Almost periodic invariant tori for the NLS on the circle

In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [15] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract “counter-term theorem” à la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find “many more” almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs.     

In-flight antiproton annihilation on nuclei at low energies

The results of the annihilation cross sections measurement of 5.3 MeV antiprotons on nickel, tin, platinum and Mylar targets performed by the ASACUSA Collaboration at CERN are presented and compared with the existing data and models. From the experimental point of view the presented data are the first measurement of antinucleon annihilation cross sections at low energies obtained with a pulsed beam. This results open the road for the next measurements at the very low energies of the order of 100 keV that are in progress by the ASACUSA Collaboration. The experimental method foreseen for the 100 keV measurement is illustrated.     

Li4C60: a polymeric fulleride with a two-dimensional architecture and mixed interfullerene bonding motifs

All known fullerene polymers have interfullerene connections via either [2 + 2] cycloaddition or single C-C bonds. The high-resolution synchrotron X-ray powder diffraction technique was employed here to determine the crystal structure of the Li4C60 fulleride. We find that the ground state of Li4C60 is a two-dimensional polymer with monoclinic crystal symmetry and an unprecedented architecture, combining both the [2 + 2] cycloaddition and the single C-C bridging motifs. The small size of the Li+ cations is crucial in stabilizing the resulting tightly packed polymeric structure.     

Low-Temperature Crystal Structure of S-camphor Solved from Powder Synchrotron X-ray Diffraction Data by Simulated Annealing

The ordered, low-temperature crystal structure of the pure enantiomer of camphor (C₁₀H₁₆O) has been solved from high-resolution powder synchrotron X-ray diffraction data. The structure is orthorhombic, space group P2₁2₁2₁, Z=8, with a=8.9277(2) Å, b=27.0359(5) Å, and c=7.3814(1) Å at 100 K. The structure was solved by autoindexing of the pattern, space group determination, and then optimization of the positions and orientations of the two independent molecules in the unit cell by simulated annealing. The molecular structure obtained from the restrained Rietveld refinement shows reasonable agreement with that optimized from ab initio molecular orbital calculations. In the crystal structure, the molecules are aligned antiferroelectrically and weak C-H…O hydrogen bonds link together the independent molecules.     

Determination of some β‐Blockers by Electrochemical Detection on Polycrstalline Gold Electrode after Solid Phase Extraction (SPE)

The electroanalytical characterization and determination of three selected β‐blocker agents, namely propranolol, atenolol and nadolol using cyclic voltammetry and differential pulse voltammetry (DPV) in phosphate buffer solution (pH 2.5) plus 22 % acetonitrile (ACN), was described. The analytes were characterized through their electrooxidation processes on polycrystalline gold electrodes. The analytical determination of the selected molecules was performed using the differential pulse voltammetry (DPV) at pH 2.5. Under DPV conditions, the detection limits (LODs) ranged between 5 μM and 20 μM for propranolol and atenolol, respectively. For all investigated molecules, two well‐defined ranges of linearity Ip vs analyte concentration have been identified which correspond to specific calibration parameters. Calibration graphs (Ip vs concentration) considered in the first interval of linearity, shown correlation coefficients >0.99. A solid phase extraction (SPE) procedure using a polymeric mixed‐mode cationic sorbent (Strata‐X‐C), was studied and optimized. The proposed DPV‐SPE method was successfully applied for the determination of propranolol in several pharmaceutical formulations and urine sample, with results in close agreement with those obtained using traditional liquid chromatography technique coupled with spectrophotometric detection.     

Uniqueness and decay estimates for a class of parabolic partial differential equations with hysteresis and convection

We deal in detail with the question of existence, uniqueness and asymptotic behaviour of solutions to a parabolic equation with hysteresis and convection. This equation is part of a model system which describes the magnetohydrodynamic (MHD) flow of a conducting fluid between two ferromagnetic plates. The result of this paper complements the content of a previous paper of the first author, where existence of the solution has been proved under fairly general assumptions on the hysteresis operator and the uniqueness was only obtained for a restricted class of hysteresis operators.     

Extrapolation methods for PageRank computations

The mathematical problem behind Web search is the computation of the nonnegative left eigenvector of a stochastic matrix P corresponding to the dominant eigenvalue 1. This vector is called the PageRank vector. Since the matrix P is ill-conditioned, the computation of PageRank is difficult and the matrix P is replaced by $P(c)=cP+(1?c)E$, where E is a rank one matrix and c a parameter. The dominant left eigenvector of $P(c)$ is denoted by PageRank$(c)$. This vector can be computed for several values of c and then extrapolated at the point $c=1$. In this Note, we construct special extrapolation methods for this problem. They are based on the mathematical analysis of the vector PageRank$(c)$. To cite this article: C. Brezinski et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).