Some geometric aspects of the calculus of fractions

We give necessary and sufficient conditions on class S of maps of a category C so that a good calculus of fractions is possible in C[S ^-1], and a geometric characterization of the communitative diagrams in the category of fractions. These conditions are also described simply in terms of Grothendiek topologies.We characterize the categories which arise in this manner by the fact that the functor C C[S ^-1] is uniformly flat and give some applications of this result.     

An Arithmetic Reduction of Finite Rank 3 Geometries with Linear Spaces as Plane Residues and with Dual Linear Spaces as Point Residues

Let be a rank three incidence geometry of points, lines and planes whose planes are linear spaces and whose point residues are dual linear spaces (notice that we do not require anything on the line residues). We assume that the residual linear spaces of belong to a natural class of finite linear spaces, namely those linear spaces whose full automorphism group acts flag-transitively and whose orders are polynomial functions of some prime number. This class consists of six families of linear spaces. In the amalgamation of two such linear spaces imposes an equality on their orders leading, in particular, to a series of diophantine equations, the solutions of which provide a reduction theorem on the possible amalgams of linear spaces that can occur in .We prove that one of the following holds (up to a permutation of the words point and plane):A) the planes of and the dual of the point residues belong to the same family and have the same orders,B) the diagram of is in one of six families,C) the diagram of belongs to a list of seven sporadic cases.Finally, we consider the particular case where the line residues of are generalized digons.     

The law of the solution to a nonlinear hyperbolicSPDE

Let { _s,t ,(s,t ₊ ² } be a white noise on ₊ ² . We consider the hyperbolic stochastic partial differential equation {ie863-3} The purpose of this paper is to study the law of the solution to this equation. We analyze the existence and smoothness of the density using the tools of Malliavin Calculus. Finally we prove a large deviation principle on the space of continuous functions, for the family of probabilities obtained by perturbation of the noise in the equation.This work has been partially supported by the grant of the DGICYT No. PB 930052 and the EU Science project CT 910459.     

Chain partitions of ordered sets

For natural numbers s and r and a finite poset P of height h, there exists a finite poset P of height H(s, r, h) such that for an arbitrary r-colouring of the s-chains of P, a monochromatically embedded copy of P can be found in P. A best possible upper bound for H(s, r, h) in terms of the well-known Ramsey numbers is given.