Miscellany on traces in ℓ-adic cohomology: a survey

We discuss classical questions concerning traces of elements of Galois groups or correspondences in ℓ-adic cohomology, mostly over finite or local fields, such as rationality and independence of ℓ, integrality, congruences modulo powers of ℓ or p. We report on the progress that has been made on this topic during the past ten years.     

Pd-catalyzed one-pot coupling of allylamines, activated alkenes, and unsaturated halides (or triflate): an atom efficient synthesis of highly functionalized pyrrolidines

A Pd-catalyzed three-component assembling of highly functionalized 4-benzyl-(and allyl-)pyrrolidines was achieved based on a combination of allyl amines, gem-diactivated alkenes, and unsaturated halides (or triflate).     

Robust Speed Control of a Low Damped Electromechanical System Based on CRONE Control: Application to a Four Mass Experimental Test Bench

Robust speed control of a low damped electromechanical system with backlash is studied, controlled load angular speed being not measured. The proposed control strategy combines a Luenberger observer (load angular speed and load torque disturbance estimations) and a robust CRONE controller. The observer provides estimation of the load angular speed and of the disturbance torque applied on the load. Through the computation of only three independent parameters (as many as a PID controller), the CRONE controller permits to ensure the robust speed control of the load in spite of plant parametric variations and speed observation errors. The proposed control strategy is applied to a four mass experimental test bench.     

Controlled synthesis of amphiphilic biodegradable polylactide‐grafted dextran copolymers

The whole controlled synthesis of novel amphiphilic polylactide (PLA)‐grafted dextran copolymers was achieved. The control of the architecture of such biodegradable and potentially biocompatible copolymers has required a three‐step synthesis based on the “grafting from” concept. The first step consisted of the partial silylation of the dextran hydroxyl groups. This protection step was followed by the ring‐opening polymerization of D ,L ‐lactide initiated from the remaining OH functions of the partially silylated polysaccharide. The third step involved the silylether group deprotection under very mild conditions. Based on previous studies, in which the control of the first step was achieved, this study is focused on the last two steps. Experimental conditions were investigated to ensure a controlled polymerization of D ,L ‐lactide, in terms of grafting efficiency, graft length, and transesterification limitation. After polymerization, the final step was studied in order to avoid degradation of both polysaccharide backbone and polyester grafts. The chemical stability of dextran backbone was checked throughout each step of the synthesis. PLA‐grafted dextrans and PLA‐grafted (silylated dextrans) were proved to adopt a core‐shell conformation in various solvents. Furthermore, preliminary experiments on the potential use of these amphiphilic grafted copolymers as liquid/liquid interface stabilizers were performed. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2577–2588, 2004     

Unreasonable effectiveness of symmetry in physics

We prove that in some reasonable sense, every possible physical law can be reformulated in terms of symmetries. This result explains the well-known success of the group-theoretic approach in physics.     

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     

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