Table of contents

A system of differential equations with 5 unknowns is fully investigated; this system is equivalent to the existence of a parallel Spin(7)-structure on a cone over a 3-Sasakian manifold. A continuous one-parameter family of solutions to this system is explicitly constructed; it corresponds to metrics with a special holonomy group, SU(4), which generalize Calabi's metrics.

Bibliography: 10 titles.

In the paper, we find necessary and sufficient conditions under which, if is a morphism of algebraic varieties (or, in a more general case, of stacks), the derived category of  can be recovered by using the tools of descent theory from the derived category of . We show that for an action of a linearly reductive algebraic group  on a scheme  this result implies the equivalence of the derived category of -equivariant sheaves on and the category of objects in the derived category of sheaves on with a given action of  on each object.

Bibliography: 18 titles.

It is well known that the 'Fukaya category' is actually an -precategory in the sense of Kontsevich and Soǐbel'man. This is related to the fact that, generally speaking, the morphism spaces are defined only for transversal pairs of Lagrangian submanifolds, and higher multiplications are defined only for transversal sequences of Lagrangian submanifolds. Kontsevich and Soǐbel'man made the following conjecture: for any graded commutative ring , the quasi-equivalence classes of -precategories over are in bijection with the quasi-equivalence classes of -categories over  with strict (or weak) identity morphisms.

In this paper this conjecture is proved for essentially small -(pre)categories when is a field. In particular, this implies that the Fukaya -precategory can be replaced with a quasi-equivalent actual -category.

Furthermore, a natural construction of the pretriangulated envelope for -precategories is presented and it is proved that it is invariant under quasi-equivalences.

Bibliography: 8 titles.

Several different constructions of a spectral sequence for a Serre fibration over a compact simply connected manifold  are considered in this paper. Namely, we consider the spectral sequence for the minimal model of the fibration, along with the spectral sequences arising from the Čech filtration in the complexes and , where is a covering of the base . It is known that all these spectral sequences have the same terms and converge to the cohomology of the total space . A new natural isomorphism of these spectral sequences is constructed in every term  with . It is also proved that in the case of a smooth locally trivial fibration these spectral sequences are isomorphic to the spectral sequences of the complex of smooth forms  and of the Čech-de Rham complex. It is therefore established that all these constructions give the same spectral sequence, starting from the term.

Bibliography: 9 titles.

An initial-value problem for a linear ordinary differential equation of noninteger order with Riemann-Liouville derivatives is stated and solved. The initial conditions of the problem ensure that (by contrast with the Cauchy problem) it is uniquely solvable for an arbitrary set of parameters specifying the orders of the derivatives involved in the equation; these conditions are necessary for the equation under consideration. The problem is reduced to an integral equation; an explicit representation of the solution in terms of the Wright function is constructed. As a consequence of these results, necessary and sufficient conditions for the solvability of the Cauchy problem are obtained.

Bibliography: 7 titles.

This paper is concerned with the problem of minimizing an integral functional with control-nonconvex integrand over the class of solutions of a control system in a Hilbert space subject to a control constraint given by a phase-dependent multivalued map with closed nonconvex values. The integrand, the subdifferential operators, the perturbation term, the initial conditions and the control constraint all depend on a parameter. Along with this problem, the paper considers the problem of minimizing an integral functional with control-convexified integrand over the class of solutions of the original system, but now subject to a convexified control constraint. By a solution of a control system we mean a 'trajectory-control' pair. For each value of the parameter, the convexified problem is shown to have a solution, which is the limit of a minimizing sequence of the original problem, and the minimal value of the functional with the convexified integrand is a continuous function of the parameter. This property is commonly referred to as the variational stability of a minimization problem. An example of a control parabolic system with hysteresis and diffusion effects is considered.

Bibliography: 24 titles.