| 64/2002 |
Pre-invex functions and weak efficient solutions of the vectorial problem between Banach spaces Lucelina Batista dos Santos, Marko A. Rojas-Medar, Rafaela Osuna-Gómez, Antonio Rufián-Lizana In this work we introduce the notion of pre-invex function for functions between Banach spaces. By using these functions, we obtain necessary and sufficient conditions of optimality for vectorial problems with restrictions of inequalities. Moreover, we will show that this class of problems has the property that all local optimal solution is in fact global.  rp-2002-64.pdf |
| 63/2002 |
Discrete Newton's method with local variations for solving large-scale nonlinear systems Maria A. Diniz-Ehrhardt, Márcia A. Gomes-Ruggiero, Véra L. R. Lopes, José Mario Martínez 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 |
| 62/2002 |
Aperture effects in 2.5-D Kirchhoff migration Thomas Hertweck, Christoph Jäger, Alexander Goertz, Jörg Schleicher Seismic images obtained by Kirchhoff time or depth migration are always accompanied by some artifacts known as ``migration noise'', ``migration boundary effects'', or ``diffraction smiles'', which may severely affect the quality of the migration result. Most of these undesirable effects are caused by a limited aperture if the algorithms make no special disposition to avoid them. Likewise, strong amplitude variation along reflection events may also cause similar artifacts. All these effects can be explained mathematically by means of the Method of Stationary Phase. However, such a purely theoretical explication is not always easy to understand for applied geophysicists. By relating the terms of the stationaryphase approximation to simple geometrical situations, a more physical interpretation of the migration artifacts can be obtained. A simple numerical experiment for poststack (zerooffset) data indicates the problem and helps to develop an intuitive understanding of the effects and the methods to avoid them. rp-2002-62.pdf |
| 61/2002 |
The Relation between Maxwell, Dirac and the Seiberg-Witten Equations Waldyr A. Rodrigues Jr. 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 |
| 60/2002 |
Geometry of surfaces corresponding to O(3, 2)/O(2,1 X O(1,1) - system Martha P. Dussan In this paper we continue the study started by Terng C-L., of the geometry of the submanifolds associated to the $U/K$-system, for the particular case $U/K$ being the space $O(3, 2)/O(2,1)\times O(1,1)$. Following the ideas from Bruck-Du-Park-Terng, we conclude that these include flat time-like surfaces in pseudo-riemannian spaces and isothermic space-like surfaces in the Lorentzian space. We also obtain an explicit action of a rational map with two simple poles on the space of solutions of the $O(3, 2)/O(2,1)\times O(1,1)$-system, and conclude that these correspond to Darboux and Ribaucour transformations for space-like surfaces in pseudo-euclidean space $\mathbb R^{2,1}$. |
| 59/2002 |
Multiple solutions for elliptic problems with asymmetric nonlinearity Francisco O. V. de Paiva 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 |
| 58/2002 |
Some integrals involving a class of filtering functions Edmundo Capelas de Oliveira We discuss some properties of the function $\sin \pi x \,/\pi x$ which is (sometimes) named by the symbol ${\mbox{sinc}}\,x$. This function is associated to problems involving filtering or interpolating functions. Several integrals are presented and a general rule is discussed. rp-2002-58.pdf |
| 57/2002 |
The bundles of algebraic and Dirac-Hestenes spinor fields Ricardo A. Mosna, Waldyr A. Rodrigues Jr. 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 |
| 56/2002 |
Algebraic and Dirac-Hestenes spinors and spinor fields Waldyr A. Rodrigues Jr. Almost all presentations of Dirac theory in first or second quantization in Physics (and Mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of non homogeneous even multivectors fields) is used. However, a carefully analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields (ASF) and Dirac-Hestenes spinor fields (DHSF) on Minkowski spacetime as some equivalence classes of pairs $(\Xi _{u},\psi _{\Xi _{u}})$, where $\Xi _{u}$ is a spin frame field and $\psi _{\Xi _{u}}$ is an appropriate sum of multivectors fields (to be specified below). The necessity of our definitions are shown by a carefull analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities associated with the `bilinear covariants' (on Minkowski spacetime) made with ASF or DHSF. We believe that the present paper clarifies some misunderstandings (past and recent) appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections a vector bundle called the left spin-Clifford bundle. The bundle formulation is essential in order to be possible to produce a coherent theory for the covariant derivatives of these fields on arbitrary Riemann-Cartan spacetimes. The present paper contains also Appendices (A-E) which exhibits a truly useful collection of results concerning the theory of Clifford algebras (including many `tricks of the trade') necessary for the intelligibility of the text. rp-2002-56.pdf |
| 55/2002 |
Sharp Orders of n-Widths of Sobolev's Classes on Compact Globally Symmetric Spaces of Rank 1 Alexander Kushpel, Sérgio A. Tozoni Sharp orders of Kolmogorov's $n$-widths $d_{n}(W^{r}_{p}(M^{d}),{\}L_{q}(M^{d}))$ of Sobolev's classes $W^{r}_{p}(M^{d})$ on compact globally symmetric spaces of rank 1 are found for different $p$ and $q$. In particular, we are considering the cases $1 \leq p \leq 2 \leq q < \infty$, $2 \leq p \leq q < \infty$ and $1 < q \leq p \leq 2$. rp-2002-55.pdf |
