MSI Weekly Bulletin - Week starting Monday 11 September, 2006

The aim of the algorithm of Lenstra, Lenstra and Lovász (LLL) is to reduce bases of euclidean lattices, i.e., dicrete subgroups of a Euclidean space. Since its introduction, it proved central in many areas, including algorithmic number theory, public-key cryptanalysis and, more recently, computer arithmetic. Given as input a basis of a lattice in Z^d with integer vectors of norms smaller than B, the LLL algorithm computes a so-called LLL-reduced basis in time O(d^6 log^3 B), by using arithmetic operations on integers of lengths O(d log B). This running-time is too high to tackle lattice bases of large dimension or with large entries. Instead of the text-book LLL, one uses in practice floating-point variants, where the integer/rational arithmetic within the Gram-Schmidt orthogonalization (central and cost-dominant in LLL) is replaced by floating-point arithmetic with small mantissas. In this talk, after some reminders on the LLL algorithm, I will describe the L² algorithm. It is a natural floating-point variant of the LLL algorithm, that always outputs LLL-reduced bases in time O(d^5 (d+logB) logB). The code derived from this algorithm is the fastest available so far.

This talk is concerned with singular solutions for the equation laplacian u=h(u) in a punctured ball in R^N (N\geq 2), where the nonlinearity h is locally Lipschitz continuous on [0,\infty) and positive in (0,\infty). We give a complete classification of isolated singularities of positive solutions for any function h varying regularly at infinity of index q with 1 le q le N/(N-2). Our result extends a well-known theorem of Veron which was established for the case h(u)=u^q. This is joint work with Y. Du

Gene regulatory networks describe the interaction of genes and its products the proteins. Because relatively small numbers of copies of each substance are involved the dynamics of these networks are mainly driven by noise generated by the translation processes involving the genes and their products. Therefore these systems are best described by stochastic models. With these models, the stochastic master equations, one can follow the time development of the probability distribution for the states defined by copy numbers of each substance. As for each substance involved, the state space grows exponentially the challenges lie in the large discrete state spaces due to high dimensionality.

By a flow on a C^*-algebra $A$, we mean a one-parameter group of automorphisms of $A$ with point-wise norm-continuity. The study of flows has a long history, starting from the special cases of uniform continuity (completed by Kadison and Sakai in 1960's) and of point-wise norm-continuity on the dual. Then followed a general case, as presented as in the book of Bratteli & Robinson in 1979. The theory is far from complete. So I will concentrate on two classes of flows: Rohlin flows and approximately finite flows.