26 research outputs found
On the sum of the Voronoi polytope of a lattice with a zonotope
A parallelotope is a polytope that admits a facet-to-facet tiling of
space by translation copies of along a lattice. The Voronoi cell
of a lattice is an example of a parallelotope. A parallelotope can be
uniquely decomposed as the Minkowski sum of a zone closed parallelotope and
a zonotope , where is the set of vectors used to generate the
zonotope. In this paper we consider the related question: When is the Minkowski
sum of a general parallelotope and a zonotope a parallelotope? We give
two necessary conditions and show that the vectors have to be free. Given a
set of free vectors, we give several methods for checking if is
a parallelotope. Using this we classify such zonotopes for some highly
symmetric lattices.
In the case of the root lattice , it is possible to give a more
geometric description of the admissible sets of vectors . We found that the
set of admissible vectors, called free vectors, is described by the well-known
configuration of lines in a cubic. Based on a detailed study of the
geometry of , we give a simple characterization of the
configurations of vectors such that is a
parallelotope. The enumeration yields maximal families of vectors, which
are presented by their description as regular matroids.Comment: 30 pages, 4 figures, 4 table
Incompressible Lagrangian fluid flow with thermal coupling
In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version