| 55/2004 |
Some Remarks on the AdS Geometry, Projective Embedded Coordinates and Associated Isometry Groups Roldão da Rocha Jr., Edmundo Capelas de Oliveira This work is intended to investigate the geometry of anti-de Sitter spacetime (AdS), from the point of view of the Laplacian Comparison Theorem (LCT), and to give another description of the hyperbolical embedding standard formalism of the de Sitter and anti-de Sitter spacetimes in a pseudoeuclidian spacetime. After Witten proved that general relativity is a renormalizable quantum system in (1+2) dimensions, it is possible to point out a few interesting motivations to investigate the AdS spacetime. A lot of attempts were made to generalize the gauge theory of gravity in (1+2) dimensions to higher ones. The first one was to enlarge the Poincar\'e group of symmetries, supposing an AdS group symmetry, which contains the Poincar\'e group. Also, the AdS/CFT correspondence asserts that a maximal supersymmetric Yang-Mills theory in 4-dimensional Minkowski spacetime is equivalent to a type IIB closed superstring theory. The 10-dimensional arena for the type IIB superstring theory is described by the product manifold $S^5\times$ AdS, an impressive consequence that motivates the investigations about the AdS spacetime in this paper, together with the de Sitter spacetime. Classical results in this mathematical formulation are reviewed in a more general setting together with the isometry group associated to the de Sitter spacetime. It is known that, out of the Friedmann models that describe our universe, the Minkowski, de Sitter and anti-de Sitterspacetimes are the unique maximally isotropic ones, so they admit a maximal number of conservation laws and also a maximal number of Killing vectors. In this paper it is shown how to reproduce some geometrical properties of AdS, from the LCT in AdS, choosing suitable functions that satisfy basic properties of riemannian geometry.We also introduce and discuss the well-known embedding of a 4-sphere and a 4-hyperboloid in a 5-dimensional pseudoeuclidian spacetime, reviewing the usual formalism of spherical embedding and the way how it can retrieve the Robertson-Walker metric. With the choice of the de Sitter metric static frame, we write the so-called reduced model in suitable coordinates. We assume the existence of projective coordinates, since de Sitter spacetime is orientable. From these coordinates, obtained when stereographic projection of the de Sitter 4-hemisphere is done, we consider the Beltrami geodesic representation, which gives a more general formulation of the seminal full model described by Schr\"odinger, concerning the geometry and topology of de Sitter spacetime. Our formalism retrieves the classical one if we consider the linear metric terms over the de Sitter splitting on Minkowski spacetime. From the covariant derivatives we find the acceleration of moving particles, Killing vectors and the isometry group generators associated to de the Sitter spacetime.  rp-2004-55.pdf |
| 54/2004 |
An Existence Result for a Linear-Superlinar Elliptic System with Neumann Boundary Conditions Eugenio Massa In this work, we consider an elliptic system of two equations in dimension one (with Neumann boundary conditions) where the nonlinearities are asymptotically linear at $-\infty$ and superlinear at $+\infty$. We obtain that, under suitable hypotheses, a solution exists for any couple of forcing terms in $L^2$ .We also present a similar result in which the superlinearity is in only one of the two equations, and we discuss the resonant problem too. rp-2004-54.pdf |
| 53/2004 |
