3. Lecture III: Stable ergodicity of skew extensions Let f be a diieomorphism of the compact Riemannian manifold M. We say f is partially hyperbolic 9] if there is a continuous Tf-invariant splitting such that Tf expands E u , contracts E s and sup kT s p fk < inf m(T c p f); sup kT c p fk < inf m(T u p f): (Here m(A) = kA ?1 k ?1 .) If the center bundle E c is tangent to a C 1-foliation F of M...

Hyperbolicity or negative curvature of complex networks is the intrinsic geometric proximity of nodes in the graph metric space, which implies an improved network function. Here, we investigate hidden combinatorial geometries in brain-to-brain coordination networks arising through social communications. The networks originate from correlations among EEG signals previously recorded during spoken...

We present local extraction of bifurcation lines together with extraction of their manifolds, a topological feature that has not yet been sufficiently recognized in scientific visualization. The bifurcation lines are extracted by a modification of the vortex core line extraction techniques due to Sujudi-Haimes, and Roth-Peikert, both formulated using the parallel vectors operator. While the for...