2018
Том 70
№ 11

# Volume 13, № 4, 1961

Chronicles (Russian)

### Science, an Instrument for the Building of Communism

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 3-4

Article (Russian)

### On representation of the complete product of finite groups

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 5-12

Let $Q$ be a complete product of the permutation group $H$ and an arbitrary finite group $G$ and let $R(Q, K)$ define the group algebra of the group $Q$ over the field of the complex numbers $K$.
We determine the system of the primitive idempotents of the algebra $R(Q, K)$ to which all the irreducible representations of the group $Q$ over the field $K$ correspond.
The necessary and sufficient conditions are also obtained for the equivalency of two irreducible representations of the group $Q$ over the field $K$.
In the case when the group $H$ is cyclic, we give formulas of the number of irreducible representations of fixed degree.

Article (Russian)

### Spectral exp nsion of self-conjoint operators on generalized elements of a Hilbert space

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 13-33

The author discusses spectral expansions of generalized eigen-elements belonging to systems derived in the article (Ukrainian Mathematical Journal, vol. 12, N 1, 1960). The necessary and sufficient condition of the possibility of such expansions was obtained. Some generalizations are discussed, as well as a supplement to the estimate of the growth of eigen-functions and the interpretation of «eigen-subspaces» of self-adjoint operators.

Article (Russian)

### Zur Darstellung der zu den Orlicz-Klassen gehörenden Funktionen durch das singulare Integral

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 34-50

Es wird die Definition des Lebesgue — Orlicz-Punktes gegeben, die eine Verallgemeinerung der Definition des Zebesgue-Punktes $p$-er Ordnung ist.
Darauf wird die Darstellung der Funktionen durch das singulare Integral in diesen Punkten auseinandergesetzt. Gefunden sind eine notwendige Bedingung, eine hinreichende Bedingung und in einigen Fällen auch eine notwendige und hinreichende Bedingung der Darstellung von Funktionen durch das singulare Integral in den Lebesgue—Orlicz-Punkten.

Article (Russian)

### On the numerical realization of the method of averaging functional corrections

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 51-66

In this article the author first presents the numerical realization of the method of averaging functional corrections on solving linear Fredholm integral equations of the second kind, then an investigation of the conditons of convergence of calculational processes arising during the numerical realization of the method and an estimate of the error of the approximate solution to be found is established. Then an investigation is conducted on the applicability of the method of averaging the functional corrections to the solution of systems of linear algebraic equations.

Article (Russian)

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

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 67-78

Let $\xi_1, \xi_2,... \xi_n,...$ be independent identically distributed random variables, $s_{n0} = 0,\; s_{nk} = \cfrac1{\sqrt{n}}(\xi_1 + ... + \xi_k)$; and $\Phi_n(x_0, x_1, ..., x_r)$ the sequence of non-negative measurable functions for which $\lim_{n\rightarrow \infty}\sup_{x_0, x_1, ..., x_n}\Phi_n(x_0, x_1, ..., x_r) = 0$.
Limit theorems for random variables $\cfrac1n\sum_{k=0}^{n-r}\Phi_n(s_{nk},...,s_{nk+r})$ are obtained in the article.

Article (Russian)

### Sur une méthode de la resolution approchée des systèmes d'équations inté'grales linéaires

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 79-87

L'auteur expose dans cet mémoire l'application de la méthode des correct ons fonctionnelles moyennes à la résolution approchée du système d'équations intégrales linéaires de la forme $$y_i(x) = \varphi(x) + \lambda \sum^m_{j=1}\int^b_a K_{ij}(x,\xi)y_j(\xi)d\xi\quad (b-a = h > 0;\; i = 1,2,...m). (1)$$ En prenant pour premières valeurs approchées des $y_i(x)$ $$y_{i1}(x) = \varphi_i(x) + \lambda h \sum_i \alpha_{j1}M_{ij}(x) (2)$$ où $$M_{ij}(x) = \frac1h\int_a^bK_{ij}(x,\xi)d\xi \quad \alpha_{ij} = \frac1h\int_a^by_{i1}(x)dx,\;\;(3)$$ on pose d'une façon générale $$y_{in}(x) = \varphi_i(x) + \lambda\sum_i \int_a^bK_{ij}(x,\xi) \left[y_{\overline{jn-1}}(\xi) + \alpha_{jn}\right] d\xi\;\; (n = 2, 3,...,),\;\;(2n)$$ où $$\alpha_{jn} = \frac1h\int_a^b\left[y_{jn}(x) — y_{\overline{jn-1}}(x)\right]dx.\;\;(2n)$$ Les §§ 2, 3 contiennent quelques conditions suffisantes de convergence du procédé. On donne aussi un exemple de l'application de la méthode.

Brief Communications (Russian)

### Oil the theory of special classes of univalents in the unity circle of functions with $k$-fold symmetry of rotation

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 88-92

Brief Communications (Russian)

### On a method of constructing difference equations during the solution by the method of nets of Dirichlet's internal problem for a Poisson equation

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 92-96

Brief Communications (Russian)

### On a property of almost periodic polynomials

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 96-98

Brief Communications (Russian)

### On analytical properties of contributions from diagrams with five outer impulses

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 98-104

Brief Communications (Russian)

### On the algorithm-of determining the equivalent initial conditions for linear heterogeneous differential equations with constant coefficients

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 104-109

Brief Communications (Russian)

### On the impossibility of constructing a non-local theory of field with a positive spectrum of tile energy-impulse operator

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 109-111

Brief Communications (Russian)

### Generalization of a theorem of Y. S. Dubnov

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 111-114

Chronicles (Russian)

### International symposium on nonlinear vibrations

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 115-116

Letter (Russian)

### Leter to Oditor. Correction

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 116