Space-time paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations

We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a $3$-dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined space-time paraproducts on parabolic H\"older spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.Comment: v3, 56 pages. Different points about renormalisation matters have been clarified. Typos correcte

Magnetic phase transitions in the two-dimensional frustrated quantum antiferromagnet Cs2CuCl4

We report magnetization and specific heat measurements in the 2D frustrated spin-1/2 Heisenberg antiferromagnet Cs2CuCl4 at temperatures down to 0.05 K and high magnetic fields up to 11.5 T applied along a, b and c-axes. The low-field susceptibility chi (T) M/B shows a broad maximum around 2.8 K characteristic of short-range antiferromagnetic correlations and the overall temperature dependence is well described by high temperature series expansion calculations for the partially frustrated triangular lattice with J=4.46 K and J'/J=1/3. At much lower temperatures (< 0.4 K) and in in-plane field (along b and c-axes) several new intermediate-field ordered phases are observed in-between the low-field incommensurate spiral and the high-field saturated ferromagnetic state. The ground state energy extracted from the magnetization curve shows strong zero-point quantum fluctuations in the ground state at low and intermediate fields

Extreme sensitivity of a frustrated quantum magnet: Cs_2CuCl_4

We report a thorough theoretical study of the low temperature phase diagram of Cs_2CuCl_4, a spatially anisotropic spin S=1/2 triangular lattice antiferromagnet, in a magnetic field. Our results, obtained in a quasi-one-dimensional limit in which the system is regarded as a set of weakly coupled Heisenberg chains, are in excellent agreement with experiment. The analysis reveals some surprising physics. First, we find that, when the magnetic field is oriented within the triangular layer, spins are actually most strongly correlated within planes perpendicular to the triangular layers. This is despite the fact that the inter-layer exchange coupling in Cs_2CuCl_4 is about an order of magnitude smaller than the weakest (diagonal) exchange in the triangular planes themselves. Second, the phase diagram in such orientations is exquisitely sensitive to tiny interactions, heretofore neglected, of order a few percent or less of the largest exchange couplings. These interactions, which we describe in detail, induce entirely new phases, and a novel commensurate-incommensurate transition, the signatures of which are identified in NMR experiments. We discuss the differences between the behavior of Cs_2CuCl_4 and an ideal two-dimensional triangular model, and in particular the occurrence of magnetization plateaux in the latter. These and other related results are presented here along with a thorough exposition of the theoretical methods, and a discussion of broader experimental consequences to Cs_2CuCl_4 and other materials.Comment: 43 pages, 20 figures; typos correcte

Curvature Diffusions in General Relativity

We define and study on Lorentz manifolds a family of covariant diffusions in which the quadratic variation is locally determined by the curvature. This allows the interpretation of the diffusion effect on a particle by its interaction with the ambient space-time. We will focus on the case of warped products, especially Robertson-Walker manifolds, and analyse their asymptotic behaviour in the case of Einstein-de Sitter-like manifolds.Comment: 34 page

Single, intra-articular treatment with 6 ml hylan G-F 20 in patients with symptomatic primary osteoarthritis of the knee: a randomised, multicentre, double-blind, placebo controlled trial

International audienceObjectives: The primary objective was to compare a single, 6 ml, intra-articular injection of hylan G-F 20 with placebo in patients with symptomatic knee osteoarthritis. The safety of a repeat injection of hylan G-F 20 was also assessed. Methods: Patients with primary osteoarthritis knee pain were randomly assigned to arthrocentesis plus a 6 ml intra-articular injection of either hylan G-F 20 or placebo in a prospective, double-blind (one injector/one blinded observer) study. Results were evaluated at 4, 8, 12, 18 and 26 weeks post-injection. The primary outcome criterion was change from baseline over 26 weeks in Western Ontario and McMaster Universities (WOMAC) Osteoarthritis Index A pain. Secondary outcome measures included WOMAC A1 and C, patient global assessment (PGA) and clinical observer global assessment (COGA) and Outcome Measures in Rheumatology, Osteoarthritis Research Society International responder rates. A 4-week, open, repeat treatment phase evaluated safety only. Results: A total of 253 patients (Kellgren-Lawrence grade II or III) was randomly assigned. Patients receiving hylan G-F 20 experienced statistically significantly greater improvements in WOMAC A pain scores (-0.15, SE 0.076, p=0.047), and several of the secondary outcome measures (WOMAC A1, PGA and COGA), than patients receiving placebo. There was no difference between the safety results of the two groups. No increased risk of local adverse events was observed in the open, repeat treatment phase. Conclusions: This placebo-controlled study demonstrated that, in patients with knee osteoarthritis, a single 6 ml intra-articular injection of hylan G-F 20 is safe and effective in providing statistically significant, clinically relevant pain relief over 26 weeks, with a modest difference versus placebo