Tissus plats et feuilletages homogènes sur le plan projectif complexe

Resum

Geometry of certain foliations on the complex projective plane

Let d≥2 be an integer. The set F(d) of foliations of degree d on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension d2+4d+2 on which Aut(P2C) acts. We show that there are exactly two orbits O(Fd1) and O(Fd2) of minimal dimension 6 , necessarily closed in F(d) . This generalizes known results in degrees 2 and 3. We deduce that an orbit O(F) of an element F∈F(d) of dimension 7 is closed in F(d) if and only if Fdi∉O(F)¯¯¯¯¯¯¯¯¯¯¯¯ for i=1,2. This allows us to show that in any degree d≥3 there are closed orbits in F(d) other than the orbits O(Fd1) and O(Fd2), unlike the situation in degree 2. On the other hand, we introduce the notion of the basin of attraction B(F) of a foliation F∈F(d) as the set of G∈F(d) such that F∈O(G)¯¯¯¯¯¯¯¯¯¯¯. We show that the basin of attraction B(Fd1) , resp. B(Fd2) , contains a quasi-projective subvariety of F(d) of dimension greater than or equal to dimF(d)−(d−1) , resp. dimF(d)−(d−3) . In particular, we obtain that the basin B(F32) contains a non-empty Zariski open subset of F(3) . This is an analog in degree 3 of a result on foliations of degree 2 due to Cerveau, Déserti, Garba Belko and Meziani

Convex foliations of degree 5 on the complex projective plane

We show that, up to automorphisms of P2 C, there are fourteen homogeneous convex foliations of degree 5 on P2 C. We establish some properties of the Fermat foliation Fd 0 of degree d ≥ 2 and of the Hilbert modular foliation F5 H of degree 5. As a consequence, we obtain that every reduced convex foliation of degree 5 on P2 C is linearly conjugated to one of the two foliations F5 0 or F5 H, which is a partial answer to a question posed in 2013 by D. Marín and J. V. Pereira. We end with two conjectures about the Camacho-Sad indices along the line at infinity at the non radial singularities of the homogeneous convex foliations of degree d ≥ 2 on P2C

A priori L2-error estimates for approximations of functions on compact manifolds

Given a { mathcal{C}^{2}} -function f on a compact riemannian manifold (X,g) we give a set of frequencies { L=L_{f}(varepsilon)} depending on a small parameter { varepsilon > 0} such that the relative L2-error { frac{f-f^{L} }{f}} is bounded above by { varepsilon}, where fL denotes the L-partial sum of the Fourier series f with respect to an orthonormal basis of L2(X) constituted by eigenfunctions of the Laplacian operator Δ associated to the metric g

Involuciones, trivoluciones y foliaciones Galois

En el presente trabajo introducimos la noción de foliaciones Galois sobre P2C, definidas como aquellas cuya aplicación de Gauss restringida a un abierto Zariski es un recubrimiento Galois. Asimismo, presentamos algunos ejemplos y un criterio para identificar este tipo de foliaciones

Visualizador virtual tridimensional con tecnología de iluminación de estado sólido

El visualizador virtual tridimensional es una pantalla 3D con tecnología de iluminación de estado sólido, usando tecnología integrada de sistemas embebidos, tales como los microcontroladores modernos de alta velocidad, gran capacidad de procesamiento y almacenamiento de información, a su vez compuesto por una hilera de LED‟s RGB que giran a una velocidad de 1800 rpm con una visibilidad de 360 grados, lo cual lo hace ideal para comunicaciones, publicidad, ambientación, lanzamiento y posicionamiento de marcas, contribuyendo también con el medio ambiente por su bajo consumo de energía. Este proyecto se compone de dos etapas fundamentales para su construcción: Parte Mecánica: Consta del ensamble de un motor que por medio de poleas transmite el movimiento a un eje central el cual tiene adherido un disco en acrílico que sirve de soporte a los circuitos impresos, y una base en acero con buena adherencia a la superficie que sostiene todo el mecanismo para darle estabilidad al visualizador cuando este en movimiento. Parte Eléctrica: Se compone de los circuitos eléctricos que nos permiten visualizar mediante la programación adecuada el mensaje que se deseen mostrar y controlan la velocidad a la que debe girar el motor. Con respecto a lo anterior se dará una descripción de todos los componentes y metodologías utilizadas para la elaboración del visualizador virtual tridimensional, se explicara claramente el desarrollo del chasis y el funcionamiento de cada uno de sus componentes y se tendrán los cálculos necesarios para la elaboración del circuito y del control del motor

Topological moduli space for germs of foliations II : universal deformations

This work deals with the topological classification of singular foliation germs on (C2,0). Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we prove the existence of a topological universal deformation through which every equisingular deformation uniquely factorizes up to topological conjugacy. This is done by representing the functor of topological classes of equisingular deformations of a fixed foliation. We also describe the functorial dependence of this representation with respect to the foliation

Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres

In this paper we consider the unfolding of saddle-node X=1xUa(x,y)(x(xμ−ε)∂x−Va(x)y∂y), parametrized by (ε,a) with ε≈0 and a in an open subset A of Rα, and we study the Dulac time T(s;ε,a) of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative ∂sT(s;ε,a) tends to −∞ as (s,ε)→(0+,0) uniformly on compact subsets of A. This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles

Problemas de módulos para una clase de foliaciones holomorfas

Consultable des del TDXTítol obtingut de la portada digitalitzadaMireu els arxius en Acrobat: "resumen_castellano.pdf" i "resumen_ingles.pdf