Relatórios de Pesquisa

In this paper we present thoughtful comments on the paper `Bessel beams and signal propagation' showing that the main claims of that paper are wrong. Moreover, we take the opportunity to show the non trivial and indeed surprising result that a scalar pulse (i.e., a wave train of compact support in the time domain) that is solution of the homogeneous wave equation (vector ($\vec{E},\vec{B}$ pulse that is solution of Maxwell equations) is such that its wave front in some cases does travel with speed greater than $c$, the speed of light . In order for a pulse to posses a front that travels with speed $c$, an additional condition must be satisfied, namely the pulse must have finite energy. When this condition is fulfilled the pulse still can show peaks propagating with superluminal (or subluminal) velocities, but now its wave front travels at speed $c$. These results are important because they explain several experimental results obtained in recent experiments, where superluminal velocities have been observed, without implying in any breakdown of the Principle of Relativity.

rp-2002-23.pdf

This paper considers an adaptive finite difference scheme for the numerical solution of evolution partial differential equations. The computational domain is formed by non-overlapping blocks. Each block is a uniform grid, but step size may change from one block to another. The blocks are not predetermined, but they are dynamically constructed according to the refinement needs of the numerical solution. The decision over whether a block should be refined or unrefined is taken by looking at the magnitude of wavelet coefficients of the numerical solution on such block. The main objective of this paper is to establish a general framework for the construction and operation on such adaptive block-grids in 2D. The algorithms and data structure are formulated by using abstract concepts borrowed from quaternary trees. This procedure helps the understanding of the method and its computational implementation. The ability of the method is demonstrated by solving some typical test problems.

rp-2002-21.pdf

In this article the main theorem establishes the necessity and sufficiency of the Poincar\'{e}-Hopf inequalities in order for the Morse inequalities to hold under the hypothesis that the flow and the reverse flow satisfy the Conley index duality condition on components of the chain recurrent set. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.

rp-2002-19.pdf

Recently Zheng [1,2], in the setting of global optimization, introduced the concepts of robust set and robust function as a generalization of open set and upper semicontinuous (u.s.c) function, respectively. The aims of this paper are to study the structure of robust sets defined on a normed space $X$ as well as to extend some multivalued convergence results obtained by the author in [3,4] and Greco et al. [6] for semicontinuous functions to the class of robust functions. More precisely, we introduce the concepts of level-convergence and epigraphic convergence on $\mathcal{R}(X)$ the space of nonnegative robust functions on a normed space $X$ and, on one hand, we study its properties and relationships, and on the other, we present some results on level-approximation and epi-approximation of functions by using convolution of robust functions.

rp-2002-17.pdf