Table of contents

Let be an optional submartingale of the class . It is proved that there exist an optional martingale and a strongly predictable process such that the Doob decomposition is valid. Bibliography: 10 titles.

Let be one of the varieties , , or , and let be the set of all non-nilpotent subvarieties of . This set is provided in a natural way with a certain topology. A characterization of is given in terms of the corresponding structure on . Bibliography: 12 titles.

This paper studies the asymptotic behavior of the fundamental solution of the equation

specified on the whole space , , as . The coefficients are periodic functions which satisfy the conditions of ellipticity, symmetry, and infinite smoothness. The main result is the construction of the asymptotics of in the form

where is an arbitrary positive integer, the are homogeneous of degree in the first argument and periodic in the remaining arguments, and for the remainder term on the set , , the estimate

holds, where the constants are independent of , , and . Bibliography: 9 titles.

In this paper a new invariant of fibered knots is studied, represented as a homotopy pairing in the sense of Spanier and Whitehead. With its help, first of all, a classification of stable fibered knots is given; second, it is shown that certain well-known invariants in knot theory define the type of a stable knot up to a finite number of possibilities; and third, a complete algebraic description of certain classes of knots is given. Bibliography: 33 titles.

The paper is concerned with the construction and investigation of new classes of completely integrable Hamiltonian systems. Figures: 5. Bibliography: 7 titles.

In this paper, theorems on the existence of smooth solutions of certain control problems describable by the Navier-Stokes and Euler equations are proved. It is shown that a mixed boundary value problem for the Navier-Stokes and Euler equations of dimension is uniquely solvable for a dense set of right-hand sides. Bibliography: 13 titles.

In this paper we study the asymptotic properties of the order functions , , of holomorphic mappings in terms of the complex variations of corresponding plurisubharmonic functions. We introduce a class of mappings for which we prove a quasisurjectivity theorem of Sohockiĭ [or Casorati-Weierstrass] type and give the asymptotics of the order functions. Bibliography: 13 titles.