Théorème de Dirichlet (séries de Fourier)

En analyse, le théorème de Dirichlet (ou de Jordan-Dirichlet) est un résultat de convergence ponctuelle pour les séries de Fourier. Une première version du théorème a été prouvée par Dirichlet en 1829. Faute d'une théorie de l'intégration adéquate, la preuve de Dirichlet ne permet de traiter que des fonctions assez particulières (monotones hors des points d'une subdivision). Le théorème sera généralisé par Jordan en 1881 pour englober le cas de toutes les fonctions « localement à variation bornée ». Énoncé Soit ƒ une fonction localement intégrable sur \mathbb{R} et de période 2π. Soit x_0 \in\mathbb{R}. On suppose que :

ƒ admet des limites à droite et à gauche en x0, notées ƒ(x0+) et ƒ(x0−) ;
il existe α > 0 tel que les intégrales suivantes convergent : :\int_0^\alpha \frac{|f(x_0+t)-f(x_0^+)|}{t} {\mathrm d} t, \qquad \int_0^\alpha \frac{|f(x_0-t)-f(x_0^-)|}{t} {\mathrm d} t.

Alors, la série de Fourier de ƒ converge au point

The geometry and spectrum of the one holed torus

Klaus-Dieter Semmler

The authors examine the space of Riemann surfaces of signature (1,1) with metric of curvature -1 and geodesic boundary. They solve explicitly the moduli problem in this case and show furthermore that two surfaces of this type having the same length spectrum (this referring to smooth closed geodesics including the boundary) are isometric. They announce the same type of result for genus two surfaces without boundary. The problem whether the analogous assertion holds for the spectrum of the Laplacian with respect to the Neumann or Dirichlet conditions is open.

1988

Plasma radiation for atmospheric entry at Titan: Emission spectroscopy measurements and numerical rebuilding

Pénélope Leyland, Raffaello Sobbia

Emission spectroscopy measurements on a plasma representative of Titan atmosphere composition were obtained in the Inductively Coupled Plasma wind tunnel facility (VKI-Minitorch) at the von Karman Institute in Belgium. Temperatures ranged from 3600 to 5000 K, pressure was fixed at 300 mbar, and the molar composition was 1.9% CH4 and 98.1% N-2. The high-pressure plasma was produced to obtain conditions close to equilibrium. In conjunction, line-by-line calculations have been carried out to assess the reliability of two distinct sets of molecular electronic transition moments, recently released, by predicting the radiative signature of high-temperature N-2-CH4 plasma. The radiative transfer problem was solved by considering the plasma plume at local thermodynamic equilibrium conditions in an axisymmetric configuration. Comparisons between the synthetic and experimental spectra demonstrated good agreement for the CN Violet and high-wavelength CN Red bands, while some discrepancies were observed for the C-2 Swan bands and low-wavelength CN Red bands.

Cambridge University Press2013

Transition metal-catalyzed carbon-carbon cross-coupling reactions using sulfonyl derivatives

The goal of this thesis was to find conditions to form C-C double bond and single bond using sulfonyl derivativies, which are arising from the Vogel's oxyallylation cascades. In a second chapter, it is shown that 2-methylprop-2-ene-1-sulfonyl fluorides can be easily prepared via the ene reaction of methallylsilanes and SO2 followed by halogenosis (NCS, then KF). In the presence of a base, aldehydes and 2-methylprop-2-ene-1-sulfonyl fluorides give mixture of (1Z)- and (1E)-1-aryl-3-methylbutadienes. Their (Z)→(E) isomerization by classical means fails or leads to their polymerization. We have discovered that SO2 can isomerize 1-aryl-3-methyl-1,3-dienes at low temperature, without formation of sulfolenes (cheletropic addition/elimination). Preliminary mechanistic studies suggest that SO2 add to the 1,3-dienes forming 1,4-diradical intermediates that are responsible for the (Z)→(E) isomerizations. In the third chapter, we have shown that arenesulfonyl chlorides are versatile electrophilic reagents for C-C cross-coupling reactions. Arene-, arylmethane- and alk-2-ene-1-sulfonyl chlorides undergo desulfitative Stille, carbonylative Stille, Negishi and Suzuki-Miyaura cross-coupling reactions. In these reactions the reactivity order is ArI> ArSO2Cl> ArBr> ArCl. Similarly, desulfitative Sonogashira-Hagihara cross-couplings of arenesulfonyl chlorides with aryl- and alkylacetylenes can be catalyzed by Pd2(dba)3/P(t-Bu)3/CuI. New conditions have been found for the desulfitative Mizoroki-Heck arylation and trifluoromethylation of mono- and disubustituted olefins with arenesulfonyl and trifluoromethanesulfonyl chlorides. This procedure allows one to obtain (E)-1,2-disubstituted alkenes with high stereoselectivity and 1,1,2-disubstituted alkenes with high (E)/(Z) stereoselectivity. If phosphine- and base-free conditions are required, 1 mol% {RhCl(C2H4)2} catalyses the desulfitative cross-coupling reactions. On the contrary to what has been reported for RuCl2(PPh3)2 catalyzed coupling reactions with sulfonyl chlorides, the palladium and rhodium desulfitative Mizoroki-Heck coupling reactions are not inhibited by radical scavenging agents. Moreover, sulfones that are formed from the sulfonylation of alkenes at 60°C can theoretically be envisaged as intermediates in all cross-coupling reactions. However we have shown that they are not desulfitated at higher temperatures in the presence of the Pd or Rh-catalysts. Alk-2-ene-1-sulfonamides can also undergo desulfamylating cross-coupling reaction with Grignard reagents in the presence of a nickel catalyst. In the fouth chapter, we have evaluated the possibility that silyl sulfinates can be used as nucleophilic partners to form C-C bond. Our preliminary results indicate that their hypothesis should be explored further. In a fifth chapter, we have contributed to the development of an efficient one-pot, three component syntheses of sulfonamides and sulfonic esters. We have demonstrated that the ene-reaction of sulfur dioxide with enoxysilanes can be stereoselective under conditions of kinetic control. As others in our laboratory, we have shown that the hetero-Diels-Alder addition of sulfur dioxide to 1-oxy or 1,3-dioxy-1,3-dienes generates zwitterions that add to enoxysilanes or allylsilanes giving silyl sulfinates that can be converted in the same pot into polyfunctional sulfones, sulfonamides or sulfonic esters. Intramolecular S-allylation of intermediate silyl sulfinates has allowed one to prepare new tetrahydro-2H-thiocine derivatives. Our key contribution has been to use enantiomerically enriched amines which has allowed one to obtain enantiomerically enriched polyfunctional sulfonamides.

EPFL2005

Transition metal-catalyzed carbon-carbon cross-coupling reactions using sulfonyl derivatives

EPFL2005