We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on “generic-case complexity” we show that if a finitely generated group G has word problem solvable in subexponential time and has a subgroup of finite index which possesses a non-elementary word-hyperbolic quotient group, then t...

Abstract. We consider two new classes of examples of sup-norm blowup in finite time for strictly hyperbolic systems of conservation laws. The explosive growth in amplitude is caused either by a gradient catastrophe or by a singularity in the flux function. The examples show that solutions of uniformly strictly hyperbolic systems can remain as smooth as the initial data until the time of blowup....

We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combinatorial approach in the spirit of Haagerup and prove that for the word length metric d on a hyperbolic group, the Schur multipliers associated with r have uniformly bounded norms for 0 < r ...

The failure of uniform dependence on the data is an interesting property classical solution for a hyperbolic system. In this paper, we consider map Cauchy problem to 2D viscous shallow water equations, which hyperbolic–parabolic We give new approach studying issue nonuniform initial these equations. prove that not uniformly continuous in Sobolev spaces H s × $H^s\times H^{s}$ > 2 $s>2$ .

We construct unbounded strange attractors for vector fields in R that are robust transitive under uniformly small perturbations. Their geometry is reminiscent of geometric Lorenz and other singular hyperbolic attractors, but they contain no equilibria.

We propose and analyze an algorithm for two finite families of nonexpansive maps on a hyperbolic space. Our results refine and generalize several recent and comparable results in uniformly convex Banach spaces as well as CAT (0) spaces.

We consider continuous SL(2,R)-cocycles over a minimal homeomorphism of a compact set K of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.