The Generalised Dyson Circular Unitary Ensemble: Asymptotic Distribution of the Eigenvalues at the Origin of the Spectrum

The first part of this paper is devoted to the study of F_N{\Phi_N} the orthogonal polynomials on the circle, with respect to a weight of type f = (1 − cos θ) ^α c where c is a sufficiently smooth function and ${\alpha > -\frac{1}{2}}${\alpha > -\frac{1}{2}}. We obtain an asymptotic expansion of the coefficients F^*(p)_N(1){\Phi^{*(p)}_{N}(1)} for all integer p where F^*_N{\Phi^*_N} is defined by F^*_N (z) = z^N [`(F)]_N(\frac1z) (z 1 0){\Phi^*_N (z) = z^N \bar \Phi_N(\frac{1}{z})\ (z \not=0)}. These results allow us to obtain an asymptotic expansion of the associated Christofel–Darboux kernel, and to compute the distribution of the eigenvalues of a family of random unitary matrices. The proof of the results related to the orthogonal polynomials are essentially based on the inversion of the Toeplitz matrix associated to the symbol f.     

Analytical formulas for calculation of K X-ray production cross sections by alpha ions

In the present study, different procedures are followed to deduce the semi-empirical and the empirical K X-rayX-ray production cross sections induced by alpha ions from the available experimental data and the theoretical results of the ECPSSR model for elements with 20≤Z≤30. The deduced K X-ray production cross sections are compared with predictions from ECPSSR model and with other earlier works. Generally, the deduced K X-ray production cross sections obtained by fitting the available experimental data for each element separately give the most reliable values than those obtained by a global fit.     

Integral inequalities related to the Tchebychev semigroup

We study the semigroup (P _t ) _t≥0 generated by the operator

The Synthesis of Natural Acetylenic Compounds from Stereum hirsutum

The regioselective synthesis of two new acetylenic compounds, sterehirsutinal ( 1 ) and sterehirsutinol ( 2 ), isolated recently from culture medium of the fungus Stereum hirsutum, is reported.     

Estimateur hiérarchique robuste pour un problème de perturbation singulière

In this Note, we show that a modified and simplified version of the estimator of Bank–Weiser can be used to define a robust a posteriori error estimator for singularly perturbed problem. We prove without comparison with a residual estimator or saturation assumption, the equivalence of the estimator with the error in the energy norm and the robusteness with respect to the diffusion coefficient. To cite this article: B. Achchab et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Affinity of trypsin for amidine derivatives immobilized on dextran-coated silica supports.

The pancreas contains two very analogous enzymes: trypsin and chymotrypsin. These two enzymes are very similar in their physicochemical characteristics and are therefore quite difficult to separate by classical purification procedures. They constitute a good model for affinity chromatography. It was previously demonstrated that amidine derivatives are able to interact strongly and specifically with these serine proteases and are often used as ligand in affinity chromatography. To understand the trypsin interaction mechanism, we synthesized different amidines and immobilised them with or without spacer arm on silica beads previously coated by dextran substituted with a calculated amount of positively charged diethylaminoethyl functions, in order to minimize the non-specific interactions of silanol groups of the silica material. First the affinity constant and the adsorption capacity of these supports for trypsin were determined in batch procedures, then they were used in affinity chromatography. The effects of ionic strength, pH and competitive inhibitors on proteins desorption were also studied. Last, to demonstrate the importance of passivation, the chromatographic performances of dextran-coated silica phases and a commercial support grafted with the same amidine were compared.