There is a homotopy equivalence ϕ:M→M′ between closed smooth manifolds of an odd dimension such that ϕ*TM′, TM are stably isomorphic but not isomorphic to each other.

Two convex bodies K and K′ in Euclidean space [ ]n can be said to be in exceptional relative position if they have a common boundary point at which the linear hulls of their normal cones have a non-trivial intersection. It is proved that the set of rigid motions g for which K and gK′ are in exceptional relative position is of Haar measure zero. A similar result holds true if ‘exceptional relative position’ is defined via common supporting hyperplanes. Both results were conjectured by S. Glasauer; they have applications in integral geometry.

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

formula here

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

formula here

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.

A new coupling of one-dimensional random walks is described which tries to control the coupling by keeping the separation of the two random walks of constant sign. It turns out that among such monotone couplings there is an optimal one-step coupling which maximises the second moment of the difference (assuming this is finite), and this coupling is ‘fast’ in the sense that for a random walk with a unimodal step distribution the coupling time achieved by using the new coupling at each step is stochastically no larger than any other coupling. This is applied to the case of symmetric unimodal distributions.

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 → ∞,

formula here

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

The most important object in real singularity theory is the C∞ map germ and the most important equivalence relation among them is C∞ right left equivalence. In [7], we presented a new systematic method for the classification of C∞ map germs by characterising C∞ right left equivalence. This paper is a topological version of [7].

Two C∞ map germs f, g:(Rn, 0) → (Rp, 0) are said to be topologically equivalent if there exist homeomorphism map germs s:(Rn, 0) → (Rn 0) and t:(Rp, 0) → (Rp, 0) such that f(x) = t∘g∘s(x). The notion of topological equivalence, although it seems to be unnatural, is also important since we know the existence of C∞ moduli for the classification of C∞ map germs with respect to C∞ right left equivalence. However, we had only one method to obtain topological equivalence for two given C∞ map germs, as stated in the following.

For two given C∞ map germs f, g:(Rn, 0) → (Rp, 0), take an appropriate one-parameter family F:(Rn×[0, 1], {0}×[0, 1]) → (Rp, 0) such that F(x, 0) = f(x) and F(x, 1) = g(x). Then prove that F is in fact topologically trivial.(*)

Two C∞ map germs f, g:(Rn, 0) → (Rp, 0) are said to be [Kscr ]-equivalent if there exist a C∞ diffeomorphism map germ s:(Rn, 0) → (Rn, 0) and a C∞ map germ M:(Rn, 0) → (GL(p, R), M(0)) such that f(x) = M(x)g(s(x)). The notion of [Kscr ]-equivalence was introduced by Mather [4, 5] in order to classify the C∞ stable map germs, and we know that generally in a [Kscr ]-orbit there are uncountably many C∞ right left orbits.

Hence it is significant to give an alternative systematic method for the topological classification even in a single [Kscr ]-orbit, which is the purpose of this paper. One of our results (Theorem 1.2) yields the following well-known theorem [2] as a trivial corollary.