卷 225, 编号 4 (2017)

This paper is an attempt to present, in a systematic way, a construction that establishes an interesting relationship between some ideas and notions well known in the theory of integrable systems on Lie algebras and a rather different area of mathematics studying projectively equivalent Riemannian and pseudo-Riemannian metrics.

We present new fixed point theorems for acyclic type multivalued maps, including a nonlinear alternative for a map defined on an open set.

The purpose of this paper is to generalize results of R. M. W. Wood on monomial bases for the mod 2 Steenrod algebra to the mod p Steenrod algebra, p > 2.

“Glued geodesic flows” and, in particular, “generalized billiard flows” on Riemannian manifolds with boundary, and geodesic flows on piecewise smooth Riemannian manifolds are studied. We develop the approaches of Lazutkin (1993) and Tabachnikov (1993) for proving the Poncelet type closure theorems via applying the classical Liouville theorem to the billiard flow (respectively to the billiard map). We prove that the condition on the refraction/reflection law to respect the Huygens principle is not only sufficient, but also necessary for “local Liouville integrability” of the glued geodesic flow, more precisely for pairwise commutation of the “glued flows” corresponding to a maximal collection of local first integrals in involution homogeneous in momenta. A similar criterion for “local Liouville integrability” of the succession/billiard map is obtained.

The quadratic Vieta formulas (x, y) ↦ (u, v) = (x + y, xy) in the complex geometry define a two-fold branched covering ℂ²→ ℂ² ramified over the parabola u² = 4v. Thinking about topics considered in Arnold’s paper Topological content of the Maxwell theorem on multipole representation of spherical functions, I came to a very simple idea: in fact, these formulas describe the algebraic structure, i.e., addition and multiplication, of complex numbers. What if, instead of the field of complex numbers, we consider an arbitrary ring? Namely for an arbitrary ring A (commutative, with unity) consider the mapping Φ: A²→ A² defined by the Vieta formulas (x, y) ↦ (u, v) = (x + y, xy). What kind of algebraic properties of the ring itself does this map reflect? At first, it is interesting to investigate the simplest finite rings A = ℤ_m and A = ℤ_k×ℤ_m. Recently, it has been very popular to consider graphs associated to rings (the zero-divisor graph, the Cayley graph, etc.). In the present paper, we study the directed graph defined by the Vieta mapping Φ.

This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction, and integrability. In particular, we show that an SDS that is diffusion-wise symmetric with respect to a proper Lie group action can be diffusion-wise reduced to an SDS on the quotient space. We also show necessary and sufficient conditions for an SDS to be projectable via a surjective map. We then introduce the notion of integrability of SDS’s, and extend the results on the existence and structure-preserving property of Liouville torus actions from the classical case to the case of integrable SDS’s. We also show how integrable SDS’s are related to compatible families of integrable Riemannian metrics on manifolds.

The main purpose of this paper is to give a complete description of isometry groups on the 4-dimensional simply connected nilpotent Lie groups. We distinguish between two geometrically distinct cases of degenerate and nondegenerate center of the group. Since Walker metrics appear as the underlying structure of neutral signature metrics on the nilpotent Lie groups with degenerate center, we find necessary and sufficient condition for them to locally admit the nilpotent group of isometries.