Relatórios de Pesquisa

48/2003 On Zeeman Topology in Kaluza-Klein and Gauge Theories
Ivan Struchiner, Márcio A. F. Rosa

E. C. Zeeman [1] has criticized the fact that in all articles and books until that moment (1967) the topology employed to work with the Minkowski space was the Euclidean one. He has proposed a new topology, wich was generalized for more general space-times by Goebel [2]. In the Zeeman and Goebel topologies for the space-time, the unique continuous curves arepolygonals composed by time-like straight lines and geodesics respectively. In his paper, Goebel proposes a topology for which the continuous curves are polygonals composed by motions of charged particles. Here we obtain in a very simple way a generalization of this topology, valid for any gauge fields, by employing the projection theorem of Kaluza-Klein theories (page144 of Bleecker [3] ). This approach relates Zeeman topologies and Kaluza-Klein, therefore Gauge Theories, what brings insights and points in the direction of a completely geometric theory.


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47/2003 A equação de Dirac no espaço-tempo de Robertson-Walker
D. Gomes, Edmundo Capelas de Oliveira
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46/2003 Invariant nearly-Kähler structures
Luiz A. B. San Martin, Rita de Cássia de J. Silva

This paper considers invariant almost Hermitian structures on a flag manifold $G/P=U/K$ where $G$ is a complex semi-simple Lie group, $P$ is a parabolic subgroup of $G$, $U$ is a compact real form of $G$ and $K=U\cap P$ is the centralizer of a torus. The main result shows that there are nearly-K\"{a}hler structures in $G/P$ which are not K\"{a}hler if and only if $G/P$ has height three. This proves for the flag manifolds a conjecture by J.A. Wolf and A. Gray.


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45/2003 On the classification of second order partial differential equations in two independent variables revisited
Silke Humbert, Edmundo Capelas de Oliveira

We present and discuss the classification of second order partial differential equations in two independent variables. We focus our attention on the case of non-constant coefficients, where the so-called partial differential equations of mixed type can appear. As an application we discuss a partial differential equation of mixed type associated with projective relativity.


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44/2003 On the Sonine-Bessel integral representation
R. Aleixo de Carvalho , Edmundo Capelas de Oliveira

Using an integral representation for the first kind Hankel (Hankel-Bessel) function we obtain the so-called Basset formula, an integral representation for the second kind modified Bessel function. Using the Sonine-Bessel integral representation we obtain the Fourier cosine integral transform of the zero order Bessel function. As an application we present the calculation of the Green's function associated with a second order partial differential equation , particularly a wave equation for a lossy two-dimensional medium. This application is associated to the transient electromagnetic field radiated by a pulsed source in the presence of dispersive media, which is of great importance in the theory of geophysical prospecting, lightning studies


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43/2003 On a real integral: particular case
Edmundo Capelas de Oliveira, Ary O. Chiacchio, R. Figueiredo Camargo

Using two convenient contours of integration in the complex plane we calculate a real integral, depending on two parameters. Several particular cases are discussed. As a by product we determine in a closed form a sum involving a product of trigonometric and hyperbolic functions. Particular cases are also presented.


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42/2003 Positive and Multiple Solutions for Quasilinear Problem
Francisco O. V. de Paiva

In this paper we establish the existence of positive and multiple solutions for the quasilinear elliptic problem\begin{eqnarray*}\begin{array}{ccl}-\Delta_p u = g(x,u) & {\rm in} & \Omega\\ u = 0 & {\rm on} & \partial \Omega,\end{array}\end{eqnarray*}where $\Omega \subset \mathbb{R}^N$ is an open bounded domain with smooth boundary $\partial \Omega$, $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carath\'eodory function such that $g(x,0)=0$ and which is asymptotically linear. We suppose that $g(x,t)/t$ tends to an $L^r$-function, $r>N/p$ if $1N$, which can change sign. We consider both cases, resonant and nonresonant.


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41/2003 Uma Estratégia de Geração de Colunas para o Problema de Empacotamento Bidimensional
Clovis Perin, Valéria de Podestá Gomes
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40/2003 Clifford Valued Differential Forms, Algebraic Spinor Fields, Gravitation, Electromagnetism and "Unified" Theories
Edmundo Capelas de Oliveira, Waldyr A. Rodrigues Jr.

In this paper we show how to describe the general theory of linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the algebra of Clifford bivectors is isomorphic to the algebra of $Sl(2,\mathbb{C)}$. In that way the pullback of the linear connection under a trivialization of the bundleis represented by a Clifford valued 1-form. That observation makes it possible to realize Einstein's gravitational theory can be formulated in a way which is similar to a $Sl(2,\mathbb{C)}$ gauge theory. Some aspects of such approach is discussed. Also, the theory of covariant spinor derivatives of spinor fields is introduced in a novel way, allowing for a physical interpretation of some rules postulated for that covariant spinor derivative in the standard theory of these objects. We use our methods to investigate some polemical issues in gravitational theories and in particular we scrutinize a supposedly "unified" field theory of gravitation and electromagnetism proposed by M. Sachs and recently used in a series of papers. Our results show that Sachs did not attain his objective and that recent papers based on that theory are ill conceived and completely invalid both as Mathematics and Physics.


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39/2003 A spline approach to nonparametric test of hypothesis
Ronaldo Dias, Nancy L. Garcia

We propose a test of hypothesis for the closeness of two distributions whose test statistic is asymptotically normal. The divergent is based on the estimation procedure developed in Dias (2000) using a proxy of symmetrized Kullback-Leibler distance. Simulation results show that for mixture of normal distributions this test is more powerful than Kolmogorov-Smirnov test. As an application we compare acoustic data from several languages in order to identify rhythmic classes.


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