The Codazzi Equation for Surfaces

In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms {\bb R}^3, {\bb S}^3 and {\bb H}^3. We give essentially sharp generalizations of some classical theorems of surface theory that mainly depend on the Codazzi equation, and we apply them to the study of Weingarten surfaces in space forms. In particular, we study existence of holomorphic quadratic differentials, uniqueness of immersed spheres in geometric problems, height estimates, and the geometry and uniqueness of complete or properly embedded Weingarten surfaces

Compact maximal hypersurfaces in globally hyperbolic spacetimes

Several uniqueness results on compact maximal hypersurfaces in a wide class of globally hyperbolic spacetimes are provided. They are obtained from the study of a distinguished function on the maximal hypersurface and under suitable natural _rst order conditions of the spacetime. As a consequence, several applications to Geometric Analysis are given

Spacelike hypersurfaces with functionally bounded mean curvature in Lorentzian warped products and generalized Calabi-Bernstein type problems

We study spacelike hypersurfaces with functionally bounded mean curvature in Lorentzian warped products M = I × f F , where F is a (non-compact) complete Riemannian mani- fold whose universal covering is parabolic. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. As an application, several Calabi- Bernstein type results are obtained which widely extend the previous ones in this setting

Aprendiendo de la Evaluación entre Compañeros

La evaluación continua forma parte de la metodología inductiva que ya llevamos unos años incorporando en nuestras aulas en el marco del EEES. Indudablemente, las tareas de evaluación que se derivan de esta metodología suponen un importante incremento de dedicación por parte del profesor, que además debe intentar proporcionar un feedback provechoso para el alumno. Sin embargo, a menudo constatamos que nuestros esfuerzos no se reflejan en el aprendizaje del estudiante. Una forma de intentar que el feedback que se deriva de la evaluación continua mejore el aprendizaje, es involucrar al estudiante en las tareas de evaluación. En este trabajo se describe la experiencia docente sobre evaluación entre compañeros en la asignatura de Lógica de segundo curso del grado en Ingeniería Informática.The use of continuous assessment is part of the inductive methodology which we have included in our teaching activity over the last years in the framework of the EEES. Undoubtedly, the evaluation tasks derived from this methodology mean a significant increase in the amount of work for professors. Moreover, such activity should provide beneficial effects amongst the students. However, our efforts often have not the desired repercussion in the students learning process. A way to overcome this problem is to involve the students in the evaluation tasks. In this work we describe a teaching experience about peer assessment in the subject Logic of Second Course in Computer Science

Mixture-based probabilistic graphical models for the partial label ranking problem

The Label Ranking problem consists in learning preference models from training datasets labeled with a ranking of class labels, and the goal is to predict a ranking for a given unlabeled instance. In this work, we focus on the particular case where both, the training dataset and the prediction given as output allow tied labels (i.e., there is no particular preference among them), known as the Partial Label Ranking problem. In particular, we propose probabilistic graphical models to solve this problem. As far as we know, there is no probability distribution to model rankings with ties, so we transform the rankings into discrete variables to represent the precedence relations (precedes, ties and succeeds) among pair of class labels (multinomial distribution). In this proposal, we use a Bayesian network with Naive Bayes structure and a hidden variable as root to collect the interactions among the different variables (predictive and target). The inference works as follows. First, we obtain the posterior-probability for each pair of class labels, and then we input these probabilities to the pair order matrix used to solve the corresponding rank aggregation problem. The experimental evaluation shows that our proposals are competitive (in accuracy) with the state-of-the-art Instance Based Partial Label Ranking (nearest neighbors paradigm) and Partial Label Ranking Trees (decision tree induction) algorithms