To easily generalize the maximum-principle-satisfying schemes for scalar conservation laws to convection diffusion equations, we propose a non-conventional high order finite volume weighted essentially non-oscillatory (WENO) scheme which can be proven maximum-principle-satisfying. Two-dimensional extensions are straightforward. We also show that the same idea can be used to construct high order schemes preserving the maximum principle for two-dimensional incompressible Navier-Stokes equations in the vorticity stream-function formulation. Numerical tests for the fifth order WENO schemes are reported.

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Using probabilistic methods, we prove new rigidity results for groups and pseudo-groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class (i.e. between$C$^{1}and$C$^{2}). In particular, we show some generalizations of Denjoy theorem and the classical Kopell lemma for abelian groups. These techniques are also applied to the study of codimension-1 foliations. For instance, we obtain several generalized versions of Sacksteder theorem in class$C$^{1}. We conclude with some remarks about the stationary measure.

Given a domain Ω⊂ℂ^{$n$}, the Lempert function is a functional on the space $\text{Hol}(\mathbb{D},\Omega)$ of analytic disks with values in Ω, depending on a set of poles in Ω. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the local indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.