Cohomology of the complement of a free divisor

We prove that if is a ``strongly quasihomogeneous" free divisor in the Stein manifold , and is its complement, then the de Rham cohomology of can be computed as the cohomology of the complex of meromorphic differential forms on with logarithmic poles along , with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather's nice dimensions (and in particular the discriminants of Coxeter groups).

     

Arrangements of hyperplanes with property D

We denote by the complement of the complexification of a real arrangement of hyperplanes. It is known that there is a certain technical property, called property D, on real arrangements of hyperplanes such that: if a real arrangement of hyperplanes is simplicial then has property D, and if has property D then is aK(, 1) space. Our main goal is to prove that: if has property D then is simplicial. We also prove that a quasi-simplicial arrangement is always simplicial.     

Self-consistent solutions of the semiclassical Einstein equations with cosmological constant

The general solution to the semiclassical backreaction equation is found for conformally invariant free quantum fields in spatially flat homogeneous and isotropic spacetime with Cosmological constant and with no classical source when the ratio of the renormalisation parameters/=9/4. It contains a two-parameter family of bouncing solutions that avoid the singularity. There are several one-parameter families which do not have particle horizons. The stability of these solutions is investigated and it is found that they are stable when and have different signs. However, when both parameters have the same sign the set of stable solutions is restricted by the condition 0 < < 1/9. In both cases these solutions have a final de Sitter stage.     

Weyl equation in some anisotropic stiff fluid universes

The Weyl equation (massless Dirac equation) is studied in a family of exact solutions of the Einstein equations whose material content is a perfect fluid with stiff equation of state (p=) and which are of Bianchi type I. The field equation is solved exactly for some members of the family.     

Effect of particle geometry on phase transitions in two-dimensional liquid crystals

Using a version of density-functional theory which combines Onsager approximation and fundamental-measure theory for spatially nonuniform phases, we have studied the phase diagram of freely rotating hard rectangles and hard discorectangles. We find profound differences in the phase behavior of these models, which can be attributed to their different packing properties. Interestingly, bimodal orientational distribution functions are found in the nematic phase of hard rectangles, which cause a certain degree of biaxial order, albeit metastable with respect to spatially ordered phases. This feature is absent in discorectangles, which always show unimodal behavior. This result may be relevant in the light of recent experimental results which have confirmed the existence of biaxial phases. We expect that some perturbation of the particle shapes (either a certain degree of polydispersity or even bimodal dispersity in the aspect ratios) may actually destabilize spatially ordered phases thereby stabilizing the biaxial phase.     

Useful dual Diels-Alder behavior of 2-azetidinone-tethered aryl imines as azadienophiles or azadienes: a beta-lactam-based stereocontrolled access to optically pure highly functionalized indolizidine systems

Imines derived from 4-oxoazetidine-2-carbaldehydes have been found to be versatile Diels-Alder reagents in that they exhibit two reactivity patterns. 2-Azetidinone-tethered imines undergo diastereoselective reaction with Danishefsky's diene in the presence of different Lewis acids. The effect of the amount of catalyst on the conversion rate as well as on the product ratio has been studied. Under standard reaction conditions, indium(III) chloride and zinc(II) iodide provided the best yields, and indium(III) triflate the highest diastereoselectivity in the Lewis acid promoted aza-Diels-Alder cycloaddition. Treatment of the aforementioned imines with cyclopentadiene, 2,3-dimethyl-1,3-butadiene or 3,4-dihydro-2 H-pyran led to cycloadducts arising from inverse electron-demand condensation involving the beta-lactam-tethered aryl imine as the heterodiene component. In addition, the first methodology for preparing indolizidines from beta-lactams has been developed. This process involves amide bond cleavage of the beta-lactam ring in the aza-Diels-Alder cycloadducts with concomitant cyclization. Full chirality transfer occurs when the reaction is performed with an enantiomerically pure substrate.     

A short synthesis of the erythrina skeleton and of (+/-)-alpha-lycorane

[reaction: see text] A new nonchain 5-endo radical cyclization starting with xanthates was exploited in a short synthesis of (+/-)-alpha-lycorane and the erythrina ring system.     

Etude MNDO des composes conjugues enneagonaux

We present MNDO calculations of a series of nine-membered conjugated compounds. The geometries have been fully optimized. Cyclononatetraenyl and benzocyclononatetraenyl anions, nonapyrrol, and nonathiophene appear to be planar aromatic compounds. Following MNDO , nonafulvene has a planar prefered conformation, but a polyenic structure, while nonafuran and cyclononatetraene are gauche molecules. The Universal Huckel Method (UHM ), a simple all valence electron empirical technique which describes σ systems using localized MO , gives, qualitatively, the same results as the MNDO method. The quantum calculation carried out with MNDO and UHM have been used to describe the physicochemical properties of ninememberedring conjugated compounds under consideration. To our knowledge, we present here the first theoretical study dealing with nine-membered conjugated molecules.     

2-Alkyl-2-carboxy-azetidines as scaffolds for the induction of γ-turns

To investigate the ability of 2-alkyl-2-carboxy-azetidines (Azx) to induce reverse turns when incorporated into peptides, RCO-Azx-l-Ala-NHMe dipeptide derivatives were selected as simplified tetrapeptide models, in which the azetidine residue is incorporated at the i + 1 position. Molecular modelling, ¹H NMR and FTIR studies showed the high tendency of the model tetrapeptides to adopt γ-turn conformations, indicating that these azetidine-containing amino acids could serve as general γ-turn promoters.