Asymptotic Freeness for Rectangular Random Matrices and Large Deviations for Sample Covariance Matrices With Sub-Gaussian Tails

We establish a large deviation principle for the empirical spectral measure of a sample covariance matrix with sub-Gaussian entries, which extends Bordenave and Caputo's result for Wigner matrices having the same type of entries [7]. To this aim, we need to establish an asymptotic freeness result for rectangular free convolution, more precisely, we give a bound in the subordination formula for information-plus-noise matrices

Convergence to Equilibrium in the Free Fokker-Planck Equation With a Double-Well Potential

We consider the one-dimensional free Fokker-Planck equation $\frac{\partial \mu\_t}{\partial t} = \frac{\partial}{\partial x} \left[ \mu\_t \left( \frac12 V' - H\mu\_t \right) \right]$, where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x) = \frac14 x^4 + \frac{c}{2} x^2$, with $-2 \le c < 0$. We prove that the solution $(\mu\_t)\_{t \ge 0}$ of this PDE converges to the equilibrium measure $\mu\_V$ as $t$ goes to infinity, which provides a first result of convergence in a non-convex setting. The proof involves free probability and complex analysis techniques

Renal cell carcinoma induces interleukin 10 and prostaglandin E2 production by monocytes

Immunotherapy with interleukin 2 (IL-2) is not an effective anti-cancer treatment in the majority of patients with renal cell carcinoma (RCC), suggesting that the activation of cytotoxic T cells or NK cells may be impaired in vivo in these patients. The production of immunosuppressive factors by RCC was investigated. Using immunohistochemistry, IL-10 was detectable in 10 of 21 tumour samples tested. IL-10 was undetectable in the supernatant of cell lines derived from these RCCs. However, these cell lines or their conditioned medium (RCC CM), but not normal renal epithelial cells adjacent to the RCC or breastcarcinoma cell lines, were found to induce IL-10, as well as prostaglandin E2 (PGE2) and tumour necrosis factor (TNF)α production by autologous or allogeneic peripheral blood mononuclear cells (PBMCs) and monocytes. IL-10 production induced by RCC CM was found to be dependent on TNF-α and PGE2 since an anti-TNF-α antibody (Ab) inhibited 40–70% of IL-10 production by monocytes, and the combination of anti-TNF-α Ab and indomethacin, an inhibitor of PGE2 production, inhibited 80–94% of RCC CM-induced IL-10 production by monocytes. The RCC CM of the five cell lines tested were found to induce a down-regulation of the expression of HLA-DR and CD86, as well as a strong inhibition of mannose receptor-dependent endocytosis by monocytes. The blockade of HLA-DR and CD86 expression was partially abrogated by indomethacin and anti-IL-10 Ab respectively, and completely abrogated by an anti-TNF-α Ab. The inhibition of mannose receptor-dependent endocytosis was partially abrogated by an anti-IL-10 Ab and completely abrogated by an anti-TNF-α Ab. These esults indicate that RCCs induce IL-10, PGE2 and TNF-α production by monocytes, which down-regulate the expression of cell-surface molecules involved in antigen presentation as well as their endocytic capacity. © 1999 Cancer Research Campaig

The SIB Swiss Institute of Bioinformatics' resources: focus on curated databases

The SIB Swiss Institute of Bioinformatics (www.isb-sib.ch) provides world-class bioinformatics databases, software tools, services and training to the international life science community in academia and industry. These solutions allow life scientists to turn the exponentially growing amount of data into knowledge. Here, we provide an overview of SIB's resources and competence areas, with a strong focus on curated databases and SIB's most popular and widely used resources. In particular, SIB's Bioinformatics resource portal ExPASy features over 150 resources, including UniProtKB/Swiss-Prot, ENZYME, PROSITE, neXtProt, STRING, UniCarbKB, SugarBindDB, SwissRegulon, EPD, arrayMap, Bgee, SWISS-MODEL Repository, OMA, OrthoDB and other databases, which are briefly described in this article

