Volume 16, № 2, 1964

For a certain class of coefficients a, including $\sigma \in L_p$ ($p$ depending on the number of variables) a generalized solution of (I), (II) is constructed satisfying the boundary condition (II) in a certain «strong» sense.

The authors establish the existence and properties of an $s + 1$ -dimensional local integral manifold of a system of $l + m + n$ nonlinear differential equations of the form $$\frac{dx}{dt} = X(y,z)x + \varepsilon X_1(t, x, y, z),$$ $$\frac{dy}{dt} =Y(x, z), y + \varepsilon Y_1 (t, x, y, z),$$ $$\frac{dz}{dt} = \varepsilon Z_1 (t, x, y, z),$$ where $x, y$ characterize the fast, and $z$ the slow motions.

For the class of all continuous $2jt$-periodic functions, two processes are considered for the approximation of the functions of this class by trigonometric polynomials.

A certain complete system of identical relations of the algebra of commutative events is indicated in this paper. It is shown that the problem of identity in the algebra of regular commutative events is solvable algorithmically. It is proved that neither in the algebra of commutative events nor in the algebra of events in an arbitrary alphabet does there exist a finite complete system of identical relations.

The basic assertion of this paper is that a set of branch points in space $R^n$ with $n \geq 3$ cannot be zero-dimensional. The proof is based on applying the very important theorem on continuity, which is also presented in the article.

The present paper deals with the theory of compact topological groups of finite special rank.