Abdellatif Ghendir Aoun

In this paper, we study a fractional differential equation $$^{c}D^{\alpha}_{0^{+}}u(t)+f(t,u(t))=0,\quad t\in(0, +\infty)$$ satisfying the boundary conditions: $$u^{\prime}(0)=0,\quad \lim_{t\rightarrow +\infty}\,^{c}D^{\alpha-1}_{0^{+}}u(t)=g(u),$$ where $1<\alpha\leqslant2$, $^{c}D^{\alpha}_{0^{+}}$ is the standard Caputo fractional derivative of order $\alpha$. The main tools used in the paper is contraction principle in the Banach space and the fixed point theorem due to D. O''Regan. Some the compactness criterion and existence of solutions are established.     

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

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).