Real analysis related to the Monge-Ampère equation

In this paper we consider a family of convex sets in , , , , satisfying certain axioms of affine invariance, and a Borel measure satisfying a doubling condition with respect to the family The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of This is achieved by showing first a Besicovitch-type covering lemma for the family and then using the doubling property of the measure The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to

     

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.     

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.     

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.     

Enantioselective synthesis of 1-deoxy-d-gulonojirimycin from a phenylglycinol-derived lactam

Cyclocondensation of (R)-phenylglycinol with appropriately γ-substituted δ-oxo acid derivatives provides bicyclic lactams from which the enantioselective synthesis of 1-deoxy-d-gulonojirimycin has been reported.     

Use of highly efficient Draper-Lin small composite designs in the formal optimisation of both operational and chemical crucial variables affecting a FIA-chemiluminescence detection system

A new formal strategy in the multidimensional optimisation of the experimental variables affecting the chemiluminescence (CL) detection in flow injection analysis (FIA) is proposed here. The strategy implies several steps, being the most significant: selection of the variables to be studied and their experimental domain; use of a screening design to detect significant variables and interactions into the experimental region; study of the main effect of variables and second-order interactions; and finally application of a Draper-Lin small composite design (orthogonal) to obtain the optimum values of the significant variables. The methodology is applied to the determination of methylamine by FIA based on the use of the peroxyoxalate CL (PO-CL) reaction. Considering the high number of experiments required due to the different chemical and instrumental variables to be taken account and their adequate compatibility to obtain maximum sensitivity, the methodology offers a rigorous study of the main effects and interactions, achieving a reduction of experimental work.