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

Least squares problems involving generalized Kronecker products and application to bivariate polynomial regression

A method for solving least squares problems (A ⊗ Bi)x = b whose coefficient matrices have generalized Kronecker product structure is presented. It is based on the exploitation of the block structure of the Moore-Penrose inverse and the reflexive minimum norm g-inverse of the coefficient matrix, and on the QR method for solving least squares problems. Firstly, the general case where A is a rectangular matrix is considered, and then the special case where A is square is analyzed. This special case is applied to the problem of bivariate polynomial regression, in which the involved matrices are structured matrices (Vandermonde or Bernstein-Vandermonde matrices). In this context, the advantage of using the Bernstein basis instead of the monomial basis is shown. Numerical experiments illustrating the good behavior of the proposed algorithm are included.Ministerio de Economía y Competitivida

Accurate computations with collocation matrices of the Lupaş-type (p,q)-analogue of the Bernstein basis

A fast and accurate algorithm to compute the bidiagonal decomposition of collocation matrices of the Lupaş-type (p,q)-analogue of the Bernstein basis is presented. The error analysis of the algorithm and the perturbation theory for the bidiagonal decomposition are also included. Starting from this bidiagonal decomposition, the accurate and efficient solution of several linear algebra problems involving these matrices is addressed: linear system solving, eigenvalue and singular value computation, and computation of the inverse and the Moore-Penrose inverse. The numerical experiments carried out show the good behaviour of the algorithm.Agencia Estatal de Investigació

Error analysis, perturbation theory and applications of the bidiagonal decomposition of rectangular totally-positive h-Bernstein-Vandermonde matrices

A fast and accurate algorithm to compute the bidiagonal decomposition of rectangular totally positive h-Bernstein-Vandermonde matrices is presented. The error analysis of the algorithm and the perturbation theory for the bidiagonal decomposition of totally positive h-Bernstein-Vandermonde matrices are addressed. The computation of this bidiagonal decomposition is used as the first step for the accurate and efficient computation of the singular values of rectangular totally positive h-Bernstein-Vandermonde matrices and for solving least squares problems whose coefficient matrices are such matrices.Agencia Estatal de Investigació

Detection and Preliminary Characterisation of Polluted White Dwarfs from Gaia EDR3 and LAMOST

We present a catalogue of 62 polluted white dwarfs observed by the 9th Low-Resolution Data Release of the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST LRS DR9v1; R$\approx$1,800) and the Early Data Release 3 (EDR3) of the Gaia Mission. Among these stellar remnants, 30 are new discoveries with previously unknown traces of calcium pollution. To generate our catalogue, we used a database of 4,324 unique Gaia EDR3 white dwarf candidates with LAMOST LRS DR9v1 observations, many of which have been spectroscopically confirmed by other telescopes. For these stars, we developed a quantitative method to detect calcium absorption in their spectra between 3,900-4,000$\mathring {\mathrm A}$, which we then validated through visual inspection and multiple literature cross-checks. Our catalogue provides the astrometric and photometric properties of the white dwarf candidates, incorporates supplementary data (e.g. Montreal White Dwarf Database, MWDD; PanSTARRS; the Hubble Space Telescope), and indicates the possibility of calcium pollution in their atmospheres. For our final sample of polluted white dwarfs, we also determine the main atmospheric properties of 23 sources with effective temperatures $T_{\rm eff}$$\leq$25,000K and no existing calcium abundances in the MWDD. Our analysis represents a first step towards measuring the full atmospheric composition of these stars and learning about the bulk properties of their accreted material. As we venture into the era of wide-field spectroscopic surveys, our work highlights the importance of combining large-scale databases for identifying and characterising new polluted white dwarfs.Comment: 29 pages, 14 figures (+3 in the Appendix), 5 tables (+5 in the Appendix). Accepted for publication in MNRA

Simulation-based Inference for Exoplanet Atmospheric Retrieval: Insights from winning the Ariel Data Challenge 2023 using Normalizing Flows

Advancements in space telescopes have opened new avenues for gathering vast amounts of data on exoplanet atmosphere spectra. However, accurately extracting chemical and physical properties from these spectra poses significant challenges due to the non-linear nature of the underlying physics. This paper presents novel machine learning models developed by the AstroAI team for the Ariel Data Challenge 2023, where one of the models secured the top position among 293 competitors. Leveraging Normalizing Flows, our models predict the posterior probability distribution of atmospheric parameters under different atmospheric assumptions. Moreover, we introduce an alternative model that exhibits higher performance potential than the winning model, despite scoring lower in the challenge. These findings highlight the need to reevaluate the evaluation metric and prompt further exploration of more efficient and accurate approaches for exoplanet atmosphere spectra analysis. Finally, we present recommendations to enhance the challenge and models, providing valuable insights for future applications on real observational data. These advancements pave the way for more effective and timely analysis of exoplanet atmospheric properties, advancing our understanding of these distant worlds.Comment: Conference proceeding for the ECML PKDD 202

Elearning, Communication and Open-data: Massive Mobile, Ubiquitous and Open Learning

ABSTRACT: In MOOCs, learning analytics have to be addressed to the various types of learners that participate. This deliverable describes indicators that enable both teachers and learner to monitor the progress and performance as well as identify whether there are learners at risk of dropping out. How these indicators should be computed and displayed to end users by means of dashboards is also explained. Furthermore a proposal based on xAPI statements for storing relevant data and events is provided

ECO D2.5 Learning analytics requirements and metrics report

In MOOCs, learning analytics have to be addressed to the various types of learners that participate. This deliverable describes indicators that enable both teachers and learner to monitor the progress and performance as well as identify whether there are learners at risk of dropping out. How these indicators should be computed and displayed to end users by means of dashboards is also explained. Furthermore a proposal based on xAPI statements for storing relevant data and events is provided.Part of the work carried out has been funded with support from the European Commission, under the ICT Policy Support Programme, as part of the Competitiveness and Innovation Framework Programme (CIP) in the ECO project under grant agreement n° 21127

Review of Theoria Temporis

The time dilation formulas of both the Special Relativity and General Relativity could be studied using a factor dependent on specific energy. Should such factor be used to define the relativistic mass, the equation that arises is an approximation of the mass and energy relation. This mathematical definition of mass is finally compared to the equations that define Dark Matter Annihilation into charged states via loop-level processes

Teoría de Centromas

Trabajo Fin de GradoEn este artículo se propone una ecuación vectorial que relaciona las velocidades absolutas de tres cuerpos rígidos móviles con un movimiento plano de tipo general. A partir de esta ecuación, es posible obtener la relación entre las velocidades polares de los tres puntos matemáticos, relacionados entre ellos por el Teorema de Aronhold-Kennedy. La fórmula permite calcular una de las velocidades polares de las otras dos, siendo conocidas las velocidades angulares y aceleraciones de los cuerpos en movimiento. Es aplicable independientemente de si los centros instantáneos (polos) están situados en puntos físicos en el enlace o no. Se incluyen ejemplos ilustrativos de la aplicación de la fórmula sobre enlaces planos representativos. En la sección final, se obtiene la expresión analítica de la velocidad del centroma, punto matemático asociado a los centros de curvatura de la trayectoria de un punto móvil. En base a este se obtienen conclusiones cinemáticas de gran importancia