# Volume 50, № 11, 1998

### The best $L_1$-approximations of classes of functions defined by differential operators in terms of generalized splines from these classes

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1443-1451

For classes of periodic functions defined by constraints imposed on the $L_1$-norm of the result of action of differential operators with constant coefficients and real spectrum on these functions, we determine the exact values of the best $L_1$-approximations by generalized splines from the classes considered.

### On one proof of the classical solvability of the Hele-Shaw problem with free boundary

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1452–1462

We consider the Stefan problem for a parabolic equation with a small parameter as the coefficient of the derivative with respect to time. We justify the limit transition as the small parameter tends to zero, which enables us to prove the classical solvability of the Hele-Shaw problem with free boundary in the small with respect to time.

### On the Levy-Baxter theorems for shot-noise fields. I

Buldygin V. V., Mel'nik V. M., Shportyuk V. G.

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1463–1476

We consider shot-noise fields generated by countably additive stochastically continuous homogeneous random measures with independent values on disjoint sets. We establish necessary and sufficient conditions under which the shot-noise fields possess the Levy-Baxter property on fixed and increasing parametric sets.

### Uniqueness theorems for multiple lacunary trigonometric series

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1477–1481

In a many-dimensional space, we study some properties of functions with lacunary Fourier series depending only on the values of these functions in a neighborhood of a certain point.

### Properties of the fundamental solutions and uniqueness theorems for the solutions of the Cauchy problem for one class of ultraparabolic equations

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1482–1496

For one class of degenerate parabolic equations of the Kolmogorov type, we establish the property of normality, the convolution formula, the property of positivity, and a lower bound for the fundamental solution. We also prove uniqueness theorems for the solutions of the Cauchy problem for the classes of functions with bounded growth and for the class of nonnegative functions.

### Groups all proper quotient groups of which possess layer-Chernikov properties

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1497–1505

We describe solvable groups all proper quotient groups of which possess layer-Chernikov properties.

### Approximation of solutions of operator-differential equations by operator polynomials

Kashpirovskii A. I., Mytnik Yu. V.

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1506–1516

We prove theorems that characterize the classes of functions whose best approximations by algebraic polynomials tend to zero with given order. We construct approximations of solutions of operator-differential equations by polynomials in the inverse operator.

### Random oscillations in the Van der Pol system under the action of a broadband random process

Mitropolskiy Yu. A., Nguen Dyk Tin, Nguyen Dong An

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1517–1521

We construct the second approximation for random oscillations described by the Van der Pol equation which are under the action of a broadband random process.

### On the noncompactness of classes of mappings with restrictions on dilation in measure

Potemkin V. L., Ryazanov V. I.

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1522–1531

We establish that the classes of mappings with restrictions in measure cannot be compact except for certain degenerate cases. In particular, this implies that any David class ΓI is noncompact.

### Structure of locally graded CDN (]-groups

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1532–1536

We introduce the notion of a CDN(]-group *G*, namely, a group such that, for any pair of its subgroups *A* and *B* such that *A* is a proper nonmaximal subgroup of *B*, there exists a normal subgroup *N* of *G* and *A < N ≤ B*. Thirteen types of non-Dedekind nilpotent groups and 9 types of nonnilpotent locally graded groups of this kind are described.

### Linear periodic boundary-value problem for a second-order hyperbolic equation. I

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1537–1544

In three spaces, we obtain exact classical solutions of the boundary-value periodic problem *u* _{tt}−*a* ^{2} *u* _{xx}=*g*(*x*,*t*), *u*(0,*t*)=*u*(π,*t*)=0, *u*(*x*,*t*+*T*)=*u*(*x*,*t*)=0, *x*,*t*∈ĝ

### Quotient groups of locally graded groups and groups of certain Kurosh-Chernikov classes

Chernikov N. S., Trebenko D. Ya.

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1545–1553

We establish the validity of the inclusion *G/N∈* X for groups *G ∈* X under certain restrictions on *N* ⊴ *G*, where X is one of the following classes, the class of locally graded groups, the class of *RI*-groups, or the class \(\hat P\mathfrak{Y}\) for a fixed group variety \(\mathfrak{Y} \supseteq \mathfrak{A}\) .

### Space-time localization in one-dimensional problems with free boundary for nonlinear second-order equations

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1554–1558

We investigate the effect of space localization and stabilization for finite time for thermal and diffusion processes that take place in active media and are described by nonlinear evolution equations.

### A new class of compact spherical spaces

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1559–1563

We describe a class of almost symmetric spherical spaces which is absent in the known classifications made by M. Krämer and I. Mikityuk.

### Paley-Wiener-type theorem for nilpotent Lie groups

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1564–1566

A Paley-Wiener-type theorem is proved for connected and simply connected Lie groups.

### Stable solutions of a system of two connected Chua cells

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1567–1569

We investigate a nonlinear chain of two Chua oscillators and its stable solutions.

### Hausdorff diameter of a nonsymmetric class of vector functions

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1570–1573

In the space of parametrically determined *m*-dimensional curves with the Hausdorff metric, we find the diameter of a class of curves whose coordinate functions satisfy the Lipschitz condition on some segment and take fixed values at its endpoints. We obtain the dependence of relations determining the value of the diameter on the evenness of *m*.

### Riquier problem for a nonlinear equation resolved with respect to the iterated Levi Laplacian

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1574–1577

Solutions are found for the nonlinear equation *Δ* _{ L } ^{2} *U*(*x*) = *f*(*U*(*x*)) (here, *Δ* _{ L } is an infinite-dimensional Laplacian) which is solved with respect to the iterated infinite-dimensional Laplacian. The Riquier problems are stated for an equation of this sort.

### On the London theorem concerning the Borel relation for entire functions

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1578–1580

An estimate exact in a certain sense is obtained for the value of the exceptional set in the Borel relation for entire functions

### Numerical-analytic method for the investigation of multipoint boundary-value problems for systems of differential equations with transformed argument

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1581–1584

The problem of existence and approximate construction is studied for the solution of a nonlinear system of differential equations with a transformed argument and linear multipoint boundary conditions.