M.A. Sánchez-Granero

A new approach to metrization

Abstract We give a new metrization theorem on terms of a new structure introduced by the authors in [Rend. Instit. Mat. Univ. Trieste 30 (1999) 21–30] and called fractal structure. This allows us to approach some classical and new metrization theorems (due to Nagata, Smirnov, Moore, Arhangelskii, Frink, Borges, Hung, Morita, Fletcher, Lindgren, Williams, Collins, Roscoe, Reed, Rudin, Hanai, Stone, Burke, Engelking and Lutzer) from a new point of view.

Completions and compactifications of quasi-uniform spaces

Abstract A ∗-compactification of a T1 quasi-uniform space (X, U ) is a compact T1 quasi-uniform space (Y, V ) that has a T ( V ∗ ) -dense subspace quasi-isomorphic to (X, U ), where V ∗ denotes the coarsest uniformity finer than V . With the help of the notion of T1 ∗-half completion of a quasi-uniform space, which is introduced and studied here, we show that if a T1 quasi-uniform space (X, U ) has a ∗-compactification, then it is unique up to quasi-isomorphism. We identify the ∗-compactification of (X, U ) with the subspace of its bicompletion ( X , U ) consisting of all points which are closed in ( X , T ( U )) and prove that (X, U ) is ∗-compactifiable if and only if it is point symmetric and ( X , U ) is compact. Finally, we discuss some properties of locally fitting T0 quasi-uniform spaces, a large class of quasi-uniform spaces whose bicompletion is T1, and, hence, they are T1 ∗-half completable.

A comparison of three Hurst exponent approaches to predict nascent bubbles in S&P500 stocks

In this paper, three approaches to calculate the self-similarity exponent of a time series are compared in order to determine which one performs best to identify the transition from random efficient market behavior (EM) to herding behavior (HB) and hence, to find out the beginning of a market bubble. In particular, classical Detrended Fluctuation Analysis (DFA), Generalized Hurst Exponent (GHE) and GM2 (one of Geometric Method-based algorithms) were applied for self-similarity exponent calculation purposes. Traditionally, researchers have been focused on identifying the beginning of a crash. Instead of this, we are pretty interested in identifying the beginning of the transition process from EM to a market bubble onset, what we consider could be more interesting. The relevance of self-similarity index in such a context lies on the fact that it becomes a suitable indicator which allows to identify the raising of HB in financial markets. Overall, we could state that the greater the self-similarity exponent in financial series, the more likely the transition process to HB could start. This fact is illustrated through actual S&P500 stocks.

How to calculate the Hausdorff dimension using fractal structures

In this paper, we provide the first known overall algorithm to calculate the Hausdorff dimension of any compact Euclidean subset. This novel approach is based on both a new discrete model of fractal dimension for a fractal structure which considers finite coverings and a theoretical result that the authors contributed previously in 14]. This new procedure combines fractal techniques with tools from Machine Learning Theory. In particular, we use a support vector machine to decide the value of the Hausdorff dimension. In addition to that, we artificially generate a wide collection of examples that allows us to train our algorithm and to test its performance by external proof. Some analyses about the accuracy of this approach are also provided.

Compactifications of quasi-uniform hyperspaces

Abstract Several results on compactification of quasi-uniform hyperspaces are obtained. For instance, we prove that if C 0 (X) denotes the family of all nonempty closed subsets of a quasi-uniform space (X, U ) and U H the Bourbaki quasi-uniformity of U , then ( C 0 (X), U H ) is ∗-compactifiable if and only if (X, U ) is closed symmetric and ∗-compactifiable and U −1 is hereditarily precompact. We deduce that for any normal Hausdorff space X , 2 βX is equivalent to the ∗-compactification of ( C 0 (X), PN H ) , where PN denotes the Pervin quasi-uniformity ofxa0 X .

Optimal sampling patterns for Zernike polynomials

A pattern of interpolation nodes on the disk is studied, for which the interpolation problem is theoretically unisolvent, and which renders a minimal numerical condition for the collocation matrix when the standard basis of Zernike polynomials is used. It is shown that these nodes have an excellent performance also from several alternative points of view, providing a numerically stable surface reconstruction, starting from both the elevation and the slope data. Sampling at these nodes allows for a more precise recovery of the coefficients in the Zernike expansion of a wavefront or of an optical surface.