Invariant generalized functions on sl(2,R) with values in a sl(2,R)-module

AbstractLet g be a finite-dimensional real Lie algebra. Let ρ:g→End(V) be a representation of g in a finite-dimensional real vector space. Let CV=(End(V)⊗S(g))g be the algebra of End(V)-valued invariant differential operators with constant coefficients on g. Let U be an open subset of g. We consider the problem of determining the space of generalized functions φ on U with values in V which are locally invariant and such that CVφ is finite dimensional.In this article we consider the case g=sl(2,R). Let N be the nilpotent cone of sl(2,R). We prove that when U is SL(2,R)-invariant, then φ is determined by its restriction to U⧹N where φ is analytic (cf. Theorem 6.1). In general this is false when U is not SL(2,R)-invariant and V is not trivial. Moreover, when V is not trivial, φ is not always locally L1. Thus, this case is different and more complicated than the situation considered by Harish-Chandra (Amer. J. Math 86 (1964) 534; Publ. Math. 27 (1965) 5) where g is reductive and V is trivial.To solve this problem we find all the locally invariant generalized functions supported in the nilpotent cone N. We do this locally in a neighborhood of a nilpotent element Z of g (cf. Theorem 4.1) and on an SL(2,R)-invariant open subset U⊂sl(2,R) (cf. Theorem 4.2). Finally, we also give an application of our main theorem to the Superpfaffian (Superpfaffian, prepublication, e-print math.GR/0402067, 2004)

CONTRIBUTION A L'HISTOIRE DES TECHNIQUES ET DE L'ARMEMENT : ESSAIS DE RESTITUTION DU FORGEAGE DE POINTES DE FLECHE A PARTIR DE BARRES DE FER D'EPOQUE ANTIQUE

International audienceCette étude porte sur la recherche des conditions de forgeage optimales ayant conduit à la réalisation de pointes de flèche diverses, identiques à celles dont on dispose pour l'époque antique, parfois en grande abondance, comme cela est le cas pour le site du siège d'Uxellodunum, au Puy d'Issolud (Lot). L'expérimentation a été conduite avec le concours d'un taillandier spécialiste de la restitution d'armes anciennes. L'originalité du travail porte sur le choix du forgeage de barres archéologiques issues des épaves gisant au large des Saintes-Maries-de-la-Mer, en Méditerranée, et des fouilles du Couperé à Saint Bertrand de Comminges, dans les Pyrénées. La caractérisation métallographique préalable de ces barres nous a permis de suivre l'incidence des conditions de forgeage sur la microstructure des pointes de flèche réalisées. Les principaux faciès typologiques et microstructuraux des flèches antiques ont été reproduits, permettant ainsi d'affiner la connaissance technique du protocole de forgeage du fer, au début de notre ère

Conserved Charges in the Principal Chiral Model on a Supergroup

The classical principal chiral model in 1+1 dimensions with target space a compact Lie supergroup is investigated. It is shown how to construct a local conserved charge given an invariant tensor of the Lie superalgebra. We calculate the super-Poisson brackets of these currents and argue that they are finitely generated. We show how to derive an infinite number of local charges in involution. We demonstrate that these charges Poisson commute with the non-local charges of the model

Actions de polyphénols de Clusiaceae sur l’endothélium vasculaire

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Screening of polyphenolic compounds isolated from Clusiaceae plants causing endothelial nitric oxide production

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Silencing of the Violaxanthin De-Epoxidase Gene in the Diatom Phaeodactylum tricornutum Reduces Diatoxanthin Synthesis and Non-Photochemical Quenching

Diatoms are a major group of primary producers ubiquitous in all aquatic ecosystems. To protect themselves from photooxidative damage in a fluctuating light climate potentially punctuated with regular excess light exposures, diatoms have developed several photoprotective mechanisms. The xanthophyll cycle (XC) dependent non-photochemical chlorophyll fluorescence quenching (NPQ) is one of the most important photoprotective processes that rapidly regulate photosynthesis in diatoms. NPQ depends on the conversion of diadinoxanthin (DD) into diatoxanthin (DT) by the violaxanthin de-epoxidase (VDE), also called DD de-epoxidase (DDE). To study the role of DDE in controlling NPQ, we generated transformants of P. tricornutum in which the gene (Vde/Dde) encoding for DDE was silenced. RNA interference was induced by genetic transformation of the cells with plasmids containing either short (198 bp) or long (523 bp) antisense (AS) fragments or, alternatively, with a plasmid mediating the expression of a self-complementary hairpin-like construct (inverted repeat, IR). The silencing approaches generated diatom transformants with a phenotype clearly distinguishable from wildtype (WT) cells, i.e. a lower degree as well as slower kinetics of both DD de-epoxidation and NPQ induction. Real-time PCR based quantification of Dde transcripts revealed differences in transcript levels between AS transformants and WT cells but also between AS and IR transformants, suggesting the possible presence of two different gene silencing mediating mechanisms. This was confirmed by the differential effect of the light intensity on the respective silencing efficiency of both types of transformants. The characterization of the transformants strengthened some of the specific features of the XC and NPQ and confirmed the most recent mechanistic model of the DT/NPQ relationship in diatoms

Polyphenolic compounds from Clusiaceae plants modulating angiogenesis and vascular endothelium

Polyphenolic compounds have created an increasing interest for their potency about cardiovascular diseases for several years1-2. Nevertheless, most of this research had been focused on polyphenolic compound such as flavanols (e.g. catechine from tea), anthocyanin (e.g. delphinidin from blueberry) and stilbenoides (e.g. resveratrol from grape).The present study was designed to screen the potent effect of polyphenolic compounds isolated from plants belonging to Clusiaceae family on endothelium. A huge number of polyphenols such as xanthones and coumarines have been identified from those species and some of them exhibiting various biological activities such as anti-inflammatory and antioxidant properties3-4. Their effect on endothelium, more particularly on angiogenesis, is not yet well-known. Firstly, we assessed the capacity of six molecules to induce endothelium-dependent relaxation in mice aortic rings involving nitric oxide production. Isocalolongic acid (A1) and 2-deprenylrheediaxanthone (A2) are able to increase NO production on endothelial cells and to induce endothelium-dependant relaxation. Then, we investigated the effects of these compounds on in vitro and ex vivo angiogenesis. We showed that A1 treatment promoted the formation of capillary-like network contrary to A2. Endothelial cell adhesion, migration and proliferation were decreased in presence of A2 whereas endothelial migration and proliferation were improved with A1 treatment. We could explain these results with the capacity of A1 to increase VEGF expression and for A2, to decrease ICAM-1 expression. Thus, the strategy used for the screening allows the detection of active molecules from Clusiaceae family that might be of therapeutic benefit in cardiovascular diseases5