Eigenvalue bounds of mixed Steklov problems

We study bounds on the Riesz means of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalue problem on a bounded domain $\Omega$ in $\mathbb{R}^n$. The Steklov-Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov-Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov-Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the appendix which in particular show the asymptotic sharpness of the bounds we obtain.Comment: An appendix by by F. Ferrulli and J. Lagac\'e is added; some changes in the introduction are mad

Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group

We prove that for any domain in the Heisenberg group the (k+1)'th Neumann eigenvalue of the sub-Laplacian is strictly less than the k'th Dirichlet eigenvalue. As a byproduct we obtain similar inequalities for the Euclidean Laplacian with a homogeneous magnetic field.Comment: 10 page

P-CNN: Pose-based CNN Features for Action Recognition

This work targets human action recognition in video. While recent methods typically represent actions by statistics of local video features, here we argue for the importance of a representation derived from human pose. To this end we propose a new Pose-based Convolutional Neural Network descriptor (P-CNN) for action recognition. The descriptor aggregates motion and appearance information along tracks of human body parts. We investigate different schemes of temporal aggregation and experiment with P-CNN features obtained both for automatically estimated and manually annotated human poses. We evaluate our method on the recent and challenging JHMDB and MPII Cooking datasets. For both datasets our method shows consistent improvement over the state of the art.Comment: ICCV, December 2015, Santiago, Chil