Boutet de Monvel`s calculus and groupoids I

Can Boutet de Monvel`s algebra on a compact manifold with boundary be obtained as the algebra Psi(0)(G) of pseudodifferential operators on some Lie groupoid G? If it could, the kernel G of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C*(G). While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C*(G) possesses an ideal I isomorphic to G. In fact, we prove first that G similar or equal to Psi circle times K with the C*-algebra Psi generated by the zero order pseudodifferential operators on the boundary and the algebra K of compact operators. As both Psi circle times K and I are extensions of C(S*Y) circle times K by K (S*Y is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.

Nonhomogeneous subalgebras of Lie and special Jordan superalgebras

We consider polynomial identities satisfied by nonhomogeneous subalgebras of Lie and special Jordan superalgebras: we ignore the grading and regard the superalgebra as an ordinary algebra. The Lie case has been studied by Volichenko and Baranov: they found identities in degrees 3, 4 and 5 which imply all the identities in degrees <= 6. We simplify their identities in degree 5, and show that there are no new identities in degree 7. The Jordan case has not previously been studied: we find identities in degrees 3, 4, 5 and 6 which imply all the identities in degrees <= 6, and demonstrate the existence of further new identities in degree 7. our proofs depend on computer algebra: we use the representation theory of the symmetric group, the Hermite normal form of an integer matrix, the LLL algorithm for lattice basis reduction, and the Chinese remainder theorem. (C) 2009 Elsevier Inc. All rights reserved.

GENERICITY OF NONDEGENERATE GEODESICS WITH GENERAL BOUNDARY CONDITIONS

Let M be a possibly noncompact manifold. We prove, generically in the C(k)-topology (2 <= k <= infinity), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of L. Biliotti, M. A. Javaloyes and P. Piccione [6] for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P subset of M x M that satisfies an admissibility condition. Such condition holds, for example, when P is transversal to the diagonal Delta subset of M x M. Further aspects of these boundary conditions are discussed and general conditions under which metrics without degenerate geodesics are C(k)-generic are given.

Identification of an Operon, Pil-Chp, That Controls Twitching Motility and Virulence in Xylella fastidiosa

Xylella fastidiosa is an important phytopathogenic bacterium that causes many serious plant diseases, including Pierce`s disease of grapevines. Disease manifestation by X. fastidiosa is associated with the expression of several factors, including the type IV pili that are required for twitching motility. We provide evidence that an operon, named Pil-Chp, with genes homologous to those found in chemotaxis systems, regulates twitching motility. Transposon insertion into the pilL gene of the operon resulted in loss of twitching motility (pilL is homologous to cheA genes encoding kinases). The X. fastidiosa mutant maintained the type IV pili, indicating that the disrupted pilL or downstream operon genes are involved in pili function, and not biogenesis. The mutated X. fastidiosa produced less biofilm than wild-type cells, indicating that the operon contributes to biofilm formation. Finally, in planta the mutant produced delayed and less severe disease, indicating that the Pil-Chp operon contributes to the virulence of X. fastidiosa, presumably through its role in twitching motility.

Chemical variability within the marine sponge Aplysina fulva

Dibromotyrosine-derived metabolites are of common occurrence within marine sponges belonging to the order Verongida. However, previous chemical analysis of crude extracts obtained from samples of the verongid sponge Aplysina fulva collected in Brazil did not provide any dibromotyrosine-derived compounds. In this investigation, five samples of A. fulva from five different locations along the Brazilian coastline and one sample from a temperate reef in the South Atlantic Bight (SAB) (Georgia, USA) were investigated for the presence of bromotyrosine-derived compounds. All six samples collected yielded dibromotyrosine-derived compounds, including a new derivative, named aplysinafulvin, which has been identified by. analysis of spectroscopic data. These results confirm previous assumptions that dibromotyrosine-derived metabolites can be considered as chemotaxonomic markers of verongid sponges. The isolation of aplysinafulvin provides additional support for a biogenetic pathway involving an arene oxide intermediate in the biosynthesis of Verongida metabolites. It cannot yet be established if the chemical variability observed among the six samples of A.fulva collected in Brazil and the SAB is the result of different environmental factors, distinct chemical extraction and isolation protocols, or a consequence of hidden genetic diversity within the postulated morphological plasticity of this species. (C) 2007 Elsevier Ltd. All rights reserved.