Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8

We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice E8. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E8. The new lattice yields a sphere covering which is more than 12% less dense than the formerly best known given by the lattice A8*. Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a non-simplicial Delone subdivision. We hereby in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid our answer is even strongest possible in a sense.Comment: 13 pages; (v2) major revision: proof of rigidity corrected, full discussion of E8-case included, src of (v3) contains MAGMA program, (v4) some correction

3D shape based reconstruction of experimental data in Diffuse Optical Tomography

Diffuse optical tomography (DOT) aims at recovering three-dimensional images of absorption and scattering parameters inside diffusive body based on small number of transmission measurements at the boundary of the body. This image reconstruction problem is known to be an ill-posed inverse problem, which requires use of prior information for successful reconstruction. We present a shape based method for DOT, where we assume a priori that the unknown body consist of disjoint subdomains with different optical properties. We utilize spherical harmonics expansion to parameterize the reconstruction problem with respect to the subdomain boundaries, and introduce a finite element (FEM) based algorithm that uses a novel 3D mesh subdivision technique to describe the mapping from spherical harmonics coefficients to the 3D absorption and scattering distributions inside a unstructured volumetric FEM mesh. We evaluate the shape based method by reconstructing experimental DOT data, from a cylindrical phantom with one inclusion with high absorption and one with high scattering. The reconstruction was monitored, and we found a 87% reduction in the Hausdorff measure between targets and reconstructed inclusions, 96% success in recovering the location of the centers of the inclusions and 87% success in average in the recovery for the volumes

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. II. Numeric Code and Tests

Based on a second-order approximation of nonlinear force-free magnetic field solutions in terms of uniformly twisted field lines derived in Paper I, we develop here a numeric code that is capable to forward-fit such analytical solutions to arbitrary magnetogram (or vector magnetograph) data combined with (stereoscopically triangulated) coronal loop 3D coordinates. We test the code here by forward-fitting to six potential field and six nonpotential field cases simulated with our analytical model, as well as by forward-fitting to an exactly force-free solution of the Low and Lou (1990) model. The forward-fitting tests demonstrate: (i) a satisfactory convergence behavior (with typical misalignment angles of $\mu \approx 1^\circ-10^\circ$), (ii) relatively fast computation times (from seconds to a few minutes), and (iii) the high fidelity of retrieved force-free $\alpha$-parameters ($\alpha_{\rm fit}/\alpha_{\rm model} \approx 0.9-1.0$ for simulations and $\alpha_{\rm fit}/\alpha_{\rm model} \approx 0.7\pm0.3$ for the Low and Lou model). The salient feature of this numeric code is the relatively fast computation of a quasi-forcefree magnetic field, which closely matches the geometry of coronal loops in active regions, and complements the existing {\sl nonlinear force-free field (NLFFF)} codes based on photospheric magnetograms without coronal constraints.Comment: Solar PHysics, (in press), 25 pages, 11 figure