# Volume 14, № 3, 1962

### Behaviour of particularly perturbed autonomous nonlinear differential systems in the differential systems in the neighbourhood of a family of cylinders

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 235-249

The author considers the system of differential equations
$$\frac{dx}{dt} = f(x,z), \quad \varepsilon\frac{dz}{dt} = F(x, z),\quad (1)$$
and the corresponding degenerated system $$\frac{d\bar{x}}{dt} = f(\bar{x},\varphi(\bar{x})),\quad \bar{z} = \varphi(\bar{x}),\quad (2)$$
where $z = \varphi(x)$ is an isolated solution of the system $F{x,z) = 0$.

It is proved that if system (2) has a family of solutions
$$\bar{x} = \bar{x}^0(\theta, c), \quad \bar{z} = \bar{z}^0(\theta, c),$$
periodic in $\theta = \omega (c)t + \varphi_0$ with a period of $2\pi$, family (1) has a stable family of periodic solutions
$$x = x^0(\theta, c, \varepsilon), z = z^0(\theta, c, \varepsilon)$$
with the same period; furthermore $|x^0 - \bar{x}^0| \rightarrow 0, \quad |z^0 - \bar{z}^0| \rightarrow 0$ together with $\varepsilon$.

### Approximate method of solving the mixed problem for a nonlinear partial differential equation containing a small parameter.

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 250-259

A method is elaborated for reducing the mixed problem [1], [3], [4] for a partial differential equation of the hyperbolic type to Cauchy's problem for an infinite system of ordinary differential equations (16). The grounds are given for applying the averaging method of N. M. Krylov and N. N. Bogoliubov to the solution of the resulting system of differential equations. The proposed method is of interest in connection with certain problems of the theory of dynamic stability.

### Compact and discrete ring groups

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 260-270

Ring groups were introduced in note [1] as a generalization of uni-modular groups. In the present note an analogy of Pontrygin's duality theorem for compact and discrete groups is established for ring groups.

### Whole-number representations of a symmetrical group of third degree

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 271-288

The authors discuss whole-number representations to a symmetrica! group of the third degree. It is shown that there exists only a finite number, i. e. ten, prime representations of this group. The dimensions of the prime representations do not exceed the order of the group.

It is further shown that the factoring of any representation into a direct sum of primes is univalent.

Thus the first example has been given of a complete description of whole-number representations of a non-commutative group.

### On a case of continuous dependence of the solutions of differencial equations on the parameter

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 289-298

A theorem is proved on the continuous dependence on the parameter of solutions of the integral equation (1.1). It is used for the investigation of the dependence on the parameter of the solution of a differential equation in respect to which the right side is continuous in the integral sense, and for finding the limiting function to the solutions.

### Asymptotic representation of the solution of a mixed problem for one class of integro-differential equations containing a small parameter

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 299-307

An approximate method is considered for the conformal mapping of arbitrary simply connected univalent regions which may be means of some elementary function be mapped on a half-plane with an aperture of arbitrary shape cut in it (fig. 1). The mapped function is sought in the form of a section of series (H or (18). Using the method of least squares the problem in the case of series (1) is reduced to the solution of system (13) the coefficients of which are easily calculated by the recurrent formulae (10) and the formulae of numerical quadratures. In the case of series (IS) the problem is reduced to the solution of an analogous system of linear equations (26). Three examples are considered in which the results obtained are compared with exact mapping functions. In § 3 the described method is extended to the case of regions with a finite number of slits.

### On the correlation functions or vectorial processes satisfying certain linear differential equations

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 322-325

### Semi-groups with condensers

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 325-327

### On an iterative method for solving the complete eigen-values problem

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 328-329

### On the powers of a limited dissipative operator

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 329-337

### Spectral representation for the Bethe—Salpeter amplitude

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 337-339

### On consructing stationary solutions for certain oscillatory; systems with one degree of freedom

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 340-348