Cognitive abilities, sociocultural background and academic achievement

The infl uence of students’ sociocultural background on academic achievement is a well established fact. Research also points out that sociocultural background is related to students’ cognitive abilities and these have an effect on their academic achievement. However, the mediator role of cognitive abilities on the relationship between sociocultural background and academic achievement is less well known. A structural equation model that represents these relationships was tested in a sample (N= 728) of Portuguese junior high school students. Multigroup analysis of the model showed the importance of the cognitive ability mediation effect between sociocultural background and academic achievement in the 7th and 9th grades, but not in the 8th grade. This difference may be the result of the academic transition experienced in the 7th and 9th grades in the Portuguese educational system, which requires parents’ higher involvement in school.

Nonautonomous difference equations with applications

This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation. In the second part we present some concrete models that are used in ecology/biology and economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models. One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter space of the autonomous Ricker competition model. Since the dynamics of the Ricker competition model is similar to the logistic competition model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results. Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model.