La distribution des zéros des fonctions L

Selon la philosophie de Katz et Sarnak, la distribution des zéros des fonctions $L$ est prédite par le comportement des valeurs propres de matrices aléatoires. En particulier, le comportement des zéros près du point central révèle le type de symétrie de la famille de fonctions $L$. Une fois la symétrie identifiée, la philosophie de Katz et Sarnak conjecture que plusieurs statistiques associées aux zéros seront modélisées par les valeurs propres de matrices aléatoires du groupe correspondant. Ce mémoire étudiera la distribution des zéros près du point central de la famille des courbes elliptiques sur $\mathbb{Q}[i]$. Brumer a effectué ces calculs en 1992 sur la famille de courbes elliptiques sur $\mathbb{Q}$. Les nouvelles problématiques reliées à la généralisation de ses travaux vers un corps de nombres seront mises en évidence

Integrability of a linear center perturbed by a fourth degree homogeneous polynomial

In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.

Integrability of a linear center perturbed by a fifth degree homogeneous polynomial

In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.

Laguerre polynomials as Jensen polynomials of Laguerre-Polya entire functions

We prove that the only Jensen polynomials associated with an entire function in the Laguerre-Polya class that are orthogonal are the Laguerre polynomials. (C) 2009 Elsevier B.V. All rights reserved.

Combining Legendre's Polynomials and Genetic Algorithm in the Solution of Nonlinear Initial-Value Problems

This work develops a method for solving ordinary differential equations, that is, initial-value problems, with solutions approximated by using Legendre's polynomials. An iterative procedure for the adjustment of the polynomial coefficients is developed, based on the genetic algorithm. This procedure is applied to several examples providing comparisons between its results and the best polynomial fitting when numerical solutions by the traditional Runge-Kutta or Adams methods are available. The resulting algorithm provides reliable solutions even if the numerical solutions are not available, that is, when the mass matrix is singular or the equation produces unstable running processes.

Ehrhart polynomials, successive minima, and an Ehrhart theory for rational dilates of a rational polytope

Magdeburg, Univ., Fak. für Mathematik, Diss., 2011

Dominance of positivity of the Green's function associated to a perturbed polyharmonic dirichlet boundary value problem by pointwise estimates

Magdeburg, Univ., Fak. für Mathematik, Diss., 2015

Exponentially small splitting of separatrices in the perturbed McMillan map

The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coefficients involved in the expansion have a resurgent origin, as shown in [MSS08].

Nonpersistence of resonant caustics in perturbed elliptic billiards

Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics —the ones whose tangent trajectories are closed polygons— are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.

Development of spatial integration depends on top-down and interhemispheric connections that can be perturbed in migraine: a DCM analysis.

In humans, spatial integration develops slowly, continuing through childhood into adolescence. On the assumption that this protracted course depends on the formation of networks with slowly developing top-down connections, we compared effective connectivity in the visual cortex between 13 children (age 7-13) and 14 adults (age 21-42) using a passive perceptual task. The subjects were scanned while viewing bilateral gratings, which either obeyed Gestalt grouping rules [colinear gratings (CG)] or violated them [non-colinear gratings (NG)]. The regions of interest for dynamic causal modeling were determined from activations in functional MRI contrasts stimuli > background and CG > NG. They were symmetrically located in V1 and V3v areas of both hemispheres. We studied a common model, which contained reciprocal intrinsic and modulatory connections between these regions. An analysis of effective connectivity showed that top-down modulatory effects generated at an extrastriate level and interhemispheric modulatory effects between primary visual areas (all inhibitory) are significantly weaker in children than in adults, suggesting that the formation of feedback and interhemispheric effective connections continues into adolescence. These results are consistent with a model in which spatial integration at an extrastriate level results in top-down messages to the primary visual areas, where they are supplemented by lateral (interhemispheric) messages, making perceptual encoding more efficient and less redundant. Abnormal formation of top-down inhibitory connections can lead to the reduction of habituation observed in migraine patients.

