We present new theoretical results which have implications in answering one of the fundamental questions in computer vision: recognition of surfaces and surface shapes. We state the conditions under which: (i) a surface can be recovered, uniquely, from the tangent plane map, in particular from the Gauss map; (ii) a surface shape can be recovered from the metric and the deforming forces. In case...

We show that the Benjamin-Ono equation is globally well-posed in H s (R) for s ≥ 1. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in H s for any s [15]. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.

We consider the defocusing, ˙ H 1-critical Hartree equation for the radial data in all dimensions (n ≥ 5). We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term − I |x|≤A|I| 1/2 |u| 2 ∆ 1 |x| dxdt in the localized Morawetz identity to rule out the possibility of energy concentration, instead of ...

In this paper we prove that the cubic wave equation is globally well posed and scattering for radial initial data satisfying ‖|x|2ǫu0‖Ḣ1/2+ǫ(R3)+‖u0‖Ḣ1/2+ǫ(R3)+‖|x| 2ǫu1‖Ḣ−1/2+ǫ(R3)+‖u1‖Ḣ−1/2+ǫ(R3) < ∞. (0.1) This space of functions is slightly smaller than the general critical space, Ḣ× Ḣ.

In this paper, we show the global well-posedness for periodic gKdV equations in the space H(T), s ≥ 1 2 for quartic case, and s > 5 9 for quintic case. These improve the previous results of Colliander et al in 2004. In particular, the result is sharp in the quartic case.