Mod p classification of Shimura F ‐crystals

Let k be an algebraically closed field of positive characteristic p. We first classify the D ‐truncations mod p of Shimura F ‐crystals over k and then we study stratifications defined by inner isomorphism classes of these D ‐truncations. This generalizes previous works of Kraft, Ekedahl, Oort, Moonen, and Wedhorn. As a main tool we introduce and study Bruhat F ‐decompositions; they generalize the combined form of Steinberg theorem and of classical Bruhat decompositions for reductive groups over k (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Attention to meaning by algebra teachers

Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students’ treatment of symbols will be largely non-referential.     

Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems

We suggest sufficient conditions for the existence of weak limits of solutions of the Liouville equation as time increases indefinitely. The presence of the weak limit of the probability distribution density leads to a new interpretation of the second law of thermodynamics for entropy increase.     

Spectral Semi-discretisations of Weakly Non-linear Wave Equations over Long Times

The long-time behaviour of spectral semi-discretisations of weakly non-linear wave equations is analysed. It is shown that the harmonic actions are approximately conserved for the semi-discretised system as well. This permits to prove that the energy of the wave equation along the interpolated semi-discrete solution remains well conserved over long times and close to the Hamiltonian of the semi-discrete equation. Although the momentum is no longer an exact invariant of the semi-discretisation, it is shown to be approximately conserved. All these results are obtained with the technique of modulated Fourier expansions. Dedicated to Professor Arieh Iserles on the Occasion of his Sixtieth Birthday.     

Kolmogorov’s 1954 paper on nearly-integrable Hamiltonian systems

Following closely Kolmogorov’s original paper [1], we give a complete proof of his celebrated Theorem on perturbations of integrable Hamiltonian systems by including few “straightforward” estimates.      

Gibbs ensembles,equidistribution of the energy of sympathetic oscillators and statistical models of thermostat

The paper develops an approach to the proof of the “zeroth” law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the mean energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.      

Dynamical systems — Past and present

Plenary lecture at the International Congress of Mathematicians (Berlin 1998, August 18–27). Reprinted from Doc. Math. J., DMV, Extra Volume ICM I, 1998, pp. 381–402. ©J.Moser, 1998.     

Global properties of integrable Hamiltonian systems

This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems. We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. Our approach, which uses simple ideas from differential geometry and algebraic topology, reveals the fundamental role of the integer affine structure on the base space of these bundles. We provide a geometric proof of the classification of Lagrangian bundles with fixed integer affine structure by their Lagrange class.      

Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations

A direct link between a Vlasov equation and the equations of motion of a rotating fluid with an effective pressure depending only on a pseudo-density is illustrated. In this direct link, the resulting fluid equations necessarily appear in flux conservative form when there are no topographical and rotational terms. In contrast, multilayer isopycnic and isentropic equations used in atmosphere and ocean dynamics, in the absence of topographical and rotational terms, cannot be brought into a conservative flux form, and, hence, cannot be derived directly from the Vlasov equations. Another route is explored, therefore: deriving the Hamiltonian formulation of the two-layer isopycnic and isentropic equations as a restriction from a Hamiltonian formulation of two decoupled Vlasov equations. The work is motivated by our search for energy-preserving or even Hamiltonian (kinetic) numerical schemes.