Made available in DSpace on 2014-07-29T16:02:15Z (GMT). No. of bitstreams: 1 Dissertacao Wender J de Souza.pdf: 1008440 bytes, checksum: b2d3405f265353a21b9eaaad1c91d71f (MD5) Previous issue date: 2010-03-29 / In this work we are interested in the solution of the following problem: Let Y = ( f ,g) be a vector field of class C1 in R2. Suppose that (x, y) = (0,0) is a singular point of Y and assume that for any q &#8712; R2, the eigenvalues of DY have negative real part, this is, det(DY) > 0 and tr(DY) < 0. Then, the solution (x, y) = (0,0) of Y is globally asymptotically stable. To this end, we show that this problema is equivalent to the following: Let Y : R2 &#8594;R2 be a C1 vector field. If det(DY) > 0 and tr(DY) < 0, then Y is globally injective. This equivalence was proved by C. Olech [1]. So we show the injectivity of the vector field Y under the conditions det(DY) > 0 and tr(DY)<0. In fact, we present a more stronger result, which was obtained by C. Gutierrez and can be found in [4]. This result is given by: Any planar vector field X of class C2 satisfying the r-eigenvalue condition for some r &#8712; [0,¥) is injective. / Neste trabalho, estamos interessados em estudar a solução do seguinte problema: Seja Y = ( f ,g) um campo de vetores, de classe C1, em R2. Suponha que (x, y) = (0,0) é um ponto singular de Y e suponha que, para todo q &#8712; R2, os autovalores de DY tem parte real negativa, isto é, det(DY) > 0 e tr(DY) < 0. Então, a solução (x, y) = (0,0) de Y é globalmente assintoticamente estável. Para este fim, mostramos que este problema é equivalente ao seguinte: Seja Y : R2 &#8594;R2 uma campo de vetores de classe C1. Se det(DY) > 0 e tr(DY) < 0, então Y é globalmente injetora. Esta equivalência foi demonstrada por C. Olech em [1]. Desta forma, a estratégia é estudar a injetividade do campo Y sob as condições det(DY)> 0 e tr(DY) < 0. Na verdade, apresentamos um resultado um pouco mais forte, o qual foi obtido por C. Gutierrez e pode ser encontrado em [4]. Este resultado é dado por: Qualquer campo de vetores X : R2 &#8594;R2 de classe C2 satisfazendo a condição de r-autovalor, para algum r &#8712; [0,¥), é injetora.

Any set of finitely presented left modules defines a relative purity for left modules and also apurity for right modules. Purities defined by various classes are compared and investigated,especially in the contexts of modules over semiperfect rings and over tame hereditary, andmore general, finite-dimensional algebras. Connections between the indecomposable relativelypure-injective modules and closure in the full support topology (a refinement of theZiegler spectrum) are described.Duality between left and right modules is used to define the concept of a class of leftmodules and a class of right modules forming an almost dual pair. Definability of suchclasses is investigated, especially in the case that one class is the closure of a set of finitelypresented modules under direct limits. Elementary duality plays an important role here.Given a set of finitely presented modules, the corresponding proper class of relativelypure-exact sequences can be used to define a relative notion of cotorsion pair, which weinvestigate.The results of this thesis unify and extend a wide range of results in the literature.