Relatórios de Pesquisa

50/2005 Regular Time-Reproductive Solutions for Generalized Boussinesq Model with Neumann Boundary Conditions for Temperature
B. Climent-Ezquerra, Francisco Guillén-González, Marko A. Rojas-Medar

The aim of this work is to prove existence of regular time reproductive solutions for a generalized Boussinesq model (with nonlinear diffusion for velocity and temperature). The main idea is to obtain higher regularity (of $H^3$ type) for temperature than for velocity (of $H^2$ type), using specifically the Neumann boundary condition for temperature.


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49/2005 Theoretical Analysis and Control Results for the FitzHugh-Nagumo Equation
Paulo M. D. Magalhães, Adilson J. Vieira-Brandão, Enrique Fernández-Cara, Marko A. Rojas-Medar

In this paper we are concerned with some theoretical questions for the FitzHugh-Nagumo equation. First, we present a simple proof of the existence and uniqueness of strong solution. We also consider an optimal control problem for this system. We prove the existence of optimal state-control pairs and, as an application of the Dubovitski-Milyoutin formalism, we deduce the corresponding optimality system. We also connect the optimal control problem with a controllability question and we construct a sequence of controls that produce solutions that converge strongly to a desired global state. Finally, we present some open questions related to the control of this equation.


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48/2005 Polynomial Identities of Algebras in Positive Characteristic
Sérgio Mota Alves, Plamen Koshlukov

The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic $p>2$ little is known about them. In previous papers we discussed some sharp differencesbetween these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case. In this paper we study the Gelfand--Kirillov dimension of the relatively free algebras of verbally prime and related algebras. We compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras $M_{1,1}(E)$ and $E\otimes E$ are not PI equivalent in characteristic $p>2$. Furthermore we show that that the following algebras are not PI equivalent in positive characteristic: $M_{a,b}(E)\otimes E$ and $M_{a+b}(E)$;$M_{a,b}(E)\otimes E$ and $M_{c,d}(E)\otimes E$ when $a+b=c+d$, $a\ge b$, $c\ge d$ and $a\ne c$; and finally, $M_{1,1}(E)\otimes M_{1,1}(E)$ and $M_{2,2}(E)$. Here $E$ stands for the infinite dimensional Grassmann algebra with 1, and $M_{a,b}(E)$ is the subalgebra of $M_{a+b}(E)$ of the block matrices with blocks $a\times a$ and $b\times b$ on the main diagonal with entries from $E_0$, andoff-diagonal entries from $E_1$; $E=E_0\oplus E_1$ is the natural grading on $E$.


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47/2005 On Hermite Representation of Distributions and Products
Pedro J. Catuogno, Sandra Molina, Christian Olivera

The space of tempered distributions $\mathcal{S}^{\prime}$ can be realized as a sequence spaces by means of the Hermite representation theorems. In this paper we introduce and study two new products of tempered distributions based in these Hermite representation. In particular, we obtain the products $[H]\delta=\frac{\delta}{2}$, $[\delta] vp(\frac{1}{x})=-\delta^{\prime}$ and$[\delta^{(r)}]vp(\frac{1}{x})=-\frac{\delta^{(r+1)}}{r+1}$, for $r$ even.


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46/2005 A Geometric Itô Formula
Pedro J. Catuogno

Resumo: Let $M$ and $N$ be manifolds equipped with connections $\Gamma^M$ and $\Gamma^N$ respectively and $F :M\rightarrow N$ be a smooth map. Let $X$ be an $M$-valued semimartingale and $\Theta$ be an 1-form on $N$. We prove the following It\^{o} formula in the context of Schwartz (second order) geometry,\[\int \Theta~ d^{\Gamma^N}F(X)=\int F^*\Theta~d^{\Gamma^M}X+\frac{1}{2}\int \beta_F^*\Theta(dX,dX)\]where the integrals are in the It\^{o} sense, and $\beta_F$ is the fundamental form of $F$. Some applications are discussed.Mathematics Subject Classifications (2000): Primary: 60H05 ; Secondary:58J65.Keywords: Stochastic Calculus. Schwartz Geometry. It\^{o}formula.


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45/2005 Busca Direta em Minimização Irrestrita
Lucas Garcia Pedroso, Maria A. Diniz-Ehrhardt

In this work we are interested in direct search procedures: methods for unconstrained minimization that do not require derivatives of the objective function, neither their approximations. We consider an algorithm due to Lucidi and Sciandrone, in which global convergence is guaranteed by using a sufficient decrease of $f$. The algorithm combines two different subclasses of direct search methods, that is, line search andpattern search methods, taking the advantages of each different strategy. Its interesting theoretical results motivated us to report numerical experiments with the objective of analysing its computational performance.


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44/2005 Isolating Blocks for Periodic Orbits
Maria Alice Bertolim, Ketty A. de Rezende, O. Manzoli Neto

In this article we prove that the Lyapunov semi-graphs associated to periodic orbits are realizable by constructing isolating blocks $N^n$ for periodic orbits of Morse-Smale flows. We analyze the effects on the Betti numbers of a manifold after a round handle operation is performed and a variety of situations are considered. Since we are concerned in showing the existence of certain blocks we keep the complexity of the manifolds in consideration under control by considering essentially manifolds with free homology groups, in particular we consider connected sums of tori manifolds.


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43/2005 Semiflows on Topological Spaces: Chain Transitivity and Semigroups
Mauro Patrão, Luiz A. B. San Martin

This paper studies semiflows on topological spaces. A concept of chain recurrence, based on families of coverings, is introduced and related to Morse decomposition. The chain transitive components are studied via semigroup theory by the introduction of the shadowing semigroups associated to a semiflow.


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42/2005 Influence Diagnostics in the Capital Asset Pricing Model Under Elliptical Distributions
Manuel Galea, José A. Díaz-García, Filidor E. Vilca-Labra

In this paper we consider the Capital Asset Pricing Model under Elliptical (symmetric) Distributions. This class of distributions, which contain the normal distribution, t, contaminated normal and power exponential, among others, offers a more flexible framework for modelling asset prices or returns. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihoodestimators, the local influence method was implemented. The results are illustrated by using a set of shares from companies who trade in the Chilean Stock Market. Our main conclusion is that symmetric distributions having heavier tails than those of the normal distribution, especially the t distribution with small degrees of freedom, show a better fit and allow the reduction of the influence of atypical returns in the maximum likelihood estimators.


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41/2005 Lie Algebras with Complex Structures Having Nilpotent Eigenspaces
Edson Carlos Licurgo Santos, Luiz A. B. San Martin

Let $ g $ be a Lie algebra with an integrable complex structure $J$. The eigenspaces of $J$ are complex subalgebras of ${g}^{C}$ isomorphic to the algebra $({g},[*]_{J}) $. We consider here the case where these subalgebras are nilpotent and prove that the original Lie algebra $(g,[,])$ must be solvable. We consider also the $6$-dimensional case and determine explicitly the possible nilpotent * Lie algebras. Finally we produce several examples illustrating different situations, in particular we show that for each given $s$ there exists $g$ with complex structure $J$ such that its *-algebra is $s$-step nilpotent. Similar examples of hipercomplex structures are also built.


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