Migration in the EU

This document focuses on migration within the EU, in the context of both EU citizens' rights of free movement and residence, and of Member States' diverse citizenship and labour migration laws. It looks into the topic with the intention of clarifying concepts and answering a number of questions: how many EU and non-EU citizens can be counted as migrants within the EU? How do migrants impact the national labour markets and what living conditions do they encounter in their new country of residence

On the density of systems of non-linear spatially homogeneous SPDEs

In this paper, we consider a system of $k$ second order non-linear stochastic partial differential equations with spatial dimension $d \geq 1$, driven by a $q$-dimensional Gaussian noise, which is white in time and with some spatially homogeneous covariance. The case of a single equation and a one-dimensional noise, has largely been studied in the literature. The first aim of this paper is to give a survey of some of the existing results. We will start with the existence, uniqueness and H\"older's continuity of the solution. For this, the extension of Walsh's stochastic integral to cover some measure-valued integrands will be recalled. We will then recall the results concerning the existence and smoothness of the density, as well as its strict positivity, which are obtained using techniques of Malliavin calculus. The second aim of this paper is to show how these results extend to our system of SPDEs. In particular, we give sufficient conditions in order to have existence and smoothness of the density on the set where the columns of the diffusion matrix span $\R^k$. We then prove that the density is strictly positive in a point if the connected component of the set where the columns of the diffusion matrix span $\R^k$ which contains this point has a non void intersection with the support of the law of the solution. We will finally check how all these results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in dimension $d\in \{1,2,3\}$

Combinatorics in the Art of the Twentieth Century

This paper is motivated by a question I asked myself: How can combinatorial structures be used in a work of art? Immediately, other questions arose: Whether there are artists that work or think combinatorially? If so, what works have they produced in this way? What are the similarities and differences between art works produced using combinatorics? This paper presents the first results of the attempt to answer these questions, being a survey of a selection of works that use or contain combinatorics in some way, including music, literature and visual arts, focusing on the twentieth century.Postprint (published version

Genera Esfera: Interacting with a trackball mapped onto a sphere to explore generative visual worlds

Genera Esfera is an interactive installation that allows the audience to interact and easily become a VJ (visual DJ) in a world of generative visuals. It is an animated and generative graphic environment with a music playlist, a visual spherical world related with and suggested by the music, which reacts and evolves. The installation has been presented at MIRA Live Visual Arts Festival 2015, in Barcelona. Genera Esfera was envisioned, developed and programmed on the basis of two initial ideas: first, to generate our spherical planets we need to work with spherical geometry and program 3D graphics; second, the interaction should be easy to understand, proposing a direct mapping between the visuals and the interface. Our main goal is that participants can focus on exploring the graphic worlds rather than concentrate on understanding the interface. For that purpose we use a trackball to map its position onto sphere rotations. In this paper, we present the interactive installation Genera Esfera, the design guidelines, the mathematics behind the generative visuals and its results.Postprint (published version

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise

The Landau Equation for Maxwellian molecules and the Brownian Motion on SO_R(N)

In this paper we prove that the spatially homogeneous Landau equation for Maxwellian molecules can be represented through the product of two elementary processes. The first one is the Brownian motion on the group of rotations. The second one is, conditionally on the first one, a Gaussian process. Using this representation, we establish sharp multi-scale upper and lower bounds for the transition density of the Landau equation, the multi-scale structure depending on the shape of the support of the initial condition.Comment: 3

Estimates for the density of a nonlinear Landau process

The aim of this paper is to obtain estimates for the density of the law of a specific nonlinear diffusion process at any positive bounded time. This process is issued from kinetic theory and is called Landau process, by analogy with the associated deterministic Fokker-Planck-Landau equation. It is not Markovian, its coefficients are not bounded and the diffusion matrix is degenerate. Nevertheless, the specific form of the diffusion matrix and the nonlinearity imply the non-degeneracy of the Malliavin matrix and then the existence and smoothness of the density. In order to obtain a lower bound for the density, the known results do not apply. However, our approach follows the main idea consisting in discretizing the interval time and developing a recursive method. To this aim, we prove and use refined results on conditional Malliavin calculus. The lower bound implies the positivity of the solution of the Landau equation, and partially answers to an analytical conjecture. We also obtain an upper bound for the density, which again leads to an unusual estimate due to the bad behavior of the coefficients