2019
Том 71
№ 6

# Volume 14, № 1, 1962

Article (Russian)

### On primarily factorable groups

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 3-9

In the present paper the author solves the question of the structure of groups with complemented $p$-subgroups (a subgroup $\mathfrak{U}$ of a group $\mathfrak{G}$ is said to be complemented in $\mathfrak{G}$ if there exists in $\mathfrak{G}$ such a subgroup $\mathfrak{B}$ that $\mathfrak{UB} = \mathfrak{G}$ and $\mathfrak{U} \bigcap \mathfrak{B} = 1$ which was suggested to the author by S. N. Chernikov. If the group is periodical, the necessary and sufficient conditions are given for the p-subgroups of the group to be complemented in it. It is also shown that in general not every subgroup of a periodical group is complemented if all its $p$-subgroups are complemented.

Article (Russian)

### Application of the method of majorant region for determining the filtration discharge with unknown depth of bedding of the impermeable soil

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 10-17

The method of majorant regions is used to determine the filtration in three essentially different cases:
1) the depth of bedding of the impermeable soil is taken as infinity;
2) the impermeable soil is at a finite depth;
3) the depth of bedding of the impermeable soil is unknown. An example is given of the first case.

Article (Russian)

### Infinitesimal bends of «glide» of rotation surfaces with negative curvature

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 18-29

The author investigated infinitesimal bends of «glide» of surfaces of rotation with negative curvature.
For some rotation surfaces of negative curvature the necessary and sufficient conditions of non-rigidity are indicated in respect to infinitesimal bends of «glide».

Article (Russian)

### On the integral representation of hermitian definite matrices with negative squares

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 30-39

In this article the integral representation of hermitian matrices $F_{j, k}$ $(j, k = 0, ± 1, ...)$ for which all the sums $\sum^n_{j, k = -n} F_{j, k} \xi_j, \xi_k$ have exactly $\mathfrak{x}$ negative squares for large n is given in terms of eigenfunctions of difference equations.

Article (Russian)

### Application de certains analogos naturels du second algorithme polynomial aux problèmes du minimax tchebychevien, ordinaire ou généralisé, à paramètres entrant linéairement

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 40-56

Dans un article récent de Ed. Stiefel [16] ont été elucidées quelques relations importantes entre certains procédés, employés dans la résolution des problèmes de meilleure approximation uniforme pour un système d'équations linéaires incompatibles, et la sipmlexe-méthode de programmation linéaire.
Dans le présent article on considère des analogos de la variante semioptimale du second algorithme polynomial de l'auteur [2, 4, 8], appliqués aux problèmes le plus généraux du minimax tchebychevien et quasitchebyche-vien, libre ou conditionné (problèmes $(A), (B), (C), (D)$ dans §§ 2, 3), pour un système fini de fonctions linéaires réelles. Ils se montrent réductibles à la phase principale («$2-n$ de phase» [19]) de la simplexe-méthode qui permet l'élaboration de programmes standardisés des calculs à l'aide des machines électroniques à rapide action.
Pour le cas particulier du premier problème (7) — $(A)$ où l'on suppose remplie la condition de Haar—Vallée Poussin, une question semblable concernant la «méthode d'échange» de E. Stiefel [15], analogue à quelque variante moins précisée «admissible» du second algorithme polynomial (cf. la fin du § 1)] a été considérée dans [16].

Article (Russian)

### Some criteria of the limitation of solutions of systems of nonlinear differential equations

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 57-68

The systems of nonlinear differential equations 1, 2, 3 are considered in this paper.
These systems are not always solved with respect to the highest derivatives. For systems 1 and 2 some criteria of limitation or tendency to zero of the solutions for $t \rightarrow \infty$ are proved. For the system 1 the criterion of limitation of the solutions and their derivatives is proved. The theorem on the average value is used to establish these theorems.
The estimations of the solutions depend on the initial conditions, characteristic numbers of symmetric parts and the norms of some matrices.

Article (Russian)

### On systems of ordinary differential equations with explicit periodic dependence on the argument

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 68-86

The following problem is considered. Given functions $u, v$ are depending on the arguments $x, y, z$ partially differentiate with respect to z and periodic with respect to this argument with a period equal to unity. In addition the functions $u, v$ are analytical in respect to the two other arguments, the Jacobian of these variables is positive with any $z$ for all values of $x, у$ of the region under consideration. Then, we may construct functions $u, v$ satisfying certain conditions depending on $x, y, z, t$, analytical in $x, у$ and differentiable a sufficient number of times with respect to $t$, the Jacobian of these functions in $x, у$ differing from zero for all real values of $z, t$ and the considered $x, y$.

Brief Communications (Russian)

### On the existence and properties of an organic solution of a system of quasilinear differential difference equations

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 87-92

Brief Communications (Russian)

### Nonautonomous quasilinear system with many degrees of freedom in the general resonance region

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 92-96

Brief Communications (Russian)

### On a generalization of subprojecting spaces

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 96-100

Brief Communications (Russian)

### On some computation schemes of iterative processes

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 100-108

Brief Communications (Russian)

### On the sufficient conditions of stability of the solutions of some nonlinear equations of the second order

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 109-113

Chronicles (Russian)

### Seminar on the history of mathematical sciences of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR for five years

Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 113-115