We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation iut+∆u = ±|u|4/du for large spherically symmetric Lx(R d) initial data in dimensions d ≥ 3. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solutio...

In the early 1980s Yau posed the problem of establishing the rigidity of the Hawking–Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry of their level sets—the horospheres. The regularity theory in the Lorentzian case is considerably more complicated and less complete than in the Riemannian case....

We introduce a new formulation for the motion of curves in R (easily extendable to the motion of surfaces in R), when the original motion generally corresponds to an ill-posed problem such as the Cauchy–Riemann equations. This is, in part, a generalization of our earlier work in [6], where we applied similar ideas to compute flows with highly concentrated vorticity, such as vortex sheets or dip...

We give a unification of several different speaker recognition problems in terms of the general speaker partitioning problem, where a set of N inputs has to be partitioned into subsets according to speaker. We show how to solve this problem in terms of a simple generative model and demonstrate performance on NIST SRE 2006 and 2008 data. Our solution yields probabilistic outputs, which we show h...