Relatórios de Pesquisa

In this paper we describe the bifurcation diagram of a boundary cusp of codimension three, i.e, a Bogdanov-Takens singular point in the boundary of the semi plane $\{(x,y)\in {\mathbb R}^2:\; x\geq 0\}$. This study is applied to the analysis of the behavior of singularity of the germ of vector field $X_{0}(x,y)=(y,2x(x^4+x^2y))$ in the class of reversible vector fields. We classify the generic three parameter families of reversible vector fields $X_{a,b,c}$ with $(a,b,c) \in ({\mathbb R^3},0)$ and $X_{a,b,c}=X_0$.

rp-2004-6.pdf

Let $(M,d)$ be a locally compact metric space. Since the work of van Dantzig and van der Waerden, it is well known that if $M$ is connected then its group of isometries $I(M,d)$ is locally compact with respect to the compact-open topology. In this paper we prove some extensions of this result to the group of homotheties $H(M,d)$. It is proved that when $(M,d)$ is aHeine-Borel metric space, its group of homotheties $H(M,d)$ is also a Heine-Borel metric space. We also prove that, when $(M,d)$ is a Heine-Borel ultrametric space, its group of isometries is an increasing union of compact subgroups and if $G$ is a finitely generated subgroup of $I(M,d)$ then $G$ is compact. It is also proved that when the space $\Sigma(M)$ of the connected components of $M$ is quasi-compact with respect to quotient topology, its group of homotheties $H(M,d)$ is locally compact with respect to the compact-open topology. An other main result is a generalization of a classical result in Riemannian geometry for Finsler manifolds: if $(M,d)$ is a connected Finsler manifold then its group of homotheties is a Lietransformation group with respect to the compact-open topology.

rp-2004-4.pdf

We demonstrate a sufficient condition for some modules $M$ over the group algebra $\BZ[G]$ to be of homological type $FP_2$, where $G$ is a finitely generated split extension of abelian groups. This generalises a result of Bieri-Strebel when $M$ is the trivial module $\BZ$ \cite{B-S1} and is a special case of a conjecture suggested in \cite[Conj.~7]{K4}.

rp-2004-2.pdf