Optimal rigidity estimates for nearly umbilical surfaces

A C Estimate for Nearly Umbilical Surfaces

Computational Error Estimates for Born-oppenheimer Molecular Dynamics with Nearly Crossing Potential Surfaces

The difference of the values of observables for the time-independent Schrödinger equation, with matrix valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states and the electron/nuclei mass ratio. The paper first proves an error estimate (depending on the electron/nuclei mass ratio ...

Nearly Optimal Classification for Semimetrics

We initiate the rigorous study of classification in semimetric spaces, which are point sets with a distance function that is non-negative and symmetric, but need not satisfy the triangle inequality. We define the density dimension dens and discover that it plays a central role in the statistical and algorithmic feasibility of learning in semimetric spaces. We compute this quantity for several w...

Rigidity of Polyhedral Surfaces

We study the rigidity of polyhedral surfaces using variational principle. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a non-triangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover...

A Rigidity Theorem for Convex Surfaces

Theorem 1. Let (S2, g) be the two dimensional sphere with a metric of class C1,1 whose Gaussian curvature satisfies 0 ≤ K ≤ 1. Then any simple closed geodesic γ on (S2, g) has length at least 2π. If the length of γ is 2π, then (S2, g) is isometric to the standard round sphere (S2, g0) and γ is a great circle on (S2, g0) or (S 2, g) is isometric to a circular cylinder of circumference 2π capped ...