Clustering of Eigenvalues on Translation Surfaces

We consider Riemannian surfaces in which there are many embedded Euclidean cylinders. We show that quantitative assumptions on the number of such cylinders provide estimates for the counting function of eigenvalues. These results in particular apply to translation surfaces. Communicated by Jens Marklof Submitted: 3/03/ 2005 Accepted: 18/11/2005     

Rational approximation in the complex plane using a τ-method and computer algebra

Numerous versions of the Lanczos τ-methods have been extensively used to produce polynomial approximations for functions verifying a linear differential equation with polynomial coefficients. In the case of an initial-value problem, an adapted τ-method based on Chebyshev series and the use of symbolic computation lead to a rational approximation of the solution on a region of the complex plane. Numerical examples show that the simplicity of the method does not prevent a high accuracy of results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Lamellar Organo-Bridged Silicones

The acid-hydrolysis of an organo-bridged bisdiethoxysilylated molecular precursor bearing urea groups, (EtO)₂MeSi(CH₂)₃NHCONH(CH₂)₁₂NHCONH(CH₂)₃SiMe(OEt)₂, has been performed in pure aqueous medium. Scanning electron microscopy (SEM) analysis of the resulting insoluble solid revealed plate-like forms with a lamellar structure as determined by powder X-ray diffraction (PXRD) studies with a sharp peak at 28.5 Å. The solid state ²⁹Si MAS-NMR spectrum of this bridged siloxane hybrid is consistent with a moderately condensed material with complete preservation of the Si–C bonds throughout the hybrid network. In comparison, the classical sol–gel hydrolysis-condensation of the molecular precursor in ethanol with stoichiometric amount of water and fluoride anion as catalyst produced an amorphous featureless solid.     

A systematic formula for the asymptotic expansion of singular integrals

Asymptotic theories like the lifting-line, the slender body or the slender ship lead to lineintegrals with singular kernels. Sometimes these integrals are improper, that is to say that they are defined only by their Finite Part. To find asymptotic expansions of these integrals, the Matched Asymptotic Expansion Method is widely used along with other more specific methods depending on the kernel type. The first method is laborious and not systematic, and the other methods are sometimes too much specific to treat general cases. Moreover, all of them are not well adapted to deal with Finite Part integrals.Here, a new method is proposed to avoid the previous difficulties. This method is systematic for homogeneous kernels and gives approximations up to any order, as long as the derivative of the weight function exists at this given order. Moreover the occurrence of logarithmic terms in the expansion is explained and easily predictable. An elliptic integral and the classical lifting-line theory are treated to illustrate the ease of this method.

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Résumé Les théories asymptotiques telles que la ligne portante, le corps élancé ou le navire de grand allongement conduisent à des intégrales curvilignes à noyaux singuliers. Parfois, ces intégrales sont impropres c'est à dire qu'elles sont définies en Parties Finies. Différentes méthodes ont été mises au point pour trouver les développements asymptotiques de ces intégrales. Généralement elles dépendent fortement de la nature du noyau, et c'est finalement la méthode des développements raccordés qui est utilisées quand le noyau est trop compliqué. Cependant, cette méthode est laborieuse et comme les précèdentes non adaptée aux intégrales défines par leur Partie Finie.Une nouvelle méthode est proposée pour surmonter ces difficultés. Cette méthode est systématique pour les noyaux homogènes et donne les approximations à tout ordre pourvu que les dérivées de la fonction poids existent jusqu'à cet ordre. De plus la présence de termes logarithmiques dans le développement est expliquée et aisément prédictible.Une intégrale elliptique, ainsi que la fameuse théorie de la ligne portante sont traités pour illustrer les possibilités de la méthode.

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Nomenclature D domain of integration - f(x) weight function - FP Finite Part - h(x) weight function - I ,I _o bounded intervals - j, J integers - K(x, ) singular kernel - L, L integers - M integer defining the approximation order - P _k (x) Legendre polynomial - R set of real numbers - R(ß) equals 1 if is an integer and 0 if not - R _{f, J} ,R _{K, L} remainders of Taylor developments - S () equals either 1 or the sign function:sgn() - t, u, v, x variable of integration - , real numbers - homogeneity order of the kernel - F () Euler's integral (gamma function) - small parameter - [.] integer part of     

Rapid analysis of triterpenic acids by liquid chromatography using porous graphitic carbon and evaporative light scattering detection

An original system which uses Porous Graphitic Carbon as support and a mixture of organic solvents as mobile phase is proposed for the analysis of triterpenic acids by liquid chromatography. The separation of betulinic acid, ursolic acid, oleanolic acid, and 18alpha- and 18beta-glycyrrhetinic acids was carried out within a short time and monitored by evaporative light scattering detection as universal detection method. Molecular modelling studies show that the main contribution to the selectivity comes from the electrostatic interaction characterised by the dipole moment of the products.     

Investigation of the formation process of MCs+-molecular ions during sputtering

In secondary ion mass spectrometry, the detection of MCs+ clusters (with M an element of the specimen) under a Cs bombardment is frequently used for the quantification of major elements. Despite some very good results obtained by this method, some problems still remain. In order to gain some more insight into these problems, the formation mechanism of the MCs+ clusters is investigated using a Monte Carlo model. It is shown that the majority of the constituent particles of the formed clusters are initially first or second neighbor atoms at the surface and that the velocity distribution of the MCs+ clusters becomes broader and peaked at higher velocities with increasing surface binding energy of the M atom. In addition, it is demonstrated that the interaction potential between the M and Cs+ particle has no influence on the velocity distribution of the MCs+ clusters. On the other hand, the cluster formation probability, defined as the probability that a sputtered M and Cs+ particle will form a MCs+ cluster, is extremely sensitive to this interaction potential. It is also shown that the cluster formation probability decreases with increasing surface binding energy. Finally, a good correspondence is obtained between the calculated and experimental velocity distributions of MCs+ clusters sputtered from different monoatomic materials. As a consequence, the Monte Carlo model and the discussed results can be validated.     

Third-order nonlinear optical figure of merits for conjugated TTF-quinone molecules

We have revealed a substantial enhancement of third-order optical figure of merits by the synthesis of a compact molecule possessing the tetrathiafulvalene (TTF) group with two backside C=O groups. Addition of the saturated methylene chain substantially suppresses the third-order optical figure of merits and even local optical hyperpolarizabilities at lambda = 532 nm. Another TTF-derivative molecule possessing ethylenic and acetylenic chains demonstrates large hyperpolarizabilities; however, generally, the figure of merit factor decreases due to the increasing optical losses as a consequence of enhanced linear absorption. At the same time, both of the chromophores have a large nonlinear optical response. General approaches for search and design of the third-order optical materials with improved properties are given.