Anion complexation. A ditriphenylphosphonium calix[4]arene derivative as a novel receptor for anions

A novel anionic receptor 2 consisting of a calix[4]arene bearing two alkytriphenylphosphonium has been prepared by two different procedures. The complexation occurred at the phosphonium sites probably due to electrostatic and/or π-anion forces.     

Double-T metamaterial for parallel and normal transverse electric incident waves

Electric and magnetic hybridized plasmonic modes are obtained by stacking two T-shaped resonators. We show that head-to-toe configuration leads to inverse the hybridization. The frequency shift between the resonances is finely controlled by adjusting the gap between the two resonators. A negative refractive index close to -1 is numerically and experimentally demonstrated at 4.3 GHz for TE waves. This left-handed behavior is similar for parallel and normal TE incident wave vectors. The proposed double-T unit cell is well adapted for developing terahertz and IR metamaterials.     

Remarks on a lemma by Jacques-Louis Lions

Let Ω   be a bounded and connected open subset of R^N${\mathbb{R}}^N$ with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on one side of ∂Ω  . It is shown in this Note that a fundamental characterization of the space L²(Ω)$L^2(\Omega )$ due to Jacques-Louis Lions is in effect equivalent to a variety of other properties. One of the keys for establishing these equivalences is a specific “approximation lemma”, itself one of these equivalent properties.     

The stationary Oseen equations in an exterior domain: An approach in weighted Sobolev spaces

In this work, we study the linearized Navier–Stokes equations in an exterior domain of R³${\mathbb{R}}^3$ at the steady state, that is, the Oseen equations. We are interested in the existence and the uniqueness of weak, strong and very weak solutions in L^p$L^p$-theory which makes our work more difficult. Our analysis is based on the principle that linear exterior problems can be solved by combining their properties in the whole space R³${\mathbb{R}}^3$ and the properties in bounded domains. Our approach rests on the use of weighted Sobolev spaces.     

Sodium hydrogen telluride as a useful nucleophilic reagent for the cleavage of epoxides and of quaternary ammonium salts

Sodium hydrogen telluride opens many epoxides cleanly by an SN₂ process to give telluro-alcohols, which by reduction with nickel boride afford alcohols. An intermediate telluro-alcohol was converted to olefin in high yield by treatment with $\text{p}$-toluene-sulphonyl chloride in pyridine. Quaternary ammonium salt are also cleaved efficiently by sodium hydrogen telluride.     

Selective reduction of the N-O bond in n-oxides and in nitrones by sodium hydrogen telluride

Sodium hydrogen telluride reduces $\text{tert}$.-amine $\text{N}$-oxides to the corresponding amines in very high yield, but does not reduce sulphoxides. This reagent also reduces nitrones to $\text{sec}$.-amines at pH6 and to imines at alkaline pH (10–11).     

New formulations of linearized elasticity problems,based on extensions of Donati's theorem

The classical Donati theorem is used for characterizing smooth matrix fields as linearized strain tensor fields. In this Note, we give several generalizations of this theorem, notably to matrix fields whose components are only in $H^{\mathrm{?1}}$. We then show that our extensions of Donati's theorem allow to reformulate in a novel fashion linearized three-dimensional elasticity problems as quadratic minimization problems with the strains as the primary unknowns. To cite this article: C. Amrouche et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Existence Result for a Superlinear Fractional Navier Boundary Value Problems

In this paper, we study the following fractional Navier boundary value problem

$$\begin{aligned} \left\{ \begin{array}{lllc} D^{\beta }(D^{\alpha }u)(x)=u(x)g(u(x)),\quad x\in (0,1), \\ \displaystyle \lim _{x\longrightarrow 0}x^{1-\beta }D^{\alpha }u(x)=-a,\quad \,\,u(1)=b, \end{array} \right. \end{aligned}$$

where \(\alpha ,\beta \in (0,1]\) such that \(\alpha +\beta >1\), \(D^{\beta }\) and \(D^{\alpha }\) stand for the standard Riemann–Liouville fractional derivatives and a, b are nonnegative constants such that \(a+b>0\). The function g is a nonnegative continuous function in \([0,\infty )\) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.