A natural Finsler--Laplace operator

We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--Ziller metrics that this Finsler--Laplace operator admits explicit representations and computations of spectral data.Comment: 25 pages, v2: minor modifications, changed the introductio

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

On "Dotsenko-Fateev" representation of the toric conformal blocks

We demonstrate that the recent ansatz of arXiv:1009.5553, inspired by the original remark due to R.Dijkgraaf and C.Vafa, reproduces the toric conformal blocks in the same sense that the spherical blocks are given by the integral representation of arXiv:1001.0563 with a peculiar choice of open integration contours for screening insertions. In other words, we provide some evidence that the toric conformal blocks are reproduced by appropriate beta-ensembles not only in the large-N limit, but also at finite N. The check is explicitly performed at the first two levels for the 1-point toric functions. Generalizations to higher genera are briefly discussed.Comment: 10 page

Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two

We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover).Comment: 32 pages, one .eps figure. The version v1 has a misprint in Theorem 1: I forgot to write the assumption that the degree of mobility is greater than two. The versions v3, v4 have only cosmetic changes wrt v

Static and dynamic polystyrene dimensions in relation to the blob model and size-exclusion chromatography

Dynamic measurements (sedimentation, viscosity, size exclusion chromatography) were carried out on a series of polystyrene samples dissolved in a good solvent, tetrahydrofuran. Using the Weill-des Cloizeaux theory we show that it is possible, from these data, to study how the coil radius of gyration varies with molecular weight. The validity of the thermal-blob model is confirmed by our results, and the expected power laws under and above the cross-over domain are indeed observed. In size-exclusion chromatography, we observe that several calibration curves may be obtained from the experimental data.Nous avons mesuré certains paramètres de transport (sédimentation, viscosité, chromatographie d'exclusion stérique) pour une série de polystyrènes linéaires en solution dans un bon solvant : le tétrahydrofuranne. Partant de la théorie récente de Weill-des Cloizeaux, nous montrons qu'il est possible d'étudier, à partir de nos résultats expérimentaux, les variations du rayon de giration RG avec la masse du polymère. Le bien-fondé de l'hypothèse du modèle de blob thermique est établi, et nous retrouvons les lois de puissance prévues théoriquement de part et d'autre du domaine de transition (ou cross-over spatial). En chromatographie d'exclusion stérique, nous montrons qu'il est possible d'établir plusieurs lois d'étalonnage à partir de nos données, sans qu'il soit possible d'affirmer laquelle est la plus « universelle »

Etude de la gélification par couplage GPC - Diffusion de lumière.

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