Noção de simetria na cosmologia de Anaximandro de Mileto

Through the philosophical movements in Ionia and through researches made on phýsis (started by the first wise men in Miletus), we elected Anaximander‟s cosmology as a fertile ground for exploring the Greek notion of symmetry. From a geographical perspective, initially, the Greeks‟ originality towards the notion of a due measure will be questioned, since the astrological and mathematical knowledge were common in Babylon and Egypt. Although the cultural environment on the Milesian commercial ports and its architecture show evidences of a possible eastern impact on the Greek thought, it will be noted (from fragments validated by the Doxography tradition) that the problem with the birthplace of the notion of harmony and of due measure is something specific of the Greek culture and inherent to its remote religiosity. The resumption of these notions refers to the issue of arché and to its divinity assumed by Thales, Anaximander and Anaximenes. The divine was not an extrinsic notion to the Milesian thought towards the first element. Therefore, we will have as a result of this investigation some assumptions for which the notion of symmetry in Anaximander, stated in his ápeiron, could be, from the dialogue between philosophy and Orphism, an assimilation of the One, as witnessed in the Derveni papyrus.

Algorithms for super-resolution of images based on Sparse Representation and Manifolds

lmage super-resolution is defined as a class of techniques that enhance the spatial resolution of images. Super-resolution methods can be subdivided in single and multi image methods. This thesis focuses on developing algorithms based on mathematical theories for single image super­ resolution problems. lndeed, in arder to estimate an output image, we adopta mixed approach: i.e., we use both a dictionary of patches with sparsity constraints (typical of learning-based methods) and regularization terms (typical of reconstruction-based methods). Although the existing methods already per- form well, they do not take into account the geometry of the data to: regularize the solution, cluster data samples (samples are often clustered using algorithms with the Euclidean distance as a dissimilarity metric), learn dictionaries (they are often learned using PCA or K-SVD). Thus, state-of-the-art methods still suffer from shortcomings. In this work, we proposed three new methods to overcome these deficiencies. First, we developed SE-ASDS (a structure tensor based regularization term) in arder to improve the sharpness of edges. SE-ASDS achieves much better results than many state-of-the- art algorithms. Then, we proposed AGNN and GOC algorithms for determining a local subset of training samples from which a good local model can be computed for recon- structing a given input test sample, where we take into account the underlying geometry of the data. AGNN and GOC methods outperform spectral clustering, soft clustering, and geodesic distance based subset selection in most settings. Next, we proposed aSOB strategy which takes into account the geometry of the data and the dictionary size. The aSOB strategy outperforms both PCA and PGA methods. Finally, we combine all our methods in a unique algorithm, named G2SR. Our proposed G2SR algorithm shows better visual and quantitative results when compared to the results of state-of-the-art methods.