Raquel Viaña | Searcharxiv

arXiv2023

Total positivity and least squares problems in the Lagrange basis

The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange-Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore-Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included.

math.NA↗

arXiv2008

Análisis de distancias temporales y espaciales entre el Lugar de La Mancha y cuatro puntos de referencia

The identity of the famous place of La Mancha appearing at the Quijote is an unknown with a history almost as long as that of the famous book by Miguel de Cervantes. This work analyzes data obtained from a Geographic Information System and compares the results with those of the previous works. Three different variables with two possible values each are considered: time or space data, 3 or 4 reference points, and the commonly used distances to the place of La Mancha or a set of recently proposed ones. The village in the Campo de Montiel which is closest to be the place of La Mancha happens to be Carrizosa or Villanueva de los Infantes, depending on the configuration, with the latter being the solution for the configuration in which the relative errors are the smallest and the second candidate village is furthest from the first. ----- La identidad del famoso lugar de la Mancha que aparece en El Quijote es una incógnita con una historia casi tan larga como la publicación de la famosa obra de Miguel de Cervantes. Este trabajo analiza datos obtenidos mediante un Sistema de Información Geográfica y compara los resultados con los de los trabajos anteriores. Se consideran tres variables diferentes con dos posibles valores cada una: datos temporales o espaciales, 3 ó 4 puntos de referencia y las distancias al lugar de La Mancha usadas habitualmente o unas recientemente introducidas. La localidad del Campo de Montiel más cercana a ser el lugar de La Mancha resulta ser Carrizosa o Villanueva de los Infantes, dependiendo de la configuración, siendo ésta última la solución para la configuración en que los errores relativos son los más pequeños y la segunda localidad candidata está más lejos de la primera.

stat.AP↗

arXiv2008

Depth of segments and circles through points enclosing many points: a note

Neumann-Lara and Urrutia showed in 1985 that in any set of n points in the plane in general positionthere is always a pair of points such that any circle through them contains at least (n-2)/60 points. In a series of papers, this result was subsequently improved till n/4.7, which is currently the best known lower bound. In this paper we propose a new approach to the problem that allows us, by using known results about j-facets of sets of points in $R^3$, to give a simple proof of a somehow stronger result: there is always a pair of points such that any circle through them has, both inside and outside, at least n/4.7 points.

math.CO↗