Sobolev regularity of the Beurling transform on planar domains

Consider a Lipschitz domain $\Omega$ and the Beurling transform of its characteristic function $\mathcal{B} \chi_\Omega(z)= - {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z)$. It is shown that if the outward unit normal vector $N$ of the boundary of the domain is in the trace space of $W^{n,p}(\Omega)$ (i.e., the Besov space $B^{n-1/p}_{p,p}(\partial\Omega)$) then $\mathcal{B} \chi_\Omega \in W^{n,p}(\Omega)$. Moreover, when $p>2$ the boundedness of the Beurling transform on $W^{n,p}(\Omega)$ follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.Comment: 33 pages, 8 figures. arXiv admin note: text overlap with arXiv:1507.0433

Generic Vanishing Index and the Birationality of the Bicanonical Map of Irregular Varieties

We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety X with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibered by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.Comment: 10 pages, to appear in Mathematische Zeitschrif

Linearly dependent vectorial decomposition of clutters

This paper deals with the question of completing a monotone increasing family of subsets of a finite set to obtain the linearly dependent subsets of a family of vectors of a vector space. Specifically, we demonstrate that such vectorial completions of the family of subsets ¿ exist and, in addition, we show that the minimal vectorial completions of the family ¿ provide a decomposition of the clutter of the inclusion-minimal elements of ¿. The computation of such vectorial decomposition of clutters is also discussed in some cases.Peer ReviewedPostprint (author’s final draft

Potentials of mean force in acidic proton transfer reactions in constrained geometries

Free energy barriers associated with the transfer of an excess proton in water and related to the potentials of mean force in proton transfer episodes have been computed in a wide range of thermodynamic states, from low-density amorphous ices to high-temperature liquids under the critical point for unconstrained and constrained systems. The latter were represented by set-ups placed inside hydrophobic graphene slabs at the nanometric scale allocating a few water layers, namely one or two in the narrowest case. Water–proton and carbon–proton forces were modelled with a Multi-State Empirical Valence Bond method. As a general trend, a competition between the effects of confinement and temperature is observed on the local hydrogen-bonded structures around the lone proton and, consequently, on the mean force exerted by its environment on the water molecule carrying the proton. Free energy barriers estimated from the computed potentials of mean force tend to rise with the combined effect of increasing temperatures and the packing effect due to a larger extent of hydrophobic confinement. The main reason observed for such enhancement of the free energy barriers was the breaking of the second coordination shell around the lone proton.Postprint (author's final draft

Automation of the Continuous Integration (CI) - Continuous Delivery/Deployment (CD) Software Development

Continuous Integration (CI) is a practice in software development where developers periodically merge code changes in a central shared repository, after which automatic versions and tests are executed. CI entails an automation component (the target of this project) and a cultural one, as developers have to learn to integrate code periodically. The main goal of CI is to reduce the time to feedback over the software integration process, allowing to locate and fix bugs more easily and quickly, thus enhancing it quality while reducing the time to validate and publish new soIn traditional software development, where teams of developers worked on the same project in isolation, often led to problems integrating the resulting code. Due to this isolation, the project was not deliverable until the integration of all its parts, which was tedious and generated errors. The Continuous Integration (CI ) emerged as a practice to solve the problems of traditional methodology, with the aim of improving the quality of the code. This thesis sets out what is it and how Continuous Integration is achieved, the principles that makes it as effective as possible and the processes that follow as a consequence, to thus introduce the context of its objective: the creation of a system that automates the start-up and set-up of an environment to be able to apply the methodology of continuous integration