Consider $N$ bosons in a finite box  $ \Lambda = [0,L]^3 \subset R^3$ interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle $$ \bar{lim|_{\rho \to 0} \bar{lim|_{L \to \infty, N/L^3 \to \rho} ( \ frac{e_0 (\rho) - 4\pi a \rho }{ (4\pi a)^{5/2} (\rho)^{3/2} } ) \le \frac{16}{15 \pi ^2} , $$ where $a$ is the scattering length of the potential. Previously, an upper bound of the form $C16/15\pi^2$ for some constant $C >1$ was obtained in (Erdös et al. in Phys. Rev. A 78:053627, 2008). Our result proves the upper bound of the prediction by Lee and Yang (Phys. Rev. 105(3):1119–1120, 1957) and Lee et al. (Phys. Rev. 106(6):1135–1145, 1957).

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The computation of a rigid body transformation which optimally aligns a set of measurement points with a surface and related registration problems are studied from the viewpoint of geometry and optimization.We provide a convergence analysis for widely used registration algorithms such as ICP, using either closest points (Besl and McKay [2]) or tangent planes at closest points (Chen and Medioni [4]), and for a recently developed approach based on quadratic approximants of the squared distance function [24]. ICP based on closest points exhibits local linear convergence only. Its counterpart which minimizes squared distances to the tangent planes at closest points is a Gauss-Newton iteration; it achieves local quadratic convergence for a zero residual problem and - if enhanced by regularization and step size control - comes close to quadratic convergence in many realistic scenarios. Quadratically convergent algorithms are based on the approach in [24]. The theoretical results are supported by a number of experiments; there, we also compare the algorithms with respect to global convergence behavior, stability and running time.

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