ar X iv : 0 70 4 . 07 20 v 1 [ m at h . N A ] 5 A pr 2 00 7 C r - Lohner algorithm

ar X iv : 0 70 4 . 39 02 v 1 [ m at h . PR ] 3 0 A pr 2 00 7 On the number of collisions in Λ - coalescents ∗

We examine the total number of collisions Cn in the Λ-coalescent process which starts with n particles. A linear growth and a stable limit law for Cn are shown under the assumption of a power-like behaviour of the measure Λ near 0 with exponent 0 < α < 1.

ar X iv : 0 70 4 . 01 15 v 2 [ m at h . G T ] 9 A pr 2 00 7 Smooth maps with singularities of bounded K - codimensions ∗ † Yoshifumi ANDO

ar X iv : 0 70 4 . 15 37 v 1 [ m at h . C A ] 1 2 A pr 2 00 7 ESTIMATES FOR SINGULAR INTEGRALS AND EXTRAPOLATION

In this note, we study singular integrals with rough kernels, which belong to a class of singular Radon transforms. We prove certain estimates for the singular integrals that are useful in an extrapolation argument. As an application, we prove L p boundedness of the singular integrals under a certain sharp size condition on their kernels.

ar X iv : 0 70 4 . 09 45 v 1 [ m at h . PR ] 6 A pr 2 00 7 Gibbs fragmentation trees ∗

We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs type fragmentation tree with Aldous’s beta-splitting model, which has an extended parameter range β > −2 with respect to the Beta(β + 1, β + 1) probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter P...

ar X iv : 0 70 4 . 35 30 v 1 [ m at h . D G ] 2 6 A pr 2 00 7 Invariant forms , associated bundles and hyperkähler metrics

We develop a method, initially due to Salamon, to compute the space of " invariant " forms on an associated bundle X = P ×G V , with a suitable notion of invariance. We determine sufficient conditions for this space to be d-closed. We apply our method to the construction of hyperkähler metrics on T CP 1 and T CP 2 .