The aim of this note is to show that weak relative hyperbolicity of a group relative to a subgroup (or relative hyperbolicity in the sense of Farb) does not imply any natural analogues of some well-known algebraic properties of ordinary hyperbolic groups. Our main tools are combination theorems for weakly relatively hyperbolic groups.

Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change nature original model. In this work, we present a hyperbolicity-preserving stochastic formulation by carefully selecting polynomial chaos approximations to nonlinear terms in equations. We derive sufficient condition preserve hyperbolicity sy...

We prove the so-called Unitary Hyperbolicity Theorem, a result on hyperbolicity of unitary involutions. The analogous previously known results for the orthogonal and symplectic involutions are formal consequences of the unitary one. While the original proofs in the orthogonal and symplectic cases were based on the incompressibility of generalized Severi-Brauer varieties, the proof in the unitar...

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them, previously predicted hyperbolicity is confirmed. The third one provides an example of a ...

We consider C Hénon-like families of diffeomorphisms of R and study the boundary of the region of parameter values for which the nonwandering set is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the loss of hyperbolicity is caused by a first homoclinic or heteroclinic tangency and that uniform hyperbolicity estimates hold uniformly in the parameter up to this bifurcation...