861 research outputs found
The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices
In this paper we report on the full classification of Dirichlet-Voronoi
polyhedra and Delaunay subdivisions of five-dimensional translational lattices.
We obtain a complete list of affine types (L-types) of Delaunay
subdivisions and it turns out that they are all combinatorially inequivalent,
giving the same number of combinatorial types of Dirichlet-Voronoi polyhedra.
Using a refinement of corresponding secondary cones, we obtain
contraction types. We report on details of our computer assisted enumeration,
which we verified by three independent implementations and a topological mass
formula check.Comment: 16 page
Schottky Algorithms: Classical meets Tropical
We present a new perspective on the Schottky problem that links numerical
computing with tropical geometry. The task is to decide whether a symmetric
matrix defines a Jacobian, and, if so, to compute the curve and its canonical
embedding. We offer solutions and their implementations in genus four, both
classically and tropically. The locus of cographic matroids arises from
tropicalizing the Schottky-Igusa modular form.Comment: 17 page
Classification of eight dimensional perfect forms
In this paper, we classify the perfect lattices in dimension 8. There are
10916 of them. Our classification heavily relies on exploiting symmetry in
polyhedral computations. Here we describe algorithms making the classification
possible.Comment: 14 page
Computational Approaches to Lattice Packing and Covering Problems
We describe algorithms which address two classical problems in lattice
geometry: the lattice covering and the simultaneous lattice packing-covering
problem. Theoretically our algorithms solve the two problems in any fixed
dimension d in the sense that they approximate optimal covering lattices and
optimal packing-covering lattices within any desired accuracy. Both algorithms
involve semidefinite programming and are based on Voronoi's reduction theory
for positive definite quadratic forms, which describes all possible Delone
triangulations of Z^d.
In practice, our implementations reproduce known results in dimensions d <= 5
and in particular solve the two problems in these dimensions. For d = 6 our
computations produce new best known covering as well as packing-covering
lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our
approach leads to new best known covering lattices. Although we use numerical
methods, we made some effort to transform numerical evidences into rigorous
proofs. We provide rigorous error bounds and prove that some of the new
lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in
Discrete and Computational Geometry, see also
http://fma2.math.uni-magdeburg.de/~latgeo
The decomposition of the hypermetric cone into L-domains
The hypermetric cone \HYP_{n+1} is the parameter space of basic Delaunay
polytopes in n-dimensional lattice. The cone \HYP_{n+1} is polyhedral; one
way of seeing this is that modulo image by the covariance map \HYP_{n+1} is a
finite union of L-domains, i.e., of parameter space of full Delaunay
tessellations.
In this paper, we study this partition of the hypermetric cone into
L-domains. In particular, it is proved that the cone \HYP_{n+1} of
hypermetrics on n+1 points contains exactly {1/2}n! principal L-domains. We
give a detailed description of the decomposition of \HYP_{n+1} for n=2,3,4
and a computer result for n=5 (see Table \ref{TableDataHYPn}). Remarkable
properties of the root system are key for the decomposition of
\HYP_5.Comment: 20 pages 2 figures, 2 table
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