Refined inequalities on graded Lie groups

We construct a Littlewood–Paley decomposition associated to a Rockland operator on graded Lie groups, which allows us to deduce refined Gagliardo–Nirenberg, Sobolev and Hardy inequalities.     

Investigation of the vibrational modes of ZnO grown by MOCVD on different orientation planes

Wurtzite ZnO thin films were prepared on sapphire substrate by metal organic chemical vapor deposition (MOCVD). Raman scattering studies on different crystallographic textures were performed in the backscattering geometry, and polarization effect is investigated in different configurations and . ZnO Raman modes are investigated in each texture. In the case of ZnO thin film deposed on r‐() sapphire plane and using backscattering geometry, new Raman line was observed at 390 cm⁻¹ because this mode has not been noticed in this geometry. It is shown that the frequencies of the quasi‐phonon modes of the examined thin film are in good agreement with the theoretical values calculated within the framework of Loudon model. Copyright © 2014 John Wiley & Sons, Ltd.     

A posteriori error analysis for nonconforming approximations of an anisotropic elliptic problem

We develop in this article an a posteriori error estimator for the P₁‐nonconforming finite element approximation, for a diffusion‐reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015     

Robust hierarchical a posteriori error estimators for stabilized convection–diffusion problems

We construct a hierarchical a posteriori error estimator for a stabilized finite element discretization of convection‐diffusion equations with height Péclet number. The error estimator is derived without the saturation assumption and without any comparison with the classical residual estimator. Besides, it is robust, such that the equivalence between the norm of the exact error and the error estimator is independent of the meshsize or the diffusivity parameter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Fixed point results on a metric space endowed with a finite number of graphs and applications

In this paper, we consider self-mappings defined on a metric space endowed with a finite number of graphs. Under certain conditions imposed on the graphs, we establish a new fixed point theorem for such mappings. The obtained result extends, generalizes and improves many existing contributions in the literature including standard fixed point theorems, fixed point theorems on a metric space endowed with a partial order and fixed point theorems for cyclic mappings.     

Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group

It was proved by Bahouri et al. [9] that the Schrödinger equation on the Heisenberg group $\mathbb{H}^d,$ involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schrödinger equation, by a refined study of the Schrödinger kernel $S_t$ on $\mathbb{H}^d.$ The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by Gaveau [19], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d.$     

Investigation of the dissociation pathways of metolachlor, acetochlor and alachlor under electron ionization - application to the identification of ozonation products

With the future aim of using gas chromatography coupled with mass spectrometry to characterize the transformation products of ozonated herbicides: metolachlor, acetochlor and alachlor, an interpretation of their electron ionization mass spectra is presented. Fragmentation mechanisms are proposed on the basis of isotopic labelling and multiple-stage mass spectrometry experiments carried out on an ion trap mass spectrometer. We also give examples in order to demonstrate how the elucidation of such fragmentation mechanisms for herbicides may simplify the characterization of their ozonation products.