Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u) = - \int_{\rr^d} K(x,y) (u(y)-u(x)) \, dy. Here we consider a kernel $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$ where $\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ means a diffeomorphism on \rr^d. A simple example being a linear function $a(x)= Ax$. The upper and lower bounds that we obtain are given in terms of the Jacobian of $a$ and the integral of $\psi$. Indeed, in the linear case $a(x) = Ax$ we obtain an explicit expression for the first eigenvalue in the whole \rr^d and it is positive when the the determinant of the matrix $A$ is different from one. As an application of our results, we observe that, when the first eigenvalue is positive, there is an exponential decay for the solutions to the associated evolution problem. As a tool to obtain the result, we also study the behaviour of the principal eigenvalue of the nonlocal Dirichlet problem in the ball $B_R$ and prove that it converges to the first eigenvalue in the whole space as $R\to \infty$

A splitting method for the augmented Burgers equation

In this paper we consider a splitting method for the augmented Burgers equation and prove that it is of first order. We also analyze the large-time behavior of the approximated solution by obtaining the first term in the asymptotic expansion. We prove that, when time increases, these solutions be have as the self-similar solutions of the viscous Burgers equation

Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals

We study a singular nonlinear ordinary differential equation on intervals {[}0, R) with R <= +infinity, motivated by the Ginzburg-Landau models in superconductivity and Landau-de Gennes models in liquid crystals. We prove existence and uniqueness of positive solutions under general assumptions on the nonlinearity. Further uniqueness results for sign-changing solutions are obtained for a physically relevant class of nonlinearities. Moreover, we prove a number of fine qualitative properties of the solution that are important for the study of energetic stability

Symmetry and Multiplicity of Solutions in a Two-Dimensional Landau–de Gennes Model for Liquid Crystals

We consider a variational two-dimensional Landau–de Gennes model in the theory of nematic liquid crystals in a disk of radius R. We prove that under a symmetric boundary condition carrying a topological defect of degree k 2 for some given even non-zero integer k, there are exactly two minimizers for all large enough R. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points

Detecting Inspiring Content on Social Media

Inspiration moves a person to see new possibilities and transforms the way they perceive their own potential. Inspiration has received little attention in psychology, and has not been researched before in the NLP community. To the best of our knowledge, this work is the first to study inspiration through machine learning methods. We aim to automatically detect inspiring content from social media data. To this end, we analyze social media posts to tease out what makes a post inspiring and what topics are inspiring. We release a dataset of 5,800 inspiring and 5,800 non-inspiring English-language public post unique ids collected from a dump of Reddit public posts made available by a third party and use linguistic heuristics to automatically detect which social media English-language posts are inspiring.Comment: accepted at ACII 202