Large deviations of random matrices and free Fokker-Planck equation

Cette thèse s'inscrit dans le domaine des probabilités et des statistiques, et plus précisément des matrices aléatoires. Dans la première partie, on étudie les grandes déviations de la mesure spectrale de matrices de covariance XX*, où X est une matrice aléatoire rectangulaire à coefficients i.i.d. ayant une queue de probabilité en exp(-at^α), α∈]0,2[. On établit un principe de grandes déviations analogue à celui de Bordenave et Caputo, de vitesse n^{1+α/2} et de fonction de taux explicite faisant intervenir la convolution libre rectangulaire. La démonstration repose sur un résultat de quantification de la liberté asymptotique dans le modèle information-plus-bruit. La seconde partie de cette thèse est consacrée à l'étude du comportement en temps long de la solution de l'équation de Fokker-Planck libre en présence du potentiel quartique V(x) = 1/4 x^4 + c/2 x² avec c≥-2. On montre que quand t→+∞, la solution µ_t de cette équation aux dérivées partielles converge en distance de Wasserstein vers la mesure d'équilibre associée au potentiel V. Ce résultat fournit un premier exemple de convergence en temps long de la solution de l'équation des milieux granulaires en présence d'un potentiel non convexe et d'une interaction logarithmique. Sa démonstration utilise notamment des techniques de probabilités libres.This thesis lies within the field of probability and statistics, and more precisely of random matrix theory. In the first part, we study the large deviations of the spectral measure of covariance matrices XX*, where X is a rectangular random matrix with i.i.d. coefficients having a probability tail like exp(-at^α), α∈]0,2[. We establish a large deviation principle similar to Bordenave and Caputo's one, with speed n^{1+α/2} and explicit rate function involving rectangular free convolution. The proof relies on a quantification result of asymptotic freeness in the information-plus-noise model. The second part of this thesis is devoted to the study of the long-time behaviour of the solution to free Fokker-Planck equation in the setting of the quartic potential V(x) = 1/4 x^4 + c/2 x² with c≥-2. We prove that when t→+∞, the solution µ_t to this partial differential equation converge in Wasserstein distance towards the equilibrium measure associated to the potential V. This result provides a first example of long-time convergence for the solution of granular media equation with a non-convex potential and a logarithmic interaction. Its proof involves in particular free probability techniques

Grandes d´eviations de matrices aléatoires et équation de Fokker-Planck libre

This thesis lies within the field of probability and statistics, and more precisely of random matrix theory. In the first part, we study the large deviations of the spectral measure of covariance matrices XX*, where X is a rectangular random matrix with i.i.d. coefficients having a probability tail like exp(-at^α), α∈]0,2[. We establish a large deviation principle similar to Bordenave and Caputo's one, with speed n^{1+α/2} and explicit rate function involving rectangular free convolution. The proof relies on a quantification result of asymptotic freeness in the information-plus-noise model. The second part of this thesis is devoted to the study of the long-time behaviour of the solution to free Fokker-Planck equation in the setting of the quartic potential V(x) = 1/4 x^4 + c/2 x² with c≥-2. We prove that when t→+∞, the solution µ_t to this partial differential equation converge in Wasserstein distance towards the equilibrium measure associated to the potential V. This result provides a first example of long-time convergence for the solution of granular media equation with a non-convex potential and a logarithmic interaction. Its proof involves in particular free probability techniques.Cette thèse s'inscrit dans le domaine des probabilités et des statistiques, et plus précisément des matrices aléatoires. Dans la première partie, on étudie les grandes déviations de la mesure spectrale de matrices de covariance XX*, où X est une matrice aléatoire rectangulaire à coefficients i.i.d. ayant une queue de probabilité en exp(-at^α), α∈]0,2[. On établit un principe de grandes déviations analogue à celui de Bordenave et Caputo, de vitesse n^{1+α/2} et de fonction de taux explicite faisant intervenir la convolution libre rectangulaire. La démonstration repose sur un résultat de quantification de la liberté asymptotique dans le modèle information-plus-bruit. La seconde partie de cette thèse est consacrée à l'étude du comportement en temps long de la solution de l'équation de Fokker-Planck libre en présence du potentiel quartique V(x) = 1/4 x^4 + c/2 x² avec c≥-2. On montre que quand t→+∞, la solution µ_t de cette équation aux dérivées partielles converge en distance de Wasserstein vers la mesure d'équilibre associée au potentiel V. Ce résultat fournit un premier exemple de convergence en temps long de la solution de l'équation des milieux granulaires en présence d'un potentiel non convexe et d'une interaction logarithmique. Sa démonstration utilise notamment des techniques de probabilités libres