Noncommutative Black Holes and the Singularity Problem

A phase-space noncommutativity in the context of a Kantowski-Sachs cosmological model is considered to study the interior of a Schwarzschild black hole. Due to the divergence of the probability of finding the black hole at the singularity from a canonical noncommutativity, one considers a non-canonical noncommutativity. It is shown that this more involved type of noncommutativity removes the problem of the singularity in a Schwarzschild black hole.Comment: Based on a talk by CB at ERE2010, Granada, Spain, 6th-10th September 201

A symplectic extension map and a new Shubin class of pseudo-differential operators

For an arbitrary pseudo-differential operator $A:\mathcal{S}(\mathbb{R}^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$ with Weyl symbol $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$, we consider the pseudo-differential operators $\widetilde{A}:\mathcal{S}(\mathbb{R}^{n+k})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n+k})$ associated with the Weyl symbols $\widetilde{a}=(a\otimes1_{2k})\circ{s}$, where $1_{2k}(x)=1$ for all $x\in\mathbb{R}^{2k}$ and ${s}$ is a linear symplectomorphism of $\mathbb{R}^{2(n+k)}$. We call the operators $\widetilde{A}$ symplectic dimensional extensions of $A$. In this paper we study the relation between $A$ and $\widetilde{A}$ in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of $\widetilde{A}$ in terms of those of $A$. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes $HG_{\rho}^{m_{1},m_{0}}$ of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in $HG_{\rho }^{m_{1},m_{0}}$ but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics.Comment: 28 pages, new version, accepted for publication in JF

A educação ambiental em maricá/rj : uma visão de Darwin e do príncipe wied-neuwied no século XIX e nos dias atuais

O presente trabalho tem por objetivo descrever os relatos dos naturalistas, Príncipe Maximiliano de Wied-Newied (1815) e Charles Darwin (1832), em suas viagens pelo Município de Marica/RJ, as ações do município no que se refere às questões ambientais em Escolas no 1° Segmento do Ensino Fundamental. É uma pesquisa qualitativa de caráter exploratório-descritiva acerca da história do Município de Maricá e da legislação vigente no Município. Dados foram levantados através de questionários e entrevistas com professores de escolas municipais. Deve ser estabelecida uma política, por parte dos órgãos públicos, direcionada à preservação do patrimônio cultural e ambiental do Município e aproximando dos alunos as atividades de Educação Ambiental

Automatic camera pose initialization, using scale, rotation and luminance invariant natural feature tracking

The solution to the camera registration and tracking problem serves Augmented Reality, in order to provide an enhancement to the user’s cognitive perception of the real world and his/her situational awareness. By analyzing the five most representative tracking and feature detection techniques, we have concluded that the Camera Pose Initialization (CPI) problem, a relevant sub-problem in the overall camera tracking problem, is still far from being solved using straightforward and non-intrusive methods. The assessed techniques often use user inputs (i.e. mouse clicking) or auxiliary artifacts (i.e. fiducial markers) to solve the CPI problem. This paper presents a novel approach to real-time scale, rotation and luminance invariant natural feature tracking, in order to solve the CPI problem using totally automatic procedures. The technique is applicable for the case of planar objects with arbitrary topologies and natural textures, and can be used in Augmented Reality. We also present a heuristic method for feature clustering, which has revealed to be efficient and reliable. The presented work uses this novel feature detection technique as a baseline for a real-time and robust planar texture tracking algorithm, which combines optical flow, backprojection and template matching techniques. The paper presents also performance and precision results of the proposed technique

The singularity problem and phase-space noncanonical noncommutativity

The Wheeler-DeWitt equation arising from a Kantowski-Sachs model is considered for a Schwarzschild black hole under the assumption that the scale factors and the associated momenta satisfy a noncanonical noncommutative extension of the Heisenberg-Weyl algebra. An integral of motion is used to factorize the wave function into an oscillatory part and a function of a configuration space variable. The latter is shown to be normalizable using asymptotic arguments. It is then shown that on the hypersufaces of constant value of the argument of the wave function's oscillatory piece, the probability vanishes in the vicinity of the black hole singularity.Comment: 4 pages, revtex