We consider the anisotropic Lorentz space of 2π-periodic functions of m variables and the Nikol’skii–Besov space of functions with mixed generalized logarithmic smoothness. Embedding theorems are proved for spaces of functions with mixed generalized logarithmic smoothness.

Monotone sets have been quite actively studied in recent years in geometric approximation theory. The concept of monotone path-connected sets has proved especially useful. The purpose of the present paper is to give a short but comprehensive survey on this topic; we also illustrate relations with key properties of approximating sets, of which we consider characterizations of best approximants, and properties of uniqueness and stability.

The paper is devoted to the study of punctual structures such that any isomorphism between any of its punctual copies is primitive recursively reducible to a fixed 0, 1-valued oracle function. To estimate the complexity of such punctual structures and isomorphisms between them we introduce and investigate the weak (0, 1-valued) jump operation. In the paper we establish that there is a rigid punctual structure for which all isomorphisms are low under the weak jump, and at least one of them is not primitive recursive. Also we construct a rigid punctual structure with every isomorphism reducible to the weak jump of the zero function, and with at least one having a high degree.

We show that generalized sobrifications of approximation spaces are homeomorphic to spaces of special basic ideals of the given spaces. Using this characterization, we generalize a series of known results on sobrifications of approximation spaces.

We study the issues of chaoticity and frequently hypercyclicity of various operators in the weighted space F_φ(Cⁿ), defined as the projective limit of Banach spaces. Theorems 8–13 consider the cases of differentiation and shift operators, as well as their compositions in F_φ(Cⁿ). For linear continuous operators commuting with differentiation, Theorem 14 shows that they are chaotic in F_φ(Cⁿ). In Theorem 15, such operators are proved to be frequently hypercyclic in F_φ(Cⁿ), and also are the most important consequences of these statements are indicated.

An initial-boundary value problem in a quarter-plane for one pseudohyperbolic equation is considered. We establish conditions for boundary functions ender which the initial-boundary value problem is uniquely solvable in Sobolev spaces with an exponential weight.