Differential equations and moving frames

The purpose of the paper is to study the relationship between differential equations, Pfaffian systems and geometric structures, via the method of moving frames of E.Cartan. We show a local structure theorem. The Lie algebra aspects differential equations is studied too.Comment: 21 page

Some remarks on weighted logarithmic Sobolev inequality

We give here a simple proof of weighted logarithmic Sobolev inequality, for example for Cauchy type measures, with optimal weight, sharpening results of Bobkov-Ledoux. Some consequences are also discussed

Escape Rates and Singular Limiting Distributions for Intermittent Maps with Holes

We study the escape dynamics in the presence of a hole of a standard family of intermittent maps of the unit interval with neutral fixed point at the origin (and finite absolutely continuous invariant measure). Provided that the hole (is a cylinder that) does not contain any neighborhood of the origin, the surviving volume is shown to decay at polynomial speed with time. The associated polynomial escape rate depends on the density of the initial distribution, more precisely, on its behavior in the vicinity of the origin. Moreover, the associated normalized push forward measures are proved to converge to the point mass supported at the origin, in sharp contrast to systems with exponential escape rate. Finally, a similar result is obtained for more general systems with subexponential escape rates; namely that the Ces\`aro limit of normalized push forward measures is typically singular, invariant and supported on the asymptotic survivor set.Comment: To appear in Trans. Amer. Math. So

Homogeneous and locally homogeneous solutions to symplectic curvature flow

J. Streets and G. Tian recently introduced symplectic curvature flow, a geometric flow on almost K\"ahler manifolds generalising K\"ahler-Ricci flow. The present article gives examples of explicit solutions to this flow of non-K\"ahler structures on several nilmanifolds and on twistor fibrations over hyperbolic space studied by J. Fine and D. Panov. The latter lead to examples of non-K\"ahler static solutions of symplectic curvature flow which can be seen as analogues of K\"ahler-Einstein manifolds in K\"ahler-Ricci flow.Comment: 15 page

Computer theorem proving in math

We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003)

Hilbert space compression for free products and HNN-extensions

Given the Hilbert space compression of two groups, we find bounds on the Hilbert space compression of their free product. We also investigate the Hilbert space compression of an HNN-extension of a group relative to a finite normal subgroup or a finite index subgroup.Comment: 18 page

The Yamabe problem on Dirichlet spaces

We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with stratified spaces furnishing the key examples. The criterion for solvability there is phrased in terms of a strict inequality of the global Yamabe invariant with a `local Yamabe invariant', which captures information about the local singular structure. All of this is generalized here to the setting of Dirichlet spaces which admit a Sobolev inequality and satisfy a few other mild hypotheses. Applications include a new approach to the nonspherical part of the CR Yamabe problem.Comment: 25 page

Evaluation of the quantitative prediction of a trend reversal on the Japanese stock market in 1999

In January 1999, the authors published a quantitative prediction that the Nikkei index should recover from its 14 year low in January 1999 and reach $\approx 20500$ a year later. The purpose of the present paper is to evaluate the performance of this specific prediction as well as the underlying model: the forecast, performed at a time when the Nikkei was at its lowest (as we can now judge in hindsight), has correctly captured the change of trend as well as the quantitative evolution of the Nikkei index since its inception. As the change of trend from sluggish to recovery was estimated quite unlikely by many observers at that time, a Bayesian analysis shows that a skeptical (resp. neutral) Bayesian sees her prior belief in our model amplified into a posterior belief 19 times larger (resp. reach the 95% level).Comment: 6 pages including 2 figure

Diffusive tomography methods : special boundary conditions and characterization of inclusions

This thesis presents mathematical analysis of optical and electrical impedance tomography. We introduce papers [I-III], which study these diffusive tomography methods in the situation where the examined object is contaminated with inclusions that have physical properties differing from the background.reviewe