Том 223, № 6 (2017)

We consider a 2-dimensional Hamiltonian system describing classical electron motion in a graphene placed in a large constant magnetic field and an electric field with a periodic potential. Using the Maupertuis–Jacobi correspondence and an assumption that the magnetic field is large, we perform averaging and reduce the original system to a 1-dimensional Hamiltonian system on the torus. This allows us to describe the trajectories of both systems and classify them by means of Reeb graphs.

The aim of this note is to present a simple proof of the completeness of the Manakov integrals for a motion of a rigid body fixed at a point in ℝⁿ, as well as for geodesic flows on a class of homogeneous spaces SO(n)/SO(n₁)×· · ·×SO(n_r).

Earlier a topological characteristic of the degree type for multivalued perturbations of Fredholm mappings with zero index was constructed and it was assumed that the multivalued perturbation permits a single-valued approximation. In this paper, a similar characteristic is constructed for multivalued perturbations of Fredholm mappings of positive index, and its application is given to the problem of existence of an optimal solution for the boundary-value problem in the theory of ordinary differential equations with feedback.

The problem of the existence of universal elements in the class of all topological groups of weight ≤τ ≠ ω remains open. In this paper, it is proved that for many classes of topological groups there are so-called continuously containing spaces. Let ???? be a saturated class of completely regular spaces of weight ≤τ and ???? be the subclass of elements of ???? that are topological groups. Then there exists an element T ∈ ???? having the following property: for every G ∈ ????, there exists a homeomorphism \( {h}_{\mathrm{T}}^G \) of G into T such that if the points x, y of T belong to the set \( {h}_{\mathrm{T}}^H \) (H) for some H ∈ ????, then for every open neighborhood U of xy in T there are open neighborhoods V and W of x and y in T, respectively, such that for every G ∈ ???? we have

\( \left(V\cap {h}_{\mathrm{T}}^G(G)\right){\left(W\cap {h}_{\mathrm{T}}^G(G)\right)}^{-1}\subset U\cap {h}_{\mathrm{T}}^G(G). \)

In this case, we say that T is a continuously containing space for the class ????. We recall that as the class ???? we can consider, for example, the following classes of completely regular spaces: n-dimensional spaces, countable-dimensional spaces, strongly countable-dimensional spaces, locally finite-dimensional spaces. Therefore, in all these classes there are elements that are continuously containing spaces for the corresponding subclasses consisting of topological groups. In this paper, some open problems are considered.

We show that for a closed Riemannian manifold the quotient of the group of projective transformations by the group of isometries contains at most two elements unless the metric has constant positive sectional curvature or every projective transformation is an affine transformation.

We consider two mathematical problems that can be ascribed to the category pointed out in the title. The first one relates to geometric quantization and deals with the twistor approach to the quantization of smooth strings. The second one concerns the adiabatic limit in the Ginzburg–Landau and Seiberg–Witten equations.

This article contains a rough topological analysis of the completely integrable system with three degrees of freedom corresponding to the motion of the Kovalevskaya top in a double field. This system is not reducible to a family of systems with two degrees of freedom. We introduce the notion of a topological atlas of an irreducible system. For the Kovalevskaya top in a double field, we complete the topological analysis of all critical subsystems with two degrees of freedom and calculate the types of all critical points. We present the parametric classification of the equipped iso-energy diagrams of the initial momentum map pointing out all chambers, families of 3-tori, and 4-atoms of their bifurcations. Basing on the ideas of A. T. Fomenko, we define the simplified net iso-energy invariant. All such invariants are constructed. Using them, we establish, for all parametrically stable cases, the number of critical periodic solutions of all types and the loop molecules of all nondegenerate rank 1 singularities.