We propose a higher-order method for solving nonsmooth optimization problems on manifolds. To obtain superlinear convergence, we apply Riemannian semismooth Newton to nonlinear...

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions related initial initial-boundary value problems. In case heat ut??u=upinRn×R,p>1, nonexistence classical in subcritical range p(n?2)<n+2 has been conjectured a long time, but all know...

We establish superlinear lower bounds on the Vapnik-Chervonen-kis (VC) dimension of neural networks with one hidden layer and local receptive eld neurons. As the main result we show that every reasonably sized standard network of radial basis function (RBF) neurons has VC dimension (W log k), where W is the number of parameters and k the number of nodes. This signiicantly improves the previousl...

Recently, we gave a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the Conjugate Gradient method. Roughly speaking, one may observe superlinear convergence while solving a sequence of (symmetric positive definite) linear systems if the asymptotic eigenvalue distribution of the sequence of the corresponding matrices ...

A generalized Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed forbidden subsequence. One of the fundamental problems in this area is bounding (asymptotically) the maximum length of such sequences. Following Klazar, let Ex(σ, n) be the maximum length of a sequence over an alphabet of size n avoiding subsequences isomorphic to σ. It ha...