Unfolded singularities of analytic differential equations

La thèse est composée d’un chapitre de préliminaires et de deux articles sur le sujet du déploiement de singularités d’équations différentielles ordinaires analytiques dans le plan complexe. L’article Analytic classification of families of linear differential systems unfolding a resonant irregular singularity traite le problème de l’équivalence analytique de familles paramétriques de systèmes linéaires en dimension 2 qui déploient une singularité résonante générique de rang de Poincaré 1 dont la matrice principale est composée d’un seul bloc de Jordan. La question: quand deux telles familles sontelles équivalentes au moyen d’un changement analytique de coordonnées au voisinage d’une singularité? est complètement résolue et l’espace des modules des classes d’équivalence analytiques est décrit en termes d’un ensemble d’invariants formels et d’un invariant analytique, obtenu à partir de la trace de la monodromie. Des déploiements universels sont donnés pour toutes ces singularités. Dans l’article Confluence of singularities of non-linear differential equations via Borel–Laplace transformations on cherche des solutions bornées de systèmes paramétriques des équations non-linéaires de la variété centre de dimension 1 d’une singularité col-noeud déployée dans une famille de champs vectoriels complexes. En général, un système d’ÉDO analytiques avec une singularité double possède une unique solution formelle divergente au voisinage de la singularité, à laquelle on peut associer des vraies solutions sur certains secteurs dans le plan complexe en utilisant les transformations de Borel–Laplace. L’article montre comment généraliser cette méthode et déployer les solutions sectorielles. On construit des solutions de systèmes paramétriques, avec deux singularités régulières déployant une singularité irrégulière double, qui sont bornées sur des domaines «spirals» attachés aux deux points singuliers, et qui, à la limite, convergent vers une paire de solutions sectorielles couvrant un voisinage de la singularité confluente. La méthode apporte une description unifiée pour toutes les valeurs du paramètre.

Perturbative expansion of the QCD Adler function improved by renormalization-group summation and analytic continuation in the Borel plane

We examine the large-order behavior of a recently proposed renormalization-group-improved expansion of the Adler function in perturbative QCD, which sums in an analytically closed form the leading logarithms accessible from renormalization-group invariance. The expansion is first written as an effective series in powers of the one-loop coupling, and its leading singularities in the Borel plane are shown to be identical to those of the standard ``contour-improved'' expansion. Applying the technique of conformal mappings for the analytic continuation in the Borel plane, we define a class of improved expansions, which implement both the renormalization-group invariance and the knowledge about the large-order behavior of the series. Detailed numerical studies of specific models for the Adler function indicate that the new expansions have remarkable convergence properties up to high orders. Using these expansions for the determination of the strong coupling from the hadronic width of the tau lepton we obtain, with a conservative estimate of the uncertainty due to the nonperturbative corrections, alpha(s)(M-tau(2)) = 0.3189(-0.0151)(+0.0173), which translates to alpha(s)(M-Z(2)) = 0.1184(-0.0018)(+0.0021). DOI: 10.1103/PhysRevD.87.014008

Smooth sets for borel equivalence relations and the covering property for σ-ideals of compact sets

This thesis is divided into three chapters. In the first chapter we study the smooth sets with respect to a Borel equivalence realtion E on a Polish space X. The collection of smooth sets forms σ-ideal. We think of smooth sets as analogs of countable sets and we show that an analog of the perfect set theorem for Σ¹₁ sets holds in the context of smooth sets. We also show that the collection of Σ¹₁ smooth sets is ∏¹₁ on the codes. The analogs of thin sets are called sparse sets. We prove that there is a largest ∏¹₁ sparse set and we give a characterization of it. We show that in L there is a ∏¹₁ sparse set which is not smooth. These results are analogs of the results known for the ideal of countable sets, but it remains open to determine if large cardinal axioms imply that ∏¹₁ sparse sets are smooth. Some more specific results are proved for the case of a countable Borel equivalence relation. We also study I(E), the σ-ideal of closed E-smooth sets. Among other things we prove that E is smooth iff I(E) is Borel.

In chapter 2 we study σ-ideals of compact sets. We are interested in the relationship between some descriptive set theoretic properties like thinness, strong calibration and the covering property. We also study products of σ-ideals from the same point of view. In chapter 3 we show that if a σ-ideal I has the covering property (which is an abstract version of the perfect set theorem for Σ¹₁ sets), then there is a largest ∏¹₁ set in I^int (i.e., every closed subset of it is in I). For σ-ideals on 2^ω we present a characterization of this set in a similar way as for C₁, the largest thin ∏¹₁ set. As a corollary we get that if there are only countable many reals in L, then the covering property holds for Σ¹₂ sets.

Consumo de água e tarifa social em áreas de baixa renda: estudo de caso das comunidades de Santa Marta, Complexo do Borel/Casa Branca e Complexo da Mangueira.

O município do Rio de Janeiro, desde sua fundação luta com o grave problema de abastecimento de água. Com o passar do tempo todas as pequenas captações no entorno da cidade foram exauridas tanto em termos de quantidade como em qualidade, afetadas pela poluição decorrente do lançamento de efluentes in natura. Solução foi a busca por novas fontes de abastecimento em outros municípios. Atualmente 80% (oitenta por cento) do abastecimento da região metropolitana do Rio de Janeiro, RMRJ, é proveniente de uma única fonte: o rio Guandu. Isto foi possível devido à transposição da bacia do rio Paraíba do Sul para o rio Guandu ocorrida na década de 50. Esta fonte essencial de abastecimento, onde foi construída a maior Estação de Tratamento de Água do mundo - ETA Guandu - está à beira da exaustão. Somente uma pequena parcela de água bruta foi captada e tratada nesta última década. Visando minimizar o grave problema que já se apresenta e enquanto investimentos em novas alternativas não forem alcançados, o consumo com responsabilidade e sustentabilidade passa a ser a tônica da discussão. Neste contexto, as áreas de baixa renda do município com suas 801 favelas e mais de 1.500 loteamentos irregulares representando em 2010 aproximadamente 1/3 (um terço) da população total, consumindo de 10% a 15% de toda a produção de água tratada da região metropolitana deve ser permanentemente estimulada a contribuir com esta redução de consumo. Estudos apresentados nesta dissertação em três metodologias distintas apontam para um consumo acima da média nacional deste mesmo perfil de população. Os resultados obtidos indicam a necessidade real da redução de consumo observando, entretanto que este é um trabalho extremamente árduo e difícil uma vez que exige mudança de hábitos e a envolvimento de todos, desde a população até a Companhia de Saneamento local.

Catalogue d'une importante collection de vignettes par ou d'après Borel, Choffard, Eisen, Gravelot, Lebarbier, Marillier, Monnet, Moreau le jeune, Queverdo et autres, Portraits des XVIe, XVIIe et XVIIIe siècles, Estampes, dessins des XVIIIe et XIXe siècles, autographes, suites de figures, composant la collection de Mr H. de G. La vente aura lieu du lundi 27 février au mercredi 8 mars 1893... 28 rue des Bons-Enfants... / Me Maurice Delestre, commissaire-priseur ; [expert] M. Ch. Porquet, libraire, M. P. Roblin, marchand d'estampes

[Vente. Art. 1893-02-27 - 1893-03-08. Paris]