Topological singular set of vector-valued maps, I: application to manifold-constrained Sobolev and BV spaces

We introduce an operator $\mathbf{S}$ on vector-valued maps $u$ which has the ability to capture the relevant topological information carried by $u$. In particular, this operator is defined on maps that take values in a closed submanifold $\mathcal{N}$ of the Euclidean space $\mathbb{R}^m$, and coincides with the distributional Jacobian in case $\mathcal{N}$ is a sphere. More precisely, the range of $\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\mathbf{S}$ to characterise strong limits of smooth, $\mathcal{N}$-valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with $\mathcal{N}$-well potentials

Time-like lorentzian minimal submanifolds as singular limits of nonlinear wave equations

We consider the sharp interface limit $\epsilon \to 0$ of the semilinear wave equation $u_{tt} - \Delta u + \nabla W(u)/ \epsilon^2 = 0$ in $\mathbf R^{1+n}$, where $u$ takes values in $\mathbf R^k$, $k = 1,2$, and $W$ is a double-well potential if $k = 1$ and vanishes on the unit circle and is positive elsewhere if $k = 2$. For fixed $\epsilon > 0$ we find some special solutions, constructed around minimal surfaces in $\mathbf R^n$. In the general case, under some additional assumptions, we show that the solutions converge to a Radon measure supported on a time-like $k$-codimensional minimal submanifold of the Minkowski space-time. This result holds also after the appearence of singularities, and enforces the observation made by J. Neu that this semilinear equation can be regarded as an approximation of the Born-Infeld equation

A variational scheme for hyperbolic obstacle problems

We consider an obstacle problem for (possibly non-local) wave equations, and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme. We provide the corresponding numerical implementation and raise some open questions

Colourgrams GUI: A graphical user-friendly interface for the analysis of large datasets of RGB images

Colourgrams GUI is a graphical user-friendly interface developed in order to facilitate the analysis of large datasets of RGB images through the colourgrams approach. Briefly, the colourgrams approach consists in converting a dataset of RGB images into a matrix of one-dimensional signals, the colourgrams, each one codifying the colour content of the corresponding original image. This matrix of signals can be in turn analysed by means of common multivariate statistical methods, such as Principal Component Analysis (PCA) for exploratory analysis of the image dataset, or Partial Least Squares (PLS) regression for the quantification of colour-related properties of interest. Colourgrams GUI allows to easily convert the dataset of RGB images into the colourgrams matrix, to interactively visualize the signals coloured according to qualitative and/or quantitative properties of the corresponding samples and to visualize the colour features corresponding to selected colourgram regions into the image domain. In addition, the software also allows to analyse the colourgrams matrix by means of PCA and PLS