Unless otherwise explicitly stated all mappings and tensors in the paper are C∞. A Poisson structure on a (C∞) manifold M is a bracket operation (f, g) [map ] {f, g}, on the set of functions on M, which gives to this set a Lie algebra structure and which verifies the relation

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An equivalent way to get such a structure is to give a 2-vector (that is, an antisymmetric two times contravariant tensor) P satisfying

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where [,] is the Schouten bracket [7]. We then have

formula here

This paper is devoted to the local classification of these structures. The decomposition theorem of A. Weinstein [8, 9] reduces the problem to the case where P is a Poisson structure on Rn which vanishes at zero.

In this paper we will denote by PS(n) the set of germs at zero of Poisson structures on Rn vanishing at the origin.

Suppose that X1, X2, X3, … are independent random points in Rd with common density f, having compact support Ω with smooth boundary ∂Ω, with f[mid ]Ω continuous. Let Rni, k denote the distance from Xi to its kth nearest neighbour amongst the first n points, and let Mn, k = maxi[les ]nRni, k. Let θ denote the volume of the unit ball. Then as n → ∞,

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If instead the points lie in a compact smooth d-dimensional Riemannian manifold K, then nθMdn, k/ log n → (minKf)−1, almost surely.