Fondation Sciences mathématiques de Paris

We give a new proof of a theorem by Le Gall and Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2-sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called 1-regular convergence, that preserves topological properties of metric surfaces.

We describe the range of the attenuated ray transform of a unitary connection on a simple surface acting on functions and 1-forms. We use this description to determine the range of the ray transform acting on symmetric tensor fields.

We present an algorithm for the following problem. Given a triangulated 3-manifold M and a (possibly non-simple) closed curve on the boundary of M, decide whether this curve is contractible in M. Our algorithm is combinatorial and runs in exponential time. This is the first algorithm that is specifically designed for this problem; its running time considerably improves upon the existing bounds implicit in the literature for the more general problem of contractibility of closed curves in a 3-manifold. The proof of the correctness of the algorithm relies on methods of 3-manifold topology and in particular on those used in the proof of the Loop Theorem.

Gaël Octavia graduated from Télécom Sud-Paris (Télécom INT) in 2001. She was initially an information systems engineer and then became a scientific journalist. From 2002 to 2008, she worked as a writer and sub-editor for Tangente, a magazine of mathematical content for the general public. In February 2008, she joined the FSMP as a communication manager. She is also a playwright and novelist.

Let $K_q(n,R)$ denote the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Recently the authors gave a new proof of a classical lower bound of Rodemich on $K_q(n,n-2)$ by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich's original proof, the method generalizes to lower-bound $K_q(n,n-k)$ for any $k>2$. The approach is best-understood in terms of a game where a winning strategy for one of the players implies the non-existence of a code. This proves to be by far the most efficient method presently known to lower-bound $K_q(n,R)$ for large $R$ (i.e. small $k$). One instance: the trivial sphere-covering bound $K_{12}(7,3)\geq 729$, the previously best bound $K_{12}(7,3)\geq 732$ and the new bound $K_{12}(7,3)\geq 878$.

This paper considers the fractional limitation lim satisfying the condition g(x)- and the same conclusion as L'Hospital Rule is given.