We consider situations where rays are reflected according to geometrical optics by a set of unknown obstacles. The aim is recover information about the obstacles from travelling-time data using methods and observations singularities. Suppose that, for disjoint union finitely many strictly convex smooth in Euclidean plane, no line meets more than two them. then give construction complete recover...

Assume a single reaction-diffusion equation has zero as an asymptotically stable stationary point. Then we prove that there exist no localized travelling waves with non-zero speed. That is, if [lim inf |x|→∞ u(x), lim sup|x|→∞ u(x)] is included in an open interval of zero that does not include other stationary points, then the speed has to be zero or the travelling profile u has to be identical...

We investigate the spectrum of the linear operator coming from the sine-Gordon equation linearized about a travelling kink-wave solution. Using various geometric techniques as well as some elementary methods from ODE theory, we find that the point spectrum of such an operator is purely imaginary provided the wave speed c of the travelling wave is not ±1. We then compute the essential spectrum o...