Network-guided analysis of genes with altered somatic copy number and gene expression reveals pathways commonly perturbed in metastatic melanoma.

Cancer genomes frequently contain somatic copy number alterations (SCNA) that can significantly perturb the expression level of affected genes and thus disrupt pathways controlling normal growth. In melanoma, many studies have focussed on the copy number and gene expression levels of the BRAF, PTEN and MITF genes, but little has been done to identify new genes using these parameters at the genome-wide scale. Using karyotyping, SNP and CGH arrays, and RNA-seq, we have identified SCNA affecting gene expression ('SCNA-genes') in seven human metastatic melanoma cell lines. We showed that the combination of these techniques is useful to identify candidate genes potentially involved in tumorigenesis. Since few of these alterations were recurrent across our samples, we used a protein network-guided approach to determine whether any pathways were enriched in SCNA-genes in one or more samples. From this unbiased genome-wide analysis, we identified 28 significantly enriched pathway modules. Comparison with two large, independent melanoma SCNA datasets showed less than 10% overlap at the individual gene level, but network-guided analysis revealed 66% shared pathways, including all but three of the pathways identified in our data. Frequently altered pathways included WNT, cadherin signalling, angiogenesis and melanogenesis. Additionally, our results emphasize the potential of the EPHA3 and FRS2 gene products, involved in angiogenesis and migration, as possible therapeutic targets in melanoma. Our study demonstrates the utility of network-guided approaches, for both large and small datasets, to identify pathways recurrently perturbed in cancer.

Deletion of the RNA-binding proteins ZFP36L1 and ZFP36L2 leads to perturbed thymic development and T lymphoblastic leukemia.

ZFP36L1 and ZFP36L2 are RNA-binding proteins (RBPs) that interact with AU-rich elements in the 3' untranslated region of mRNA, which leads to mRNA degradation and translational repression. Here we show that mice that lacked ZFP36L1 and ZFP36L2 during thymopoiesis developed a T cell acute lymphoblastic leukemia (T-ALL) dependent on the oncogenic transcription factor Notch1. Before the onset of T-ALL, thymic development was perturbed, with accumulation of cells that had passed through the beta-selection checkpoint without first expressing the T cell antigen receptor beta-chain (TCRbeta). Notch1 expression was higher in untransformed thymocytes in the absence of ZFP36L1 and ZFP36L2. Both RBPs interacted with evolutionarily conserved AU-rich elements in the 3' untranslated region of Notch1 and suppressed its expression. Our data establish a role for ZFP36L1 and ZFP36L2 during thymocyte development and in the prevention of malignant transformation.

Random regression models to estimate genetic parameters for milk production of Guzerat cows using orthogonal Legendre polynomials

The objective of this work was to compare random regression models for the estimation of genetic parameters for Guzerat milk production, using orthogonal Legendre polynomials. Records (20,524) of test-day milk yield (TDMY) from 2,816 first-lactation Guzerat cows were used. TDMY grouped into 10-monthly classes were analyzed for additive genetic effect and for environmental and residual permanent effects (random effects), whereas the contemporary group, calving age (linear and quadratic effects) and mean lactation curve were analized as fixed effects. Trajectories for the additive genetic and permanent environmental effects were modeled by means of a covariance function employing orthogonal Legendre polynomials ranging from the second to the fifth order. Residual variances were considered in one, four, six, or ten variance classes. The best model had six residual variance classes. The heritability estimates for the TDMY records varied from 0.19 to 0.32. The random regression model that used a second-order Legendre polynomial for the additive genetic effect, and a fifth-order polynomial for the permanent environmental effect is adequate for comparison by the main employed criteria. The model with a second-order Legendre polynomial for the additive genetic effect, and that with a fourth-order for the permanent environmental effect could also be employed in these analyses.