Research Reports

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these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case.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 $E$.

rp-2005-48.pdf

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.

rp-2005-46.pdf

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.

rp-2005-44.pdf

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.

rp-2005-42.pdf