Hausdorff volume in non equiregular sub-Riemannian manifolds

In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures)

Randomly growing braid on three strands and the manta ray

Consider the braid group $B_3=$ and the nearest neighbor random walk defined by a probability $\nu$ with support $\{a,a^{-1},b,b^{-1}\}$. The rate of escape of the walk is explicitly expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitly describe the harmonic measure of the induced random walk on $B_3$ quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups.Comment: Published at http://dx.doi.org/10.1214/105051606000000754 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rotating disk electrodes to assess river biofilm thickness and elasticity

The present study examined the relevance of an electrochemical method based on a rotating disk electrode (RDE) to assess river biofilm thickness and elasticity. An in situ colonisation experiment in the River Garonne (France) in August 2009 sought to obtain natural river biofilms exhibiting differentiated architecture. A constricted pipe providing two contrasted flow conditions (about 0.1 and 0.45 m s−1 in inflow and constricted sections respectively) and containing 24 RDE was immersed in the river for 21 days. Biofilm thickness and elasticity were quantified using an electrochemical assay on 7 and 21 days old RDE-grown biofilms (t7 and t21, respectively). Biofilm thickness was affected by colonisation length and flow conditions and ranged from 36 ± 15 μm (mean ± standard deviation, n = 6) in the fast flow section at t7 to 340 ± 140 μm (n = 3) in the slow flow section at t21. Comparing the electrochemical signal to stereomicroscopic estimates of biofilms thickness indicated that the method consistently allowed (i) to detect early biofilm colonisation in the river and (ii) to measure biofilm thickness of up to a few hundred μm. Biofilm elasticity, i.e. biofilm squeeze by hydrodynamic constraint, was significantly higher in the slow (1300 ± 480 μm rpm1/2, n = 8) than in the fast flow sections (790 ± 350 μm rpm1/2, n = 11). Diatom and bacterial density, and biofilm-covered RDE surface analyses (i) confirmed that microbial accrual resulted in biofilm formation on the RDE surface, and (ii) indicated that thickness and elasticity represent useful integrative parameters of biofilm architecture that could be measured on natural river assemblages using the proposed electrochemical method