Showing papers in "Doklady Mathematics in 2009"

Any sub-Riemannian metric has points of smoothness

Abstract: We prove the result stated in the title that is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. In the case of a complete real-analytic sub-Riemannian manifold, we prove that the metric is analytic on an open everywhere dense subset.

On the Statistical Properties of Elements of Continued Fractions

Abstract: tribution of , , a s − K , ..., a s + K ( K is a constant) when α is a random number in the interval (0, 1) (see [1, 4, 5]). The existence of this distribution was proved by ergodic theory methods in [4]. It is also important to know that this distribution is also obtained when α is a random rational number of the form , where 1 ≤ a ≤ b ≤ R 2 and ( a , b ) = 1 (see [1]). In this paper, this distribution is written explicitly and we prove that it has the same form in the continuous and discrete cases. Denote by M the set of all integer matrices S =

Algorithmic Analysis of Local Integrability

Abstract: We consider an autonomous system of ordinary differential equations (ODEs) solved for the derivatives. Its formal and analytical integrals near its finite stationary point are sought from its normal form. A stationary point at infinity is first mapped to a finite point by a power transformation. This approach is applied to the Yang‐Mills system. It is found that, in the neighborhood of one stationary point at infinity, this system is locally nonintegrable, which supports the well-known result about its global nonintegrability. 1. Consider the autonomous system of ordinary differential equations (1)

Embeddings of 4-valent framed graphs into 2-surfaces

Abstract: This paper considers framed 4-valent graphs and studies the question of estimating the genus of 2-surfaces (by the genus of a nonorientable closed surface we mean , where χ is the Euler characteristic) into which such graphs can be embedded. A graph is said to be framed if, at each vertex of this graph, the four outgoing half-edges are divided into two families; halfedges from the same family are called (formally) opposite; by an embedding we mean an embedding under which the formal relation of being opposite coincides with the relation of being opposite induced by the embedding. We consider only the case where the complement to the graph in the surface is a disconnected union of 2-cells. In what follows, all graphs are assumed to be connected. The class of all embeddings contains the natural subclass of Z 2 -homologically trivial embeddings, which is characterized as follows: If a framed graph Γ is Z 2 homologically trivially embedded in a surface S , then the cells from S \ Γ admit a 2-coloring under which any cells with a common edge have different colors. We refer to such embeddings as checkerboard embeddings. Any embedding of a graph reduces to a checkerboard embedding by means of an appropriate covering. We prove the following theorem. Theorem 1. A framed graph Γ on n vertices admits a checkerboard embedding into a surface of genus g if and only if the index set of the matrix M Γ defined below admits a partition {1, 2, …, n } = I J such that the sum of ranks of the square block matrices is rk M I + rk M J = 2 g (the block matrices are determined by the rows and columns corresponding to the chosen sets of indices). Moreover, if all diagonal elements of the matrix M Γ are zero, then all surfaces into which the graph Γ can be embedded in the checkerboard manner 2 χ ‐ 2

Generalized Lévy Laplacians and Cesàro means

Abstract: In this paper, we show that the composition of the generalized Levy Laplacian of order p (which is gener- ated by the corresponding Cesaro mean) and a certain linear transformation of the domain of the function to which it is applied is proportional to the generalized Levy Laplacian of smaller or larger order; this makes it possible to transform the generalized Levy Laplacian of any order into the classical Levy Laplacian. The con- struction is based on an expression of generalized Cesaro means in terms of ordinary ones, which is also suggested in this paper. By virtue of results of (5), this implies that all results on the classical Levy Laplacian have analogues for the generalized Levy Laplacian. In particular, we consider a relationship between the gen- eralized Levy Laplacian and quantum random pro- cesses.

A local approximation theorem on Carnot manifolds under minimal smoothness conditions

Abstract: dcc (i) X 1 ( v ), X 2 ( v ), …, X N ( v ) form a basis in T v � ; (ii) H i ( v ) = s pan{ X 1 ( v ), X 2 ( v ), …, ( v )} is a subspace of dimension dim H i in T v � ; (iii) ( X i , X j )( v ) = ( v ) X k ( v ) ,

Force acting on a rough disk spinning in a flow of noninteracting particles

Abstract: 1. Consider a flow of point particles impinging on a body spinning around a fixed point. The particles do not interact with one another, and their collisions with the body are elastic. The goal is to determine the pressure force exerted by the flow on the body. The problem is considered in two dimensions. In the Euclidean space 2 , we introduce an orthonormal frame of reference Ox 1 x 2 . The flux density ρ is a constant. Initially, the particles move at the identical veloc-