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There are many unsolved problems in discrete mathematics and mathematical cybernetics. Writing a comprehensive survey of such problems involves great difficulties. First, such problems are rather numerous and varied. Second, they greatly differ from each other in degree of completeness of their solution. Therefore, even a comprehensive survey should not attempt to cover the whole variety of such problems; only the most important and significant problems should be reviewed. An impersonal choice of problems to include is quite hard. This paper includes 13 unsolved problems related to combinatorial mathematics and computational complexity theory. The problems selected give an indication of the author's studies for 50 years; for this reason, the choice of the problems reviewed here is, to some extent, subjective. At the same time, these problems are very difficult and quite important for discrete mathematics and mathematical cybernetics.

Bibliography: 74 items.

This paper is a survey of concepts and results connected with generalizations of the notion of a periodic sequence, both classical and new. The topics discussed relate to almost periodicity in such areas as combinatorics on words, symbolic dynamics, expressibility in logical theories, computability, Kolmogorov complexity, and number theory.

Bibliography: 124 titles.

This paper contains results relating to billiards and their applications to various resistance optimization problems generalizing Newton's aerodynamic problem. The results can be divided into three groups. First, minimum resistance problems for bodies moving translationally in a highly rarefied medium are considered. It is shown that generically the infimum of the resistance is zero, that is, there are almost `perfectly streamlined' bodies. Second, a rough body is defined and results on characterization of billiard scattering on non-convex and rough bodies are presented. Third, these results are used to reduce some problems on minimum and maximum resistance of moving and slowly rotating bodies to special problems on optimal mass transfer, which are then explicitly solved. In particular, the resistance of a 3-dimensional convex body can be at most doubled or at most reduced by 3.05% by grooving its surface.

Bibliography: 27 titles.