New formulas for decreasing rearrangements and a class of Orlicz-Lorentz spaces

Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces $M_{\varphi,w}$ and $m_{\varphi,w}$ respectively, introduced earlier in \cite{HKM} for describing the K\"othe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases ($\varphi$ is an Orlicz function and $w$ a {\it decreasing} weight). We study these $M_{\varphi,w}$ classes in the most general setting, where they may even not be linear, and identify their K\"othe duals with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces $\mathcal{M}_{\varphi,w}$ which proves to be the K\"othe biduals of the $M_{\varphi,w}$ classes. In the case when the class $M_{\varphi,w}$ is a separable quasi-Banach space, $\mathcal{M}_{\varphi,w}$ is its Banach envelope.Comment: 25 page

Convective dynamos: Symmetries and modulation

International audienc

Deligne's duality for de Rham realizations of 1-motives

We show that the pairing on de Rham realizations of 1-motives in "Theorie di Hodge III", IHES 44, can be defined over any base scheme and we prove that it gives rise to a perfect duality if one is working with a 1-motive and its Cartier dual. Furthermore we study universal extensions of 1-motives and their relation with $\natural$-extensions.Comment: 21 pages. Last section rewritten. New proof

Free-form, form finding and anisotropic grid shell

p. 966-876The new geometrical developments open new perspectives for free-form design, making it possible to escape from planar triangular or quadrilateral discretizations. Recent advances in theory algorithms allow for the discretization of any surface using only single curvature panels thus allowing the realisation of smooth double curvature glazed envelops of any form. Grid shell structures usually present a nearly in plane uniform behaviour, but previous realisations have shown that grid shells can be designed also according to an anisotropic inplane arrangement. The control of principal direction and the fine tuning of the stiffness of the different structural elements (arcs, cables etc.) is a tool for adjusting the form-finding thus controlling the resulting geometry. Moreover, the form-finding can also be performed without researching a constant stress (self weight); in this case an even wider range of forms become possible. These new geometrical and structural approaches have been coupled together and tested in re-designing, as a case study, the glazed roof of the Neumunster Abbey in Luxembourg. Such approach allowed for the conception of an efficient structure supporting a smooth double curvature glass skin, made out of only single curvature panels, perfectly coherent with the perimeter of the courtyard i.e. matching all the edges without any gaps.Baldassini, N.; Raynaud, J. (2010). Free-form, form finding and anisotropic grid shell. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/696

Convergent Numerical Schemes for the Compressible Hyperelastic Rod Wave Equation

We propose a fully discretised numerical scheme for the hyperelastic rod wave equation on the line. The convergence of the method is established. Moreover, the scheme can handle the blow-up of the derivative which naturally occurs for this equation. By using a time splitting integrator which preserves the invariants of the problem, we can also show that the scheme preserves the positivity of the energy density