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).     

Syntheses and crystal structures of two magnesium–tetrazole compounds in salt and polymeric forms

Two alkaline earth–tetrazole compounds, namely catena‐poly[[[triaquamagnesium(II)]‐μ‐5,5′‐(azanediyl)ditetrazolato‐κ³N¹,N^1′:N⁵] hemi{bis[μ‐5,5′‐(azanediyl)ditetrazolato‐κ³N¹,N^1′:N²]bis[triaquamagnesium(II)]} monohydrate], {[Mg(C₂HN₉)(H₂O)₃][Mg₂(C₂HN₉)₂(H₂O)₆]_0.5·H₂O}_n, (I), and bis[5‐(pyrazin‐2‐yl)tetrazolate] hexaaquamagnesium(II), (C₅H₃N₆)[Mg(H₂O)₆], (II), have been prepared under hydrothermal conditions. Compound (I) is a mixed dimer–polymer based on magnesium ion centres and can be regarded as the first example of a magnesium–tetrazolate polymer in the crystalline form. The structure shows a complex three‐dimensional hydrogen‐bonded network that involves magnesium–tetrazolate dimers, solvent water molecules and one‐dimensional magnesium–tetrazolate polymeric chains. The intrinsic cohesion in the polymer chains is ensured by N—H...N hydrogen bonds, which form R₂²(7) rings, thus reinforcing the propagation of the polymer chain along the a axis. The crystal structure of magnesium tetrazole salt (II) reveals a mixed ribbon of hydrogen‐bonded rings, of types R₂²(7), R₂²(9) and R₂⁴(10), running along the c axis, which are linked by R₂⁴(16) rings, generating a 4,8‐c flu net.