Resumen En los últimos años se ha alcanzado cierto consenso acerca del papel de la enseñanza de la Matemática en el desarrollo del pensamiento, por encima de la transferencia de conocimientos matemáticos. En este sentido, la atención al desarrollo de la capacidad para resolver problemas va cediendo terreno con respecto al desarrollo del pensamiento en la resolución de problemas. Numerosos autores han aportado métodos para resolver problemas, sin embargo, aún son escasas las propuestas concretas que ayuden a los docentes a utilizar los métodos de resolución de problemas y los recursos de la heurística para llevar a la práctica el tratamiento de la resolución de problemas con el fin de estimular el desarrollo del pensamiento matemático. Este trabajo analiza las potencialidades de los métodos de resolución de problemas para estimular el desarrollo del pensamiento matemático y propone ideas para su implementación en el aula.
The approximation of trace(f(Ω)), where f is a function of a symmetric matrix Ω, can be challenging when Ω is exceedingly large. In such a case even the partial Lanczos decomposition of Ω is computationally demanding and the stochastic method investigated by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996) is preferred. Moreover, in the last years, a partial global Lanczos method has been shown to reduce CPU time with respect to partial Lanczos decomposition. In this paper we review these techniques, treating them under the unifying theory of measure theory and Gaussian integration. This allows generalizing the stochastic approach, proposing a block version that collects a set of random vectors in a rectangular matrix, in a similar fashion to the partial global Lanczos method. We show that the results of this technique converge quickly to the same approximation provided by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996), while the block approach can leverage the same computational advantages as the partial global Lanczos. Numerical results for the computation of the Von Neumann entropy of complex networks prove the robustness and efficiency of the proposed block stochastic method.
In this paper we propose a method for computing the contour of an object in an image using a snake represented as a subdivision curve. The evolution of the snake is driven by its control points which are computed minimizing an energy that pushes the snake towards the boundary of the interest region. Our method profits from the hierarchical nature of subdivision curves, since the unknowns of the optimization process are the few control points of the subdivision curve in the coarse representation and, at the same time, good approximations of the energies and their derivatives are obtained from the fine representation. We introduce a new region energy that guides the snake maximizing the contrast between the average intensity of the image within the snake and over the complement of the snake in a bounding box that does not change during the optimization. To illustrate the performance of our method we discuss the snakes associated with two classical subdivision schemes: the four point scheme and the cubic B-spline. Our experiments using synthetic and real images confirm that the proposed method is fast and robust.
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