Instability results for the damped wave equation in unbounded domains

We extend some previous results for the damped wave equation in bounded domains in Euclidean spaces to the unbounded case. In particular, we show that if the damping term is of the form $\alpha a$ with bounded $a$ taking on negative values on a set of positive measure, then there will always exist unbounded solutions for sufficiently large positive $\alpha$. In order to prove these results, we generalize some existing results on the asymptotic behaviour of eigencurves of one-parameter families of Schrodinger operators to the unbounded case, which we believe to be of interest in their own right.Comment: LaTeX, 19 pages; to appear in J. Differential Equation

Narrowing the Gap: Random Forests In Theory and In Practice

Despite widespread interest and practical use, the theoretical properties of random forests are still not well understood. In this paper we contribute to this understanding in two ways. We present a new theoretically tractable variant of random regression forests and prove that our algorithm is consistent. We also provide an empirical evaluation, comparing our algorithm and other theoretically tractable random forest models to the random forest algorithm used in practice. Our experiments provide insight into the relative importance of different simplifications that theoreticians have made to obtain tractable models for analysis.Comment: Under review by the International Conference on Machine Learning (ICML) 201

Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).Comment: 26 pages, 20 figure