On special types of nonholonomic contact elements

Our starting point has been a recent clarification of the role of semiholonomic contact elements in the theory of submanifolds of Cartan geometries, Kolá? and Vitolo (2010) [5]. We deduce some further properties of the iterated contact elements by using the general concept of contact (n,F)-element for a regular subcategory F of the category of nonholonomic r-jets. Special attention is paid to the incidence relation of contact F-elements of different dimensions.     

Jets of closed orbits of Mañé’s generic Hamiltonian flows

We prove a perturbation theorem for the k-jets, k ≥ 2, of the Poincaré map of a closed orbit of the Hamiltonian flow of a Tonelli Hamiltonian H: T *M → ?, on a closed manifold M. As a consequence we obtain Mañé generic properties of Hamiltonian and Lagrangian flows.     

Generation of k-jets on toric varieties

The notion of a k-convex -support function for a toric variety is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the -support function . Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k. Received October 22, 1997; in final form January 12, 1998     

Principal bundle structure on jet prolongations of frame bundles

In this paper, we introduce the structure of a principal bundle on the r-jet prolongation J ^r FX of the frame bundle FX over a manifold X. Our construction reduces the well-known principal prolongation W ^r FX of FX with structure group G _n ^r . For a structure group of J ^r FX we find a suitable subgroup of G _n ^r . We also discuss the structure of the associated bundles. We show that the associated action of the structure group of J ^r FX corresponds with the standard actions of differential groups on tensor spaces.     

Whitney’s jets for Sobolev functions

We present two fundamental facts from the jet theory for Sobolev spaces W ^{m, p} . One of these facts is that the formal differentiation of the k-jets theory is compatible with the pointwise definition of Sobolev (m − 1)-jet spaces on regular subsets of the Euclidean spaces ℝⁿ. The second result describes the Sobolev imbedding operator of Sobolev jet spaces increasing the order of integrability of Sobolev functions up to the critical Sobolev exponent. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 345–358, March, 2007.     

Contact Symmetries and Variational PDE’s

Symmetries of the contact ideal on the r-jet bundle over a fibred manifold are studied, and transformation properties under contact symmetries of different objects in the variational sequence related with systems of partial differential equations are investigated. This paper is dedicated to Valentin Lychagin on the occasion of his 60th birthday.     

Genealized poly-Cauchy and poly-Benoulli numbes by using incomplete $${\vavec{}}$$-Stiling numbes

In this paper we introduce restricted r-Stirling numbers of the first kind. Together with restricted r-Stirling numbers of the second kind and the associated r-Stirling numbers of both kinds, by giving more arithmetical and combinatorial properties, we introduce a new generalization of incomplete poly-Cauchy numbers of both kinds and incomplete poly-Bernoulli numbers.     

Two proofs of the Bermond-Thomassen conjecture for tournaments with bounded minimum in-degree

The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r−1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r=2, and very recently the conjecture was proved for the case where r=3. It is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we present two proofs of this conjecture for tournaments with minimum in-degree at least 2r−1. In particular, this shows that the conjecture is true for (almost) regular tournaments. In the first proof, we prove auxiliary results about union of sets contained in another union of sets, that might be of independent interest. The second one uses a more graph-theoretical approach, by studying the properties of a maximum set of vertex-disjoint directed triangles.     

Two modularity lifting conjectures for families of Siegel modular forms

For a prime p and a positive integer n, using certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus n. Describing L-functions attached to Siegel modular forms and their analytic properties, we formulate two conjectures on the existence of the modularity liftings from GSp _r × GSp_2m to GSp _r+2m for some positive integers r and m.     

About some cyclic properties in digraphs

In this article we are concerned with digraphs in which any two vertices are on a common cycle. For example, we prove that, in a strong digraph of order n and half degrees at least 2 with at least n² ? 5n + 15 arcs, any two vertices are on a common cycle. We also consider related properties and give sufficient conditions on half degrees and the number of arcs to insure these properties. In particular, we show that every digraph of order n with half degrees at least r and with at least n² ? rn + r² arcs is 2-linked.     

On differential-geometric structures on a manifold of nonholonomic (<Emphasis Type="Italic">n</Emphasis> + 1)-web

We consider a nonholonomic (n + 1)-web NW on an n-dimensional manifold M, i.e., n + 1 distributions of codimension 1. We prove that there exists an invariant pencil of projective connections on M. To an ordered nonholonomic (n + 1)-web on M, there corresponds a unique curvilinear (n + 1)-web on M and vice versa. This correspondence is defined by the polarity with respect to a certain invariant multilinear n-form or the barycentric subdivision of some (n - 1)- dimensional simplex. In conclusion, we consider, as a special case, a nonholonomic (n + 1)-web ANW of hyperplanes in the affine space. A web ANW generates an invariant pencil of affine connections. We also study the case when these connections are projective.     

Concentration at a radius for Hardy class functions

In this paper we establish the fundamental properties of concentration at a radius for functions in the classical Hardy space on the unit disk. For f(z) which is not identically zero and given r, 0<r<1, the concentration is defined via the ratio of the norm of f(rz) to the norm of f(z). Using the Mahler measure of f(z), we obtain information on the distribution of the zeros of f(z) in terms of the concentration ratio. In the last section of the paper, we examine the sharpness of concentration estimates for the Blaschke factor.     

On r-stacked triangulated manifolds

The notion of r-stackedness for simplicial polytopes was introduced by McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this paper, we define the r-stackedness for triangulated homology manifolds and study its basic properties. In addition, we find a new necessary condition for face vectors of triangulated manifolds when all the vertex links are polytopal.     

Curvature and tachibana numbers

The purpose of this paper is to define the rth Tachibana number t _r of an n-dimensional closed and oriented Riemannian manifold (M,g) as the dimension of the space of all conformal Killing r-forms for r = 1, 2, . . . , n ? 1 and to formulate some properties of these numbers as an analog of properties of the rth Betti number b _r of a closed and oriented Riemannian manifold.     

On associated and co-recursive d-orthogonal polynomials

The associated sequence of order r for a given d-OPS (i.e. a sequence of orthogonal polynomials satisfying a (d + 1)-order recurrence relation), is again a d-OPS. In this paper we are interested in the determination of the corresponding dual sequence. The explicit form of the dual sequence of the first associated sequence and the corresponding formal Stieltjes function are given. Indeed, we construct by recurrence the dual sequence of the r-associated sequence and we give some properties of the corresponding Stieltjes function. Second, we give the definition of co-recursive polynomials of dimension d and some relations in the particular cases d = 3 and d = 4. Some properties of the dual sequence as well as of the corresponding Stieltjes functions are given.     

Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.     

A note on the congruence \left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right) (mod p r )

In the paper we discuss the following type congruences: $$\left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right)(\bmod p^r ),$$ where p is a prime, n, m, k and r are various positive integers with n ? m ? 1, k ? 1 and r ? 1. Given positive integers k and r, denote by W(k, r) the set of all primes p such that the above congruence holds for every pair of integers n ? m ? 1. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets W(k, r) and inclusion relations between them for various values k and r. In particular, we prove that W(k + i, r) = W(k ? 1, r) for all k ? 2, i ? 0 and 3 ? r ? 3k, and W(k, r) = W(1, r) for all 3 ? r ? 6 and k ? 2. We also noticed that some of these properties may be used for computational purposes related to congruences given above.