Relatórios de Pesquisa

A globally convergent discrete Newton method is proposed for solving large-scale nonlinear systems of equations. Advantage is taken from discretization steps so that the residual norm can be reduced while the Jacobian is approximated, besides the reduction at Newtonian iterations. The Curtis-Powell-Reid (CPR) scheme for discretization is used for dealing with sparse Jacobians. Global convergence is proved and numerical experiments are presented.

rp-2002-63.pdf

In this paper we discuss some unusual and unsuspected relations between Maxwell, Dirac and the Seiberg-Witten equations. First we investigate what is now known as the Maxwell-Dirac equivalence (MDE) of the first kind. Crucial to that proposed equivalence is the possibility of solving for $\psi$ (a representative on a given spinorial frame of a Dirac-Hestenes spinor field) the equation $F=\psi\gamma_{21}\tilde{\psi}$, where $F$ is a given electromagnetic field. Such non trivial task is presented in this paper and it permits to clarify some possible objections to the MDE which claims that no MDE may exist, because $F$ has six (real) degrees of freedom and $\psi$ has eight (real) degrees of freedom. Also, we review the generalized Maxwell equation describing charges and monopoles. The enterprise is worth even if there is no evidence until now for magnetic monopoles, because there are at least two faithful field equations that have the form of the generalized Maxwell equations. One is the generalized Hertz potential field equation (which we discuss in detail) associated with Maxwell theory and the other is a (non linear) equation (of the generalized Maxwell type) satisfied by the 2-form field part of a Dirac-Hestenes spinor field that solves the Dirac-Hestenes equation for a free electron. This is a new and surprising result, which can also be called MDE of the second kind. It strongly suggests that the electron is a composed system with more elementary "charges" of the electric and magnetic types. This finding may eventually account for the recent claims that the electron has been splited into two electrinos. Finally, we use the MDE of the first kind together with a reasonable hypothesis to give a derivation of the famous Seiberg-Witten equations on Minkowski spacetime. A suggestive physical interpretation for those equations is also given.

rp-2002-61.pdf

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem\begin{eqnarray*}\begin{array}{ccccc}-\Delta u = g(x,u) & {\rm in} & \Omega\\u = 0\ \ & {\rm on} & \partial \Omega,\end{array}\end{eqnarray*}where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $g:\Omega \times\mathbb{R}\to \mathbb{R}$ is a function of class $C^1$ such that $g(x,0)=0$ and which is asymptotically linear at infinity with jumping nonlinearities. We considered both cases resonant and nonresonant with respect to Fu$\check{\rm c}$ik Spectrum. We use critical groups to distinguish the critical points.

rp-2002-59.pdf

The main objective of this paper is to clarify the ontology of Dirac-Hestenes spinor fields (DHSF) and its relationship with even multivector fields, on a Riemann-Cartan spacetime (RCST) $\mathfrak{M}=$($M,g,\nabla,\tau_{g},\uparrow$) admitting a spin structure, and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE) in the case where $\mathfrak{M}$ is a Lorentzian spacetime (the general case when $\mathfrak{M}$ is a RCST will be discussed in another publication). To this aim we introduce the Clifford bundle of multivector fields ($\mathcal{C\ell}(M,g)$) and the left ($C\ell_{\mathrm{Spin}_{1,3}^{e}}^{l}(M)$) and right $\mathcal{C \ell}_{\mathrm{Spin}_{1, 3}^{e}}^{r} (M)$) spin-Clifford bundles on the spin manifold $(M,g)$. The relation between left ideal algebraic spinor fields (LIASF) and Dirac-Hestenes spinor fields (both fields are sections of $\mathcal{C \ell}_{\mathrm{Spin}_{1, 3}^{e}}^{l}(M)$) is clarified. We study in DHSF $\mathbf{\Psi}\in\sec\mathcal{C\ell}_{\mathrm{Spin}_{1,3}^{e}}^{l}(M)$ (denoted DE $\mathcal{C\ell}^{l})$ on a Lorentzian spacetime is found. We also obtain a representation of the DE $\mathcal{C\ell}^{l}$ in the Clifford bundle $\mathcal{C\ell}(M,g)$. It is such equation that we call the DHE and it is satisfied by Clifford fields $\mathit{\psi}_{\Xi}\in\sec\mathcal{C\ell}(M,g)$. This means that to each DHSF $\mathbf{\Psi}\in\sec$ $\mathcal{C\ell}_{\mathrm{Spin}_{1,3}^{e}}^{l}(M)$ and $\Xi\in\sec P_{\mathrm {Spin}_{1, 3}^{e}}(M)$, there is a well defined sum of even multivector fields $\mathit{\psi}_{\Xi}\in \sec \mathcal {C\ell} (M,g)$ (EMFS) associated with $\Psi$. Such a EMFS is called a representative of the DHSF on the given spin frame. And, of course, such a EMFS (the representative of the DHSF) is not a spinor field. With this crucial distinction between a DHSF and its representatives on the Clifford bundle, we provide a consistent theory for the covariant derivatives of Clifford andspinor fields of all kinds. We emphasize that the DE $\mathcal{C\ell}^{l}$ and the DHE, although related, are equations of different mathematical natures. We study also the local Lorentz invariance and the electromagnetic gauge invariance and show that only for the DHE such transformations are of the same mathematical nature, thus suggesting a possible link between them.

rp-2002-57.pdf