2017
Том 69
№ 9

# Volume 13, № 1, 1961

Article (Russian)

### On a class of computation algorithms for approximate integration of ordinary differential equations with initial conditions

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 3-21

The author first introduces the concepts of computation and real computation algorithms for approximate integration, by the method of finite differences, of systems of ordinary differential equations with initial conditions and investigates the general properties of these algorithms, in particular the property of their stability.
Then a study is made of the class of computation algorithms based on methods of the Euler, Runge and Adams type for numerical integration of these equations, and it is shown that this class of computation algorithms makes it possible to determine with sufficient precision the guaranteed interval of existence of the solution sought for, to estimate the position of the graph of this solution and to find a guaranteed real estimate of the error of the numerical solution obtained.

Article (Russian)

### A generalization of Haag's theorem

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 22-27

An analogue of Haag's theorem is proved for the case of a neutral scalar field. The vacuum expectation apparatus developed by A. Wightman and G. Källen is used lor the proof.

Article (Russian)

### Approximate solution of singular integral equations of the convolution type by Galerkn's method

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 28-38

This paper discusses the question of finding an approximate solution of equation (2) by a procedure of the Galerkin type for any index. A theorem on the convergence ol the approximate solution to the exact solution in space $L_2$ is proved. An example on the shore refraction of a plane electromagnetic wave is solved.

Article (Russian)

### Approximate solution of linear operator eqmtions in a Banach space by Yu. D. Sokolov's Method

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 39-52

Yu. D. Sokolov's method was applied to obtain approximate solutions (16) of the linear operator equation (2) in a Banach space. The sufficient condition $L_k < 1$ (29) for the convergence of the process is derived; estimates of the error (41), (42) and (44) are given; and the efficacy of Yu. D. Sokolov's method is illustrated by examples.

Article (Russian)

### Élaboration de quelques procédés de calcul pour la construction approximative des solutions des problèmes tchebycheviens à paramètres entrant non linéairement II

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 53-62

Dans ce qui suit (§ 4 de la présente partie II, ainsi que §§ 5 ,6 de la partie III qui terminera le travail) nous envisageons, plus spécialement, ces problèmes tchebycheviens d'approximation non linéaire de l'espèce de (27) qui permettent l'adaptation de généralisations convenables de la méthode des interpolations tchebyclieuiennes, successives basée sur le 2-nd algorithme de E. J. Rémès [3, 2, 4]. En comparant les nouveaux procédés ici élaborés avec le mode d'agir formulé dans la partie I (§ 2) du travail (ce Journal, t. XII, N 3, 1960), on constatera ici une utilisation bien plus limitée du principe de linéarisation approximative [(6)—(33)], en tant que l'énoncé non linéaire précis du problème (27) reste maintenant, à chaque phase du processus calculatoire, d'une manière plus directe dans le champ visuel.

Article (Russian)

### Proof of Haag's asymptotic condition for a two-dimensional quantum field theory

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 63-71

The quantum field theory is considered in this paper on an axiomatic basis in a two-dimensional space-time. Calculating the vacuum expectation of the product of the field operators, the author shows that in the two-dimensional case the generalized Kallen-Willhelmsson functions of all orders are expressed by those of the third order. An explicit form of this function is found. A simple proof is given of Haag's asymptotic condition, which also holds for an unfixed time.

Article (Russian)

### Determination of the constants of the Christoffel-Schwarz integral by simulating on resistance paper

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 72-79

A very simple procedure is described for the experimental determination of the constants of the Christoffel-Schwarz integral, which ensures sufficient precision for solving many technical problems.

Article (Russian)

### Application of the method of reflections to certain biharmonic problems

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 80-90

In this paper we discuss the problem of bending of rectangular strips of concentrated forces. The challenge is to integrate the biharmonic equation with boundary conditions of three types: hinged edge, rigid clamping and free edge.

Brief Communications (Russian)

### Numerical solution of linear differential equations of the second order with Sturm boundary conditions

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 91-95

Brief Communications (Russian)

### On a problem of G. Borg

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 95-100

Brief Communications (Russian)

### On the asymptotic estimate of the remainder in approximating periodic functions satisfying Lipshitz's condition by interpolation polynomials with equidistant nodes

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 100-106

Brief Communications (Russian)

### Perturbation of a self-adjoined operator by a finite-dimensional and the condition of completeness

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 106-111

Brief Communications (Russian)

### On the conformai mapping of a circle on a rectangular region

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 111-117

Brief Communications (Russian)

### On one example of a point set

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 117-118

Anniversaries (Russian)

### Nikolai Ivanovich Muskhelishvili (on his 70 aniversary)

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 119-123