2019
Том 71
№ 11

# Volume 16, № 1, 1964

Article (Russian)

### On a class of linear groups

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 3-12

The author considers the matrix linear groups described in 13], namely partition groups. The problem was to distinguish in the complete linear group, given as invariant, the subgroups which, with proper choice of basis, become partition groups. Three solutions of the problem are given.

Article (Russian)

### Investigation of the solutions of a system of $n + m$ nonlineai differential equations in the vicinity of an integral manifold

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 13-30

For a system of $n + m$ equations $$\frac{dx}{dt} = X(y)x + \varepsilon X*(t, x, y),$$ $$\frac{dy}{dt} = \varepsilon Y(t, x, y),$$ where $x, X*, y, Y$ are respectively $n$ and $m$ vectors, $X — n \times n$ is the matrix, $\varepsilon$ is a small parameter, the author proves the theorem of the existence and properties of a two-dimensional local integral manifold in the neighbourhood of family of periodic solutions $$x = 0,\; y = y^0(\psi, a)$$ oi the lollowing auxiliary system $$\frac{dx}{dt} = X(y)x,$$ $$\frac{dy}{dt} = \varepsilon Y_0(x, y),$$ where $$Y_0(x, y) = \lim_{T\rightarrow 0}\int_0^T Y(t, x,y)dt.$$

Article (Russian)

### Complex singular points of contributions of Feinman's diagrams and continuity theorem

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 31-40

A method is proposed, based on the application of the theorem of continuity and permitting the determination of which points of Landau's surface are singular points of contributions of Feinman's diagrams on a «physical sheet».

Article (Russian)

### Some limit theorems for additive functionals of a sequence of sums of independent random variables

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 41-60

Let $\xi_1, \xi_2, ...,\xi_n,...$ be independent identically distributed random variables, $S_{n0} = 0,\; S_{nk} = \frac1{\sqrt{n}} (\xi_1 + ...+ \xi_k)$, and $f_n(x, у)$ the sequence of measurable functions for which $\lim_{n\rightarrow \infty } \sup_{x, y} |f_n(x, у)| \rightarrow 0$. Limit theorems for random variables $\sum_{k=0}^{n—1}f_n(S_{nk}, S_{nk+1})$ are obtained.

Article (Russian)

### Coexistence of the cycles of a continuous mapping of the line into itself

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 61-71

The basic result of this investigation may be formulated as follows. Consider a set of natural numbers in which the following relationship is introduced: $n_1$ precedes $n_2$ ($n_1 \preceq n_2$) if for any continuous mappings of the real line into itself the existence of a cycle of order $n_2$ follows from the existence of a cycle of order The following theorem holds.
Theorem. The introduced relationship transforms the set of natural numbers into an ordered set, ordered in the following way:
$3 \prec 5 \prec 7 \prec 9 \prec 11 \prec ... \prec 3 \cdot 2 \prec 5 \cdot 2 \prec ... \prec 3 \cdot 2^2 \prec 5 \cdot 2^2 \prec ... \prec 2^3 \prec 2^2 \prec 2 \prec 1$

Brief Communications (Russian)

### Some problems of unsteady inertial motion of an ideal liquid

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 72-78

Brief Communications (Russian)

### Criterion of completeness of a system of root vectors of compression

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 78-82

Brief Communications (Russian)

### On representations of finite groups over a ring of classes of subtraction by modulus $m$

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 82-89

Brief Communications (Russian)

### On the curvature of curves on a smooth surface at points where no second derivatives exist

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 89-93

Brief Communications (Russian)

### On the properties of chains with complete bonds

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 93-99

Brief Communications (Russian)

### Example of an annular group generated by Lee groups

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 99-105

Brief Communications (Russian)

### Chebyshev approximations and the problem of moments

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 105-115

Brief Communications (Russian)

### On single-value solutions of nonlinear differential equations of the first order and some properties of real periodic solutions

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 110-115

Brief Communications (Russian)

### On some approximate methods of solving nonlinear equations in a coordinate Banach space

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 115-120

Brief Communications (Russian)

### Inner radius of a region and some of its properties

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 117-122

Brief Communications (Russian)

### On the determinative aggregate of relationships of the algebra of regular events

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 120-126

Brief Communications (Russian)

### On some multidimensional determinants with integer elements

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 126-132

Letter (Russian)

### Letter to the Editor

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 135

Chronicles (Russian)

### In memory of Academical N. M. Krylov

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 136