We evaluate information theoretic quantities that quantify complexity in terms of k-th order statistical dependencies that cannot be reduced to interactions among k − 1 random variables. Using symbolic dynamics of coupled maps and cellular automata as model systems, we demonstrate that these measures are able to identify complex dynamical regimes.

We prove some Bernstein type theorems for entire space-like submanifolds in pseudo-Euclidean space and as a corollary, we give a new proof of the Calabi-Pogorelov theorem for Monge-Ampère equations.

We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space R n , the hyperbolic space H n and a Riemannian manifold S n (n ≥ 3) with the Schwarzschild metric to any Riemannian manifold N .