La condensation de la matrice masse,ou mass-lumping,pour les éléments finis mixtes de Raviart–Thomas–Nédélec d'ordre 1

. We present practical means to obtain a positive diagonal matrix for the first order Raviart–Thomas–Nédélec finite elements. Applying it to the second order Maxwell equations allows us to use explicit time schemes to solve the resulting differential system. To cite this article: P. Lacoste, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Knowing and Teaching Middle School Mathematics: A Professional Development Course for In‐Service Teachers

This article describes a professional development course intended to improve the content understanding of middle school mathematics teachers. The design of the course included three professional learning strategies: problem solving, examination of student thinking, and discussion of research. The concepts studied in the course included multi‐digit subtraction, multi‐digit multiplication, operations with fractions, and concepts of area and perimeter. Results from pre‐ and post‐tests administered to the nineteen participants indicate a significant increase in the mean score for each concept and document growth in the teachers' content understanding. In particular, their solutions moved from primarily procedural to more conceptual. Responses to an open‐ended survey indicate other important aspects of the professional development. Examples of teachers' work and comments are included.     

Elliptic divisibility sequences over certain curves

Let n?5 be an integer. We provide an effective method for finding all elliptic curves in short Weierstrass form E/Q with j(E)∈{0,1728} and all P∈E(Q) such that the nth term in the elliptic divisibility sequence defined by P over E fails to have a primitive divisor. In particular, we improve recent results of Everest, Mclaren, and Ward on the Zsigmondy bounds of elliptic divisibility sequences associated with congruent number curves.     

Spinors, Inflation, and Non-Singular Cyclic Cosmologies

We consider toy cosmological models in which a classical, homogeneous, spinor field provides a dominant or sub-dominant contribution to the energy-momentum tensor of a flat Friedmann-Robertson-Walker universe. We find that, if such a field were to exist, appropriate choices of the spinor self-interaction would generate a rich variety of behaviors, quite different from their widely studied scalar field counterparts. We first discuss solutions that incorporate a stage of cosmic inflation and estimate the primordial spectrum of density perturbations seeded during such a stage. Inflation driven by a spinor field turns out to be unappealing as it leads to a blue spectrum of perturbations and requires considerable fine-tuning of parameters. We next find that, for simple, quartic spinor self-interactions, non-singular cyclic cosmologies exist with reasonable parameter choices. These solutions might eventually be incorporated into a successful past- and future-eternal cosmological model free of singularities. In an Appendix, we discuss the classical treatment of spinors and argue that certain quantum systems might be approximated in terms of such fields.     

Combinatorics of normal sequences of braids

Many natural counting problems arise in connection with the normal form of braids—and seem to have not been much considered so far. Here we solve some of them. One of the noteworthy points is that a number of different induction schemes appear. The key technical ingredient is an analysis of the normality condition in terms of permutations and their descents, in the vein of the Solomon algebra. As was perfectly summarized by a referee, the main result asserts that the size of the automaton involved in the automatic structure of Bn associated with the normal form can be lowered from n! to p(n), the number of partitions of n.     

On regularity of a weak solution to the Navier–Stokes equation with generalized impermeability boundary conditions

We consider the 3D Navier–Stokes equation with generalized impermeability boundary conditions. As auxiliary results, we prove the local in time existence of a strong solution (‘strong’ in a limited sense) and a theorem on structure. Then, taking advantage of the boundary conditions, we formulate sufficient conditions for regularity up to the boundary of a weak solution by means of requirements on one of the eigenvalues of the rate of deformation tensor. Finally, we apply these general results to the case of an axially symmetric flow with zero angular velocity.     

Calculation of number and free energy of the conformers of linear alkanes with medium and long chains. Implications for catalysis

A method for calculating the number of rotamers of a linear alkane and of the number of rotamers with a given number of gauche conformations along the chain as a function of the total number of atoms in the chain, using general equations, is presented. A graphical method for generating individual rotamers was applied to the homologs up to decane, which has 1134 rotamers. The steric energies calculated by molecular mechanics (MM2 force field) were used as measures of the heat of formation for the coiled conformations relative to the anti conformer for each molecule, whereas the statistical entropy differences were calculated for classes of coiled rotamers grouped by the number of gauche bonds and steric energy. The free energy values calculated from these components show that already at 400 K hexane exists preferentially in conformations containing gauche bonds. For larger chains the free energy advantage for the coiled chains increases very steeply. The implications for the question of reactions of linear alkanes occurring on the surface or inside the channels of small- and medium-pore zeolites are briefly examined.     

Baer orderings with noninvariant valuation ring

We construct division algebras with involution containing a Baer ordering with noninvariant order ring. This gives a negative answer to a question of Holland, whether the order ring is always invariant under inner automorphisms. Furthermore, we give examples of any index. Previously, the only known examples of division algebras containing Baer orderings were of index 2 ⁿ or of indexp forp a prime of the form 4m + 3. Supported in part by the National Science Foundation.     

Straightforward conversion of alcohols into dibenzenesulfonimides

The reaction of various alcohols with N-fluorodibenzenesulfonimide and triphenylphosphine leads to the corresponding dibenzenesulfonylated amines in high yields.     

Self-organized ZnAl2O4 nanostructures grown on -sapphire

Self-organized ZnAl₂O₄ nanostructures with the appearance (in SEM) of high aspect ratio horizontal nanowires are grown on uncatalysed c-sapphire by vapour phase transport. The nanostructures grow as three equivalent crystallographic variants on c-sapphire. Raman and cathodoluminescence spectroscopy confirm that the nanostructures are not ZnO and TEM shows that they are the cubic spinel, zinc aluminate, ZnAl₂O₄, formed by the reaction of Zn and O with the sapphire substrate.