Periodic Hamiltonean Elliptic Systems in Unbounded Domains: Superquadratic and Asymptotically Quadratic Djairo G. Figueiredo, Y. H. Ding We study the system of elliptic equations\[\left\{\begin{array}{rcl}-\Delta u + b(x)\cdot \nabla u + V (x)u &=& H_v(x; u; v) \\-\Delta v + b(x)\cdot \nabla v + V (x)v &=& H_u(x; u; v)\\\end{array}\right. \qquad \mbox{\textrm{for $x \in \mathbb{R}^N$}}\]where $z = (u; v) \colon \mathbb{R}^N \to \mathbb{R}^m \times \mathbb{R}^m$, $b\in C^1(\mathbb{R}^N;\mathbb{R}^N)$ with the gauge condition $div b = 0$, $V\in C(\mathbb{R}^N;\mathbb{R})$ and $H\in C^1(\mathbb{R}^N \times \mathbb{R}^{2m};R)$. Assume the system depends periodically on the variable $x$ and $H(x; z)$ is asymptotically quadratic or super quadratic with nonlinear conditions as $|z|\to\infty$ more general than the Ambrosetti-Rabinowitz condition.We establish the existence of solutions in $H^1(\mathbb{R}^N;\mathbb{R}^{2m})$. If moreover $H(x; z)$ is even in $z$ we obtain infinitely many such solutions. rp-2004-53.pdf |
| 52/2004 |
Consistent Estimator for Basis Selection Based on a Proxy of the Kullback-Leibler Distance Ronaldo Dias, Nancy L. Garcia Given a random sample from a continuous and positive density $f$, the logistic transformation is applied and a log density estimate is provided by using basis functions approach. The number of basis functions acts as the smoothing parameter and it is estimated by minimizing a penalized proxy of the Kullback-Leibler distance which includes as particular cases AIC and BIC criteria. We prove that this estimator is consistent. rp-2004-52.pdf |
| 51/2004 |
Continuous Time-Optimization Problems via KT-Invexity Valeriano Antunes de Oliveira, Marko A. Rojas Medar We prove that the notion of KT-invexity for continuous-time nonlinear optimization problem is a necessary and sufficient condition for global optimality of a Karush-Kuhn-Tucker point. rp-2004-51.pdf |
| 50/2004 |
The Cauchy Problem and Decay Rates for Strong Solutions of a Boussinesq System Francisco Guillén González, Márcio Santos da Rocha, Marko A. Rojas Medar The Boussinesq equations describe the motion of an incompressible viscous fluid subject to convective heat transfer. Decay Rates of derivatives of solutions of the three-dimensional Cauchy problem for a Boussinesq system are studied in this work. rp-2004-50.pdf |
| 49/2004 |
Multiple Solutions for Some Elliptic Equations With a Nonlinearity Concave in the Origin Francisco O. V. de Paiva, Eugenio Massa In this paper we establish the existence of multiple solutions for the semilinear elliptic problem\[\begin{array}{lll}-\Delta u = -\lambda |u|^{q-2}u +au+g(u) & {\rm in} & \Omega\\ \ \ \ \ u = 0 & {\rm on} & \partial \Omega,\end{array}\]where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $g:\mathbb{R}\to \mathbb{R}$ is a function of class $C^1$ such that $g(0)=g'(0)=0$, $\lambda>0$ is real parameter, $a\in\R$, and $1 rp-2004-49.pdf |
| 48/2004 |
Product of harmonic maps is harmonic: a stochastic approach Pedro J. Catuogno, Paulo R. C. Ruffino Let $\phi_j: M_j\rightarrow G$, $j=1,2,\ldots, n$, be harmonic mappings from Riemannian manifolds $M_j$ to a Lie group $G$. Then the product $\phi_1\phi_2 \ldots \phi_n$ is a harmonic mapping between $M_1\times M_2\times \ldots M_n$ and $G$. The proof is a combination of properties of Brownianmotion in manifolds and It\^{o} formulae for stochastic exponential and logarithm of product of semimartingales in Lie groups. rp-2004-48.pdf |
| 47/2004 |
Multiplicity of Solutions for Resonant Elliptic Systems Marcelo F. Furtado, Francisco O. V. de Paiva We establish the existence and multiplicity of solutions for some resonant elliptic systems. The results are proved by applying minimax arguments and Morse Theory. rp-2004-47.pdf |
| 46/2004 |
Multiplicity of Nodal Solutions for a Critical Quasilinear Equation with Symmetry Marcelo F. Furtado We consider the quasilinear problem $\mbox{div}(|\nabla u|^{p-2}\nabla u) + \lambda|u|^{q-2}u + |u|^{p^*-2}u=0$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega \subset \real^N$ is a bounded smooth domain, $N \geq p^2$, $\lambda>0$ and $p rp-2004-46.pdf |
