359–1365

Under additional conditions on a bounded normally distributed random function *X = X(* *t*), *t* ∈ T, we establish a relation of the form $$\mathop {\lim }\limits_{n \to \infty } P(b_n (||Z_n || - a_n ) \leqslant x) = \exp ( - e^{ - x} )\forall x \in R^1 $$ where \(Z_n = Z_n (t) = \mathop {\max }\limits_{1 \leqslant k \leqslant n} X_k (t),(X_n )\) are independent copies of \(X,||x(t)|| = \mathop {\sup }\limits_{1 \in T} |x(t)|\) , and (*a* _{n}) and (*b* _{n}) are numerical sequences.

### Asymptotic method for the investigation of *m*-frequency oscillation systems

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1366–1387

We describe an asymptotic method for the integration of *m*-frequency oscillation systems of order 2n, analyze averaged equations in nonresonance and resonance cases, prove a theorem on the preservation of smooth *p*-dimensionai invariant tori under perturbation for any 0 ≤ *p ≤ n*, and indicate forms of decomposable *m*-frequency oscillation systems.

### Moduli and extremal metrics on nonorientable and twisted Riemannian manifolds

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1388–1398

We present results on the moduli and extremal metrics of families of curves on certain nonorientable or twisted Riemannian manifolds. We also introduce and investigate weighted *l*-moduli of families of curves and the corresponding extremal metrics.

### Asymptotic solution of one-dimensional spectral boundary-value problems with rapidly varying coefficients: The case of multiple spectra

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1399–1408

We suggest a method for the construction of complete asymptotic expansions for eigenvalues and eigen-functions of spectral boundary-value problems for differential equations with rapidly varying coefficients in the case of multiple spectra of the averaged problem. The effect of splitting of multiple eigen-values is illustrated by an example of a special fourth-order problem.

### Control over a linear system up to a given Lyapunov function

Khusainov D. Ya., Kozhametov A. T.

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1409–1415

We obtain control conditions for a linear system whose Lyapunov functions satisfy certain conditions.

### On the 85th birthday of Alexandr Yul’evich Ishlinskii

Bogolyubov A. N., Mitropolskiy Yu. A., Pustovoytov M. O.

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1416–1418

### On the first ruin moment for a modified risk process with immediate reflection

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1419–1425

For a modified risk process with immediate reflection downward, we establish relations for an integral transformation of its characteristic function and the corresponding transformation of the limit distribution of the considered process under ergodicity conditions. The distribution is obtained for the first ruin moment of the introduced risk process.

### On the distribution of the maximum of the difference of independent renewal processes with discrete time

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1426–1432

We find the distribution of the maximum of the difference of two renewal processes with discrete time that is semicontinuous in discrete topology.

### Density of the transition probability for one class of generalized diffusion processes

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1433–1437

The existence of the density of the transition probability is investigated for a generalized diffusion process with transport that satisfies a certain condition of integrability with respect to the Gaussian measure.

### Polynomial matrices over a factorial domain and their factorization with given characteristic polynomials

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1438–1440

We establish conditions for the existence of a unital divisor with given characteristic polynomial of a polynomial matrix over a factorial domain.