2018
Том 70
№ 9

# Volume 13, № 2, 1961

Article (Russian)

### Brief essay on the life and scientific activity of Joseph Louis Lagrange (on the 225th anniversary of his birth)

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 127-140

Article (Russian)

### Number of blocks of characters of a finite group with a given defect

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 136-141

R. Brauer's problem of the group-theoretic characteristic of the number of blocks with a given defect is considered in this paper. The following theorems are proved:
Theorem 1. The number of blocks with defect $d$ of a group $G$ does not exceed the number of $p$-regular non-nilpotent classes with defect $d$.
The theorem is a reinforcement of a result of Brauer — Nesbitt. An example is given showing that this estimate is not always attained.
Corollary 3. Let $G$ be a finite group of order $p^aq ((p, q) = 1$, $p$ is a prime number) containing a normal subgroup $H$ of the order $p^{\gamma}q (0 < \gamma < a)$, some sylow $p$-subgroup of which is a normal subgroup of $H$. Then the number of blocks with defect $d$ coincides with the number of $p$-regular non-nilpotent classes of $G$ with the same defect.
Theorem 3. There exist no blocks with zero defect in the group $G$ if, and only if, all classes with defect zero are nilpotent.
A new proof is also presented for Brauer's theorem on the number of blocks with a maximum defect.

Article (Russian)

### On the theory of critical speeds of rotatings hafts

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 142-149

Vibrations of a horizontal elastic shaft with a fixed heavy disk on it is described by a system of three nonlinear differential equations of the second order. A. Stodola suggested a particular solution of this system which corresponds to the second critical speed, but a purely theoretical analysis, of the stability of the particular solution suggested by A. Stodola has not been carried out.
An attempt to find a solution of this problem is made in this article. In his analysis the writer uses the method of calculating the logarithm of the monodromy matrix suggested by [5], [7, a], [4].
In our case it enabled us
a)to state the necessary and sufficient conditions of stability,
b)to calculate the zone of stability in the space of parameters.

Article (Russian)

### Elaboration de quelques procédés de calcul pour la construction approximative des solutions des problèmes tchebycheviens à paramètres entrant non linéairement. III

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 150-172

Dans cette partie finale du travail, ainsi que dans la partie II précédente (§ 4), on s'occupe de ces problèmes tchebycheviens d'approximation non linéaire de l'espèce de (27) qui permettent une adaptation de généralisations convenables de la méthode des interpolations tehebycheviennes successives de E.J. Rémès 13, 2, 4]. Les procédures calculatoires indiquées dans § 4, même appliquées une fois pour la réalisation approchée de chacune des opérations successives d'interpolation tchebychevienne, étaient un peu compliquées en tant qu'elles exigeaient, comme phases préliminaires, la formation explicite des systèmes linéarisés (33) de $n + 1$ équations incompatibles ainsi que la détermination des multiplicateurs $\{C_v\}_0^n$ de la relation linéaire entre les formes linéaires correspondantes. Dans la présente partie III du travail on développe une méthode élaborée plus souple, considérablement simplifiée par l'élimination de ces phases préliminaires, dans laquelle le principe de linéarisation approximative n'est utilisé que d'une manière implicite par l'algorithme (41)—(42) et la précision des résultats de chacune des opérations successives susdites peut être haussée à volonté par l'application itérative du procédé des moyennes à poids (42) qui est caractéristique pour la méthode.

Article (Russian)

### Solution of a multivariate functional equation

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 173-189

Let there be a non-singular matrix $C_r$ for every natural $r$, equation (1) to be satisfied for every vector $T$. All scalar functions $H(T)$ with these properties are described, certain assumptions as to their continuity being made. This problem is suggested by one probabilistic problem.

Article (Russian)

### Convectional in stability of rarefied plasma

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 190-209

The method of approximate solution of equations, based on the assumption of the local nature of the perturbations, is applied to an investigation of the convectional stability in the case of various models accepted for thedescription of plasma: magnetic hydrodynamics, two-fluid hydrodynamics, (the adiabatic and non-adiabatic case), kinetic equations for rarefied plasma (without taking into account collisions). The stability conditions obtained do not depend on the form of the perturbations and are, consequently, of a universal nature. The results obtained for different plasma models are compared.

Article (Russian)

### Rapidly converging iterative processes

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 210-215

The author shows that every rapidly converging iterative process $F(x)$, constructed by a given iterative process $f(x)$, in a certain vicinity of a stationary point, where $F(x)$ and $f(x)$ are assumed to be sufficiently smooth, can be presented in the form (4).
The necessary and sufficient conditions of convergence of an iterative sequence generated by $F(x)$ are found. It is shown that any interval containing a stationary point of the iterative process $f(x)$, in which (8) occurs, is a region of attraction of this stationary point as a stationary point of the process $F(x)$.

Brief Communications (Russian)

### On the principle of the attenuation of correlation in the quant um theory of field

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 216-217

Brief Communications (Russian)

### On non - stationary oscillations of nonlinear systems with gyroscopic terms, taking into consideration links with a source of low energy

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 220-223

Brief Communications (Russian)

### Analytical structure of the solution of an equation of integral curves of asyst em of two differential equations in some cases

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 224-226

Brief Communications (Russian)

### On the expansibility of Sp and Wp nearly periodic functions into finite sums of the same

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 226-231

Brief Communications (Russian)

### On the existence and uniqueness of the solutions of a class of boundary problems of the theory of elasticity

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 231-235

Brief Communications (Russian)

### On the solution of certain problems of non-stationary filtration by means of the STJXA integrator

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 235-238

Letter (Russian)

### Letters to the Editor. Amendment

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 239

Chronicles (Russian)

### Сommercially available

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 241