Asymmetric synthesis of the oxygenated polycyclic system of (+)-harringtonolide

A straightforward asymmetric synthesis of the cage oxygenated structure of (+)-harringtonolide has been accomplished for the first time. The key steps involved (i) a templated stereoselective IMDA reaction to build a highly functionalized cyclohexene ring D, (ii) functionalization of the cycloadduct, (iii) ring-closing metathesis providing the five-membered ring C, and finally (iv) a challenging one-step cascade cyclization of an epoxy-alcohol toward the target structure, whose mechanism was investigated.     

Reactivity of cyclohexene epoxides toward intramolecular acid-catalyzed cyclizations for the synthesis of naturally occurring cage architectures

This account compiles our results on the reactivity of cyclohexene epoxides toward the synthesis of naturally occurring cage architectures, in particular, the one found in harringtonolide. Bicyclic and branched monocyclic functional triads (hydroxy-epoxy-esters) were synthesized with the aim of undertaking cascade processes toward the formation of lactone and/or cycloether bridges, through a central epoxide opening. This work successfully culminated in the cascade cyclization of the fully oxygenated bridge-structure of harringtonolide, by using a dual Brønsted-Lewis acid complex.     

A Laccase/Chitosan‐Lambda‐Carrageenan Based Voltammetric Biosensor for Phenolic Compound Detection

In this contribution, a new concept of voltammetric catechol biosensor, based on the encapsulation of laccase (LAC) in a chitosan+lambda‐carrageenan (CHIT+CAR) polyelectrolyte complex (PEC) employing a simple coacervation process is presented. Chitosan (CHIT) was prepared from α‐chitin extracted from shrimp shells and lambda‐carrageenan (CAR) was extracted from red algae, both polysaccharides and PEC being characterized using FTIR spectrometry and electrochemistry. Cyclic voltammetry was utilized to determine the analytical features of the laccase (LAC) biosensor for catechol detection. The linear range was from 10^?20 M to 10^?14 M with a sensitivity of 1.55 mA/p[catechol] and a limit of detection of 3×10^?21 M.The laccase biosensor exhibits good repeatability (RSD 2.38 %) and stability (four weeks). The developed biosensor was tested by applying it to the evaluation of the total polyphenolic content in natural oil samples.     

Stability by rescaled weak convergence for the Navier–Stokes equations

We study a weak stability problem for the three-dimensional Navier–Stokes system: if a sequence (u_0,n)_n∈N${(u_{0,n})}_{n\in \mathbb{N}}$ of initial data, bounded in some scaling invariant space, converges weakly to an initial data u₀$u_0$ which generates a global regular solution, does u_0,n$u_{0,n}$ generate a global regular solution? Because of the invariances of the Navier–Stokes equations, a positive answer in general to this question would imply global regularity for any data, so we introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier–Stokes equations.     

Electrophilic reactivities of 7-L-4-nitrobenzofurazans in σ-complexation processes: Kinetic studies and structure–reactivity relationships

The second-order rate constants (k) for reaction of 7-chloro-4-nitrobenzofurazan 1 and 7-methoxy-4-nitrobenzofurazan 2 with a series of nitroalkyl anions and several of para-substituted phenoxide anions in aqueous solution at 20 °C have been reported. On the basis of the linear novel approach recently designed by Mayr and coworkers, the electrophilicity parameters E at the C-5 position of the two nitrobenzofurazans 1 and 2 have been quantified and ranked on the comprehensive electrophilicity scale. Mayr's approach was found to correctly predict the rate constants for the addition of phenoxide anions at the C-5 position of 1 and 2 witting a factor of <2. Analysis of the kinetic measurements using Brønsted's model shows that β_nuc values remain remarkably constant for changes in the nature of the substituent and that the σ-complexation process is associated with high Marcus intrinsic barriers. In addition, satisfactory correlations between the log k_exp (k_exp values measured in this work for reactions of benzofurazans 1 and 2 with a series of phenoxide anions in aqueous solution at 20 °C) and log k_calcd (k_calcd values calculated from equation 1 using the electrophilicity parameters E of benzofurazans 1 and 2 and the previously published nucleophilicity parameters N and s_N of the phenoxide anions) with a slope very close to unity have been obtained and discussed.     

On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations

Let X be a suitable function space and let ${\mathcal{G} \subset X}$ be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of ${\mathcal{G}}$ belongs to ${\mathcal{G}}$ if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier–Stokes equations; it is also shown that initial data which do not belong to ${\mathcal{G}}$ (hence which produce a solution blowing up in finite time) cannot have a strong anisotropy in their frequency support.     

Analyse de l'espace des phases et calcul pseudo-differential sur le groupe de Heisenberg

We establish pseudo-differential calculus on the Heisenberg group by defining an algebra of operators acting continuously on Sobolev spaces and containing the class of differential operators. Our approach puts into light microlocal directions and completes, with the Littlewood–Paley theory developed by Bahouri et al., a microlocal analysis of the Heisenberg group. To cite this article: H. Bahouri et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the blow‐up criterion of 3D Navier‐Stokes equation in

In this paper, we prove two results about the blow‐up criterion of the three‐dimensional incompressible Navier‐Stokes equation in the Sobolev space . The first one improves the result of Cortissoz et al. The second deals with the relationship of the blow up in and some critical spaces. Fourier analysis and standard techniques are used.     

Refined Sobolev Inequalities in Lorentz Spaces

We establish refined Sobolev inequalities between the Lorentz spaces and homogeneous Besov spaces. The sharpness of these inequalities is illustrated on several examples, in particular based on non-uniformly oscillating functions known as chirps. These results are also used to derive refined Hardy inequalities.