Том 14, № 3, 1962
Поведение особо возмущенных автономных нелинейных дифференциальных систем вблизи семейства цилиндров
Укр. мат. журн. - 1962. - 14, № 3. - С. 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 $\varepsilon$.
Приближенный метод решения смешанной задачи для нелинейного уравнения в частных производных гиперболического типа, содержащего малый парлметр
Укр. мат. журн. - 1962. - 14, № 3. - С. 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.
Компактные и дискретные кольцевые группы
Укр. мат. журн. - 1962. - 14, № 3. - С. 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.
Целочисленные представления симметрической группы третьей степени
Укр. мат. журн. - 1962. - 14, № 3. - С. 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.
Об одном случае непрерывной зависимости решений дифференциальных уравнений от параметра
Укр. мат. журн. - 1962. - 14, № 3. - С. 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.
Асимптотическое представление решения смешанной задачи для одного класса интегро-дифференциальных уравнений, содержащих малый параметр
Укр. мат. журн. - 1962. - 14, № 3. - С. 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.
О корреляционных функциях векторных процессов, удов летворяющих некоторым линейным дифференциальным уравнениям
Укр. мат. журн. - 1962. - 14, № 3. - С. 322-325
Полугруппы с конденсаторами
Укр. мат. журн. - 1962. - 14, № 3. - С. 325-327
одном итерационном методе решения полной проблемы собственных чисел
Укр. мат. журн. - 1962. - 14, № 3. - С. 328-329
О степенях ограниченного диссипативного оператора
Укр. мат. журн. - 1962. - 14, № 3. - С. 329-337
Спектральное представление для амплитуды Бете—Сальпетера
Укр. мат. журн. - 1962. - 14, № 3. - С. 337-339
К вопросу о построении стационарных решений для некоторых колебательных систем с одной степенью свободы
Укр. мат. журн. - 1962. - 14, № 3. - С. 340-348