136 research outputs found
Partitioning the triangles of the cross polytope into surfaces
We present a constructive proof that there exists a decomposition of the
2-skeleton of the k-dimensional cross polytope into closed surfaces
of genus , each with a transitive automorphism group given by the
vertex transitive -action on . Furthermore we show
that for each the 2-skeleton of the (k-1)-simplex is a union
of highly symmetric tori and M\"obius strips.Comment: 13 pages, 1 figure. Minor update. Journal-ref: Beitr. Algebra Geom. /
Contributions to Algebra and Geometry, 53(2):473-486, 201
A Dense Packing of Regular Tetrahedra
We construct a dense packing of regular tetrahedra, with packing density .Comment: full color versio
Lattice-point enumerators of ellipsoids
Minkowski's second theorem on successive minima asserts that the volume of a
0-symmetric convex body K over the covolume of a lattice \Lambda can be bounded
above by a quantity involving all the successive minima of K with respect to
\Lambda. We will prove here that the number of lattice points inside K can also
accept an upper bound of roughly the same size, in the special case where K is
an ellipsoid. Whether this is also true for all K unconditionally is an open
problem, but there is reasonable hope that the inductive approach used for
ellipsoids could be extended to all cases.Comment: 9 page
Multi-resolution image analysis for vehicle detection
Proceeding of: Second Iberian Conference, IbPRIA 2005, Estoril, Portugal, June 7-9, 2005Computer Vision can provide a great deal of assistance to Intelligent Vehicles. In this paper an Advanced Driver Assistance Systems for Vehicle Detection is presented. A geometric model of the vehicle is defined where its energy function includes information of the shape and symmetry of the vehicle and the shadow it produces. A genetic algorithm finds the optimum parameter values. As the algorithm receives information from a road detection module some geometric restrictions can be applied. A multi-resolution approach is used to speed up the algorithm and work in realtime. Examples of real images are shown to validate the algorithm.Publicad
Valuations on lattice polytopes
This survey is on classification results for valuations defined on lattice polytopes that intertwine the special linear group over the integers. The basic real valued valuations, the coefficients of the Ehrhart polynomial, are introduced and their characterization by Betke and Kneser is discussed. More recent results include classification theorems for vector and convex body valued valuations. © Springer International Publishing AG 2017
Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces
Polyhedra and their arrangements have intrigued humankind since the ancient
Greeks and are today important motifs in condensed matter, with application to
many classes of liquids and solids. Yet, little is known about the
thermodynamically stable phases of polyhedrally-shaped building blocks, such as
faceted nanoparticles and colloids. Although hard particles are known to
organize due to entropy alone, and some unusual phases are reported in the
literature, the role of entropic forces in connection with polyhedral shape is
not well understood. Here, we study thermodynamic self-assembly of a family of
truncated tetrahedra and report several atomic crystal isostructures, including
diamond, {\beta}-tin, and high- pressure lithium, as the polyhedron shape
varies from tetrahedral to octahedral. We compare our findings with the densest
packings of the truncated tetrahedron family obtained by numerical compression
and report a new space filling polyhedron, which has been overlooked in
previous searches. Interestingly, the self-assembled structures differ from the
densest packings. We show that the self-assembled crystal structures can be
understood as a tendency for polyhedra to maximize face-to-face alignment,
which can be generalized as directional entropic forces.Comment: Article + supplementary information. 23 pages, 10 figures, 2 table
Rescaled coordinate descent methods for linear programming
We propose two simple polynomial-time algorithms to find a positive solution to Ax=0Ax=0 . Both algorithms iterate between coordinate descent steps similar to von Neumann’s algorithm, and rescaling steps. In both cases, either the updating step leads to a substantial decrease in the norm, or we can infer that the condition measure is small and rescale in order to improve the geometry. We also show how the algorithms can be extended to find a solution of maximum support for the system Ax=0Ax=0 , x≥0x≥0 . This is an extended abstract. The missing proofs will be provided in the full version
Majorations explicites de fonctions de Hilbert-Samuel géométrique et arithmétique
International audienceBy using the -filtration approach of Arakelov geometry, one establishes explicit upper bounds for geometric and arithmetic Hilbert-Samuel function for line bundles on projective varieties and hermitian line bundles on arithmetic projective varieties
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