The uniqueness of solutions of perturbed backward stochastic differential equations

We prove a result on the preservation of the pathwise uniqueness property for the adapted solution to backward stochastic differential equation under perturbations.     

The length of harmonic forms on a compact Riemannian manifold

We study -dimensional Riemannian manifolds with harmonic forms of constant length and first Betti number equal to showing that they are -step nilmanifolds with some special metrics. We also characterize, in terms of properties on the product of harmonic forms, the left-invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.

     

Global solutions for quasilinear parabolic problems

Results on the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Dirichlet or Neumann boundary conditions are presented. Besides quasilinear parabolic equations, the method is also applicable to some weakly-coupled reaction-diffusion systems and to elliptic equations with nonlinear dynamic boundary conditions. Received December 21, 2000; accepted August 30, 2001.     

Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve

Let G be a semi-simple group and M the moduli stack of G-bundles over a smooth, complex, projective curve. Using representation-theoretic methods, I prove the pure-dimensionality of sheaf cohomology for certain “evaluation vector bundles” over M, twisted by powers of the fundamental line bundle. This result is used to prove a Borel-Weil-Bott theorem, conjectured by G. Segal, for certain generalized flag varieties of loop groups. Along the way, the homotopy type of the group of algebraic maps from an affine curve to G, and the homotopy type, the Hodge theory and the Picard group of M are described. One auxiliary result, in Appendix A, is the Alexander cohomology theorem conjectured in [Gro2]. A self-contained account of the “uniformization theorem” of [LS] for the stack M is given, including a proof of a key result of Drinfeld and Simpson (in characteristic 0). A basic survey of the simplicial theory of stacks is outlined in Appendix B. Oblatum 17-XII-1996 & 26 VI-1997     

Singular nature of nonlinear macroscale effects in high-rate flow through porous mediaNature singulière des effets macroscopiques non linéaires dans un écoulement à hautes vitesses en milieux poreux

The quadratic law of laminar flow through porous media at high Reynolds numbers, which is well confirmed by the multiple experimental data, is shown to give rise to three fundamental paradoxes. All them can be resolved by assuming the singular structure of flow. The singularity is produced by the formation of jet brunches which invade the stagnant zones and sharply loss their kinetic energy. The numerical simulation confirms this effect. To cite this article: M. Panfilov et al., C. R. Mecanique 331 (2003).     

Shaping of intense ion beams into hollow cylindrical form

A specifically tailored plasma lens could shape a high-energy, heavy-ion beam into the form of a hollow cylinder without loss of beam intensity. It has been experimentally confirmed that both a positive as well as a negative radial gradient of the current density in the active plasma lens can be the underlying principle. Calculations were performed that yield the ideal current density distribution for both cases. A numerical simulation of an experiment with an intense ion beam highlights that the shaping of the beam increases the achievable compression in a lead sample.     

Infinite Prandtl Number Convection

We prove an inequality of the type NCR ^1/3(1+log₊ R)^2/3 for the Nusselt number N in terms of the Rayleigh number R for the equations describing three-dimensional Rayleigh–Bénard convection in the limit of infinite Prandtl number.     

Planar Dielectrochromatography on Non-Wetted Thin Layers

JPC – Journal of Planar Chromatography – Modern TLC - Some of the physical properties of dielectric liquids are influenced by electric fields. Interesting results were obtained in this...     

External Homogenization for Hopf Algebras: Applications to Maschke's Theorem

We apply to Hopf algebras a construction from graded rings, called the group ring of a graded ring. By using this construction we study the transfer of properties between certain categories of relative Hopf modules. As another application, we obtain a Maschke-type theorem for a Galois extension over a semisimple Hopf algebra.