Russian Journal of Mathematical Physics最新文献

We study magnetic geodesic flows invariant under rotations on the 2-sphere. The dynamical system is given by a generic pair of functions ((f,Lambda)) in one variable. The topology of the Liouville fibration of the given integrable system near its singular orbits and singular fibers is described. The types of these singularities are computed. The topology of the Liouville fibration on regular 3-dimensional isoenergy manifolds is described by computing the Fomenko–Zieschang invariant. All possible bifurcation diagrams of the momentum mappings of such integrable systems are described. It is shown that the bifurcation diagram consists of two curves in the ((h,k))-plane. One of these curves is a line segment (h=0), and the other lies in the half-plane (hge0) and can be obtained from the curve ((a:-1:k) = (f:Lambda:1)^*) projectively dual to the curve ((f:Lambda:1)) by the transformation ((a:-1:k)mapsto(a^2/2,k)=(h,k)).

DOI 10.1134/S1061920825600084

The classical Ky Fan theorem is a combinatorial equivalent of the Borsuk–Ulam theorem. It is a generalization and extension of Tucker’s lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of a triangulated sphere (S^n). Here we describe generalizations of Ky Fan theorem for the case when the sphere is replaced by the total space of a triangulated sphere bundle.

DOI 10.1134/S1061920825600138

In this note, we give a criterion for the existence of a fractional-linear integral for a geodesic flow on a Riemannian surface and explain that, modulo the Möbius transformations, the moduli space of such local integrals (if nonempty) is either the two-dimensional projective plane or a finite number of points. We also consider explicit examples and discuss a relation of such rational integrals to Killing vectors.

DOI 10.1134/S1061920824601836

One of the constructions for obtaining flows on a manifold is the construction of a suspension over a diffeomorphism. S. Smale showed that suspensions over conjugate diffeomorphisms are topologically equivalent. The converse is not true in the general case. A classic illustration of this fact are examples of nonconjugate diffeomorphisms of a circle whose suspensions are equivalent. In this paper, we establish relations between the invariants of topological conjugacy of Cartesian products of rough transformations of a circle and the invariants of topological equivalence of suspensions over them.

DOI 10.1134/S1061920824601794

We give an explicit formula for the index of a Lie algebra of the shape ( {mathfrak{g}} _0= {mathfrak{h}} oplus( {mathfrak{g}} / {mathfrak{h}} )^{ text{ab} }), where ( {mathfrak{g}} ) is a semisimple Lie algebra, ( {mathfrak{h}} ) is a subalgebra in ( {mathfrak{g}} ) regarded as a subalgebra in ( {mathfrak{g}} _0), and (( {mathfrak{g}} / {mathfrak{h}} )^{ text{ab} }) is an ( {mathfrak{h}} )-module ( {mathfrak{g}} / {mathfrak{h}} ) regarded as an Abelian ideal of ( {mathfrak{g}} _0). This formula has applications to Poisson commutative subalgebras in the symmetric algebra ( operatorname{S} ( {mathfrak{g}} )) and to completely integrable systems.

DOI 10.1134/S1061920825600199

By analogy with the normal subgroups related to pseudocharacters on groups, we introduce and study the properties of two normal subgroups of a group related to a one-dimensional pure pseudorepresentation on the group.

DOI 10.1134/S1061920825010133

In this paper we continue to develop the approach to constructing global uniform asymptotics of solutions of (pseudo)differential problems in terms of special functions based on the theory of the Maslov canonical operator. In particular, we show that if the corresponding Lagrangian manifold has a fold-type singularity, then the canonical operator on it is represented via the Airy function Ai and its derivative of complex arguments. This approach is illustrated by a known problem about construction of asymptotic eigenfunctions of the Laplace operator in an elliptic domain with Dirichlet boundary conditions.

DOI 10.1134/S1061920825600072

We study quotients of projective and affine spaces by various actions of the icosahedral group. Basically we concentrate on the rationality questions.

DOI 10.1134/S1061920824601800

There is a considerable collection of examples of spaces (X) equipped with an involution (tau) such that the mod 2-cohomology rings (H^{2*}(X)) and (H^*(X^tau)) are isomorphic. In [4], it was shown that such an isomorphism is a part of a certain structure on the equivariant cohomology of (X) and (X^tau), which is called an (H)-frame. An important part of the (H)-frame structure in [4] was the so-called conjugation equation. In [3], the coefficients of the conjugation equation were calculated in terms of the Steenrod squares. Later, another proofs were obtained, [9, 10]. In this paper, we develop a similar notion of a (Q)-framing, which occurs in the situation when a space (X) is equipped with two commuting involutions (tau_1,tau_2) and the mod 2-cohomology rings (H^{4*}(X)) and (H^*(X^{tau_1,tau_2})) are isomorphic. Basic examples are the quaternionic Grassmannians and the quaternionic flag manifolds equipped with two complex involutions. Our main result is the establishment of the quaternionic conjugation equations and identifying their coefficients in terms of Steenrod operations.

DOI 10.1134/S1061920825600096

We discuss the occurrence of spatial asymptotic expansions of solutions to the Navier–Stokes equation on ( {mathbb{R}} ^d). In particular, we prove that the Navier–Stokes equation is locally well-posed in a class of weighted Sobolev and asymptotic spaces. The solutions depend analytically on the initial data and time and (generically) develop nontrivial asymptotic terms as (|x|toinfty). In addition, the solutions have a spatial smoothing property that depends on the order of the asymptotic expansion.

DOI 10.1134/S1061920824601812