# Volume 49, № 12, 1997

### Many-dimensional Dirichlet and Tricomi problems for one class of hyperbolic-elliptic equations

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1587–1593

For the generalized many-dimensional Lavrent’ev-Bitsadze equation, we prove the unique solvability of the Dirichlet and Tricomi problems. We also establish the existence and uniqueness of a solution of the Dirichlet problem in the hyperbolic part of a mixed domain.

### The unity of physical theory and its mathematical formalism

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1594–1600

The relation between the physical theory and its mathematical formalism is shown.

### Strongly nonlinear degenerate elliptic equations with discontinuous coefficients. II

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1601–1609

We use energy methods to prove the existence and uniqueness of solutions of the Dirichlet problem for an elliptic nonlinear second-order equation of divergence form with a superlinear tem [i.e., *g(x, u)=v(x) a(x)⋎u⋎* ^{p−1}u,p>1] in unbounded domains. Degeneracy in the ellipticity condition is allowed. Coefficients *a* _{i,j}(x,r) may be discontinuous with respect to the variable *r*.

### On the growth of functions represented by Dirichlet series with complex coefficients on the real axis

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1610–1616. December

We establish conditions under which, for a Dirichlet series $F(z) = \sum_{n = 1}^{∞} d n \exp(λ_n z)$, the inequality $⋎F(x)⋎ ≤ y(x),\quad x ≥ x_0$, implies the relation $\sum_{n = 1}^{∞} |d_n \exp(λ_n z)| ⪯ γ((1 + o(1))x)$ as $x → +∞$, where $γ$ is a nondecreasing function on $(−∞,+∞)$.

### Asymptotic solution of a system of integro-differential equations

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1617–1623

An asymptotic solution of a system of inegro-differential equations is constructed for the case where turning points are present.

### Estimate of the maximum of modulus of generalized solutions of the first boundary-value problem for degenerate parabolic equations

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1624–1637

We study parabolic equations of the divergent form with degeneration. We have obtained an estimate for the maximum of modulus of generalized solutions of the first boundary-value problem with a zero on the parabolic boundary.

### On the optimal coding of one class of vector functions

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1638–1645

We consider the problem of coding of parametrically defined vector functions by continuous functionals which act on their coordinate functions. We obtain the optimal method for coding some class of vector functions. In the linear case, we show the relation between this method and the informativeness of functionals with respect to the class of coordinate functions.

### Radical algebras subgroups of whose adjoint groups are subalgebras

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1646–1652

We obtain the characteristic for radical algebras subgroups of whose adjoint groups are subalgebras. In particular, we prove that the algebras of this sort are nilpotent with nilpotent length at most three. We give the complete classification of those algebras under consideration which are generated by two elements.

### One class of singular complex-valued random variables of the Jessen-Wintner type

O. V. Shkol’nyi, Pratsiovytyi M. V.

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1653–1660

We study the structure of the distribution of a complex-valued random variable ξ = Σ*a* _{ k } ξ_{ k }, where ξ_{ k } are independent complex-valued random variables with discrete distribution and *a* _{k} are terms of an absolutely convergent series. We establish a criterion of discreteness and sufficient conditions for singularity of the distribution of ξ and investigate the fractal properties of the spectrum.

### Investigation of a nonlinear difference equation in a Banach space in a neighborhood of a quasiperiodic solution

Samoilenko A. M., Slyusarchuk V. E., Slyusarchuk V. V.

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1661–1676

We investigate the behavior of a diserete dynamical system in a neighborhood of a quasiperiodic trajeetory for the case of an infinite-dimensional Banach space We find conditions sufficient for the system considered to reduce, in such a neighborhood, to a system with quasiperiodic coefficients.

### On quadrature and cubature formulas for a class of multiple singular integrals

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1677–1683

We consider the problem of conditions for the existence of multiple singular integrals of a certain class at inner and boundary points of a domain. We obtain the quadrature and cubature formulas for calculating multiple singular integrals and present the corresponding estimates for the formulas.

### Hyperbolic stefan problem in a curvilinear sector

Berehova H. I., Kirilich V. M.

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1684–1689

The problem with unknown boundaries for a first-order semilinear hyperbolic system is studied in the case where the curve of definition of the initial conditions degenerates to a point. An existence and uniqueness theorem for a classical solution of the problem is proved for small *t*.

### Quasilinear periodic boundary-value problem

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1690–1693

We study a periodie boundary-value problem for the quasilinear equation *u* _{ tt } − *u* _{ xx } = *F*[*u*, *u* _{ t }, *u* _{ x }]. We find conditions under which a theorem on the uniqueness of a solution is true.

### Exact estima tes for the rate of convergence of the *s*-step method of steepest descent in eigenvalue problems

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1694–1699

We obtain exact (unimprovable) estimates for the rate of convergence of the *s*-step method of steepest descent for finding the least (greatest) eigenvalue of a linear bounded self-adjoint operator in a Hilbert space.

### Estimates of the Kolmogorov widths for classes of infinitely differentiable periodic functions

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1700–1706

Lower estimates of the Kolmogorov widths are obtained for certain classes of infinitely differentiable periodic functions in the metrics of *C* and *L.* For many important cases, these estimates coincide with the values of the best approximations of convolution classes by trigonometric polynomials calculated by Nagy, and, hence, they are exact.

### A theorem on the structure of a complete set of conformal-like series of conserved quantities for massless fields

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1707–1711

We formulate a convenient general method for construating a complets set of comformal-like series of conservation laws of the *n* th order. As examples, we give all conformal-like series which are generated by symmetric tensors of the third and fourth ranks.

### Smooth solution of one boundary-value problem

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1712–1716

We study the boundary value problem for the quasilinear equation *u* _{u} − u_{xx}=F[u, u_{t}], u(x, 0)= u(x, π)=0, u(x+w, t)=u(x, t), x ε ®, t ε [0, π], and establish conditions under which a theorem on the uniqueness of a smooth solution is true.

### A criterion of exponential dichotomy for a countable system of differential equations with quasiperiodic coefficients

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1717–1722

We establish necesary and sufficient conditions for the exponential diehetemy with matrix projeseters of a countable system of differential equations with quasiperiodic coefficients.

### Index 49th volume of "Ukrainian Mathematical Journal"

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1723-1728