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We investigate a flexible two-component semiparametric mixture of regressions model, in which one of the conditional component distributions of the response given the covariate is unknown but assumed symmetric about a location parameter, while the other is specified up to a scale parameter. The location and scale parameters together with the proportion are allowed to depend nonparametrically on covariates. After settling identifiability, we provide local M-estimators for these parameters which converge in the sup-norm at the optimal rates over Holder-smoothness classes. We also introduce an adaptive version of the estimators based on the Lepski-method. Sup-norm bounds show that the local M-estimator properly estimates the functions globally, and are the first step in the construction of useful inferential tools such as confidence bands. In our analysis we develop general results about rates of convergence in the sup-norm as well as adaptive estimation of local M-estimators which might be of some independent interest, and which can also be applied in various other settings. We investigate the finite-sample behaviour of our method in a simulation study, and give an illustration to a real data set from bioinformatics.

As shown by theoretical calculations, azabis(oxazoline)-copper complexes are considerably more stable than the analogous bis(oxazoline)-copper complexes. This enhanced stability allows them to be efficiently immobilized by means of electrostatic interactions to different anionic supports, such as clays and nafion-silica nanocomposites, without the loss of a ligand, as is observed for bis(oxazolines). As a result, enantioselectivities of around 90% ee are obtained in the cyclopropanation reaction between styrene and ethyl diazoacetate. Moreover, the solid catalyst is easily recoverable., This work was made possible by the generous financial support of the CICYT (project PPQ2002-04012), the MCYT and DAAD (Acción Integrada HA2001-0096), and the DGA. C.I.H. is indebted to the MCYT for a grant.

Aza(bisoxazoline) ligands were attached to various polymeric supports and the resulting immobilized ligands were evaluated in copper(I)-catalyzed asymmetric cyclopropanations. The efficiency of these transformations depends greatly on the polymeric support, on the protocol being applied for the immobilization of the ligands, and on the preparation of the catalysts.

This workw as supported by the Fonds der Chemischen Industrie, the DAAD (Acción Integrada HA2001 – 0096), the C.I.C.Y.T. (project PPQ2002– 04012), the DGA, and through generous chemical gifts by Degussa AG. I.P. is indebted to the MEC for a grant.

A straightforward synthesis of chiral aza-bis(oxazoline) (Azabox) ligands from commercially available amino alcohols is described. The new protocol allows access to previously reported Azabox ligands in considerably improved yields but also to new derivatives, including non-C2-symmetrical ones.

Bis(oxazoline)–copper complexes, immobilised by electrostatic interactions on Laponite and Nafion/silica nanocomposites, have been tested as catalysts in the cyclopropanation of styrene with ethyl diazoacetate. Several factors play a decisive role in the outcome of the reaction. Firstly, the nature of the solid counterion is important, with perfluorosulfonic solids being the best in this regard. Secondly, the nature of the solvent used has a marked influence as some solvents cause the solid support to modify the stereochemical course of the reaction. Finally, the nature of the chiral ligand is also very important, in particular the strength of its copper complex, in order to avoid the formation of non-chiral catalytic copper centres. This particular effect has been highlighted by using a ligand that is able to give strong complexation, namely an iminobis(oxazoline) ligand, and this leads to the best enantioselectivities being obtained for reactions using electrostatically immobilised catalysts., This work was made possible by the generous financial support of the CICYT (project MAT99-1176) and the MCYT (Acción Integrada HA2001-0096). C.I.H. is also indebted to the MCYT for a grant.