Research Reports

Let $G$ be a semisimple real Lie group of non-compact type, $K$ a maximal compact subgroup and $S\subseteq G$ a semigroup with nonempty interior. We consider the ideal boundary $\partial_{\infty }\left( G/K\right) $ of the associated symmetric space and the flag manifolds $G/P_{\Theta }$. We prove that the asymptotic image $\partial _{\infty }\left( Sx_{0}\right) \subseteq \partial _{\infty }\left( G/K\right) $, where $x_{0}\in G/K$ is any given point, is the maximal invariant control set of $S$ in $\partial_{\infty }\left( G/K\right) $. Moreover there is a surjective projection $\pi :\partial _{\infty }\left( Sx_{0}\right) \rightarrow \bigcup\limits_{\Theta \subseteq \Sigma }C_{\Theta }$, where $C_{\Theta }$ is the maximal invariant control set for the action of $S$ in the flag manifold $G/P_{\Theta }$, with $P_{\Theta }$ a parabolic subgroup. The points that project over $C_{\Theta } $ are exactly the points of type $\Theta $ in $\partial _{\infty }\left( Sx_{0}\right) $ (in the sense of the type of a cell in a Tits Building).

rp-2002-3.pdf

In this work it is introduced a new quasi-Newton method for solving large-scale nonlinear systems of equations. In this method two columns of the approximation of the inverse Jacobian are updated, in such a way that the two last secant equations are satisfied (when it is possible) at every iteration. The new method is called the Inverse Two-Columns Updating Method (ITCUM). Moreover, it is proposed a right implementation from the point of view of linear algebra and numerical stability. It is presented a local convergence analysis and several numerical tests an a comparison between the performance of this new quasi-Newton method with other quasi-Newton methods, in particular the ICUM (Inverse Column Updating Method) \cite{zam}.

rp-2002-1.pdf

The Common Reflection Surface (CRS) is a stacking process that simulates a zero-offset section and provides useful parameters sections. It can be applied to the real non horizontal measurement surface and it is necessary to remove the topographic effect to obtain a more accurate section to be interpretated. This work presents a new technique to remove the topographic effect: continuation of the information to a chosen depth datum using the CRS parameter. A synthetic example is provided to illustrate the approach.

rp-2001-47.pdf

In recent years, the phase-field methodology has achieved considerable importance in modeling and numerically simulating a range of phase transitions and complex growth structures that occur during solidification processes. In attempt to understand the mathematical aspects of such methodology, in this article we consider a simplified model of this sort for a nonstationary process of solidification/melting of a binary alloy with thermal properties. The model includes the possibility of occurrence of natural convection in non-solidified regions and, therefore, leads to a free-boundary value problem for a highly non-linear system of partial differential equations consisting of a phase-field equation, a heat equation, a concentration equation and a modified Navier-Stokes equations by a penalization term of Carman-Kozeny type, which accounts for the mushy effects, and Boussinesq terms to take in consideration the effects of variations of temperature and concentration in the flow.A proof of existence of weak solutions for the system is given. The problem is firstly approximated and a sequence of approximate solutions is obtained by Leray-Schauder's fixed point theorem. A solution of the original problem is then found by using compactness arguments.

rp-2001-45.pdf