arXiv subjects

Frank Vallentin

Publications and source records attributed to Frank Vallentin.

60 records · Page 4Linked to original sources

arXiv2007

Optimal Embeddings of Distance Regular Graphs into Euclidean Spaces

In this paper we give a lower bound for the least distortion embedding of a distance regular graph into Euclidean space. We use the lower bound for finding the least distortion for Hamming graphs, Johnson graphs, and all strongly regular graphs. Our technique involves semidefinite programming and exploiting the algebra structure of the optimization problem so that the question of finding a lower bound of the least distortion is reduced to an analytic question about orthogonal polynomials.

math.CO↗

arXiv2006

Incremental Algorithms for Lattice Problems

In this short note we give incremental algorithms for the following lattice problems: finding a basis of a lattice, computing the successive minima, and determining the orthogonal decomposition. We prove an upper bound for the number of update steps for every insertion order. For the determination of the orthogonal decomposition we efficiently implement an argument due to Kneser.

math.NT↗

arXiv2005

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.

math.MG↗

arXiv2005

Some six-dimensional rigid forms

One can always decompose Dirichlet-Voronoi polytopes of lattices non-trivially into a Minkowski sum of Dirichlet-Voronoi polytopes of rigid lattices. In this report we show how one can enumerate all rigid positive semidefinite quadratic forms (and thereby rigid lattices) of a given dimension d. By this method we found all rigid positive semidefinite quadratic forms for d = 5 confirming the list of 7 rigid lattices by Baranovskii and Grishukhin. Furthermore, we found out that for d <= 5 the adjacency graph of primitive L-type domains is an infinite tree on which GL_d(Z) acts. On the other hand, we demonstrate that in d = 6 we face a combinatorial explosion.

math.MG↗

arXiv2004

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.

math.MG↗

arXiv2004

A Note on Space Tiling Zonotopes

In 1908 Voronoi conjectured that every convex polytope which tiles space face-to-face by translations is affinely equivalent to the Dirichlet-Voronoi polytope of some lattice. In 1999 Erdahl proved this conjecture for the special case of zonotopes. A zonotope is a projection of a regular cube under some affine transformation. In 1975 McMullen showed several equivalent conditions for a zonotope to be a space tiling zonotope, i.e. a zonotope which admits a face-to-face tiling of space by translations. Implicitly, he related space tiling zonotopes to a special class of oriented matroids (regular matroids). We will extend his result to give a new proof of Voronoi's conjecture for zonotopes using oriented matroids. This enables us to distinguish between combinatorial and metrical properties and to apply the fact that oriented matroids considered here have an essentially unique realization. Originally, this is a theorem due to Brylawski and Lucas. By using oriented matroid duality we interpret a part of McMullen's arguments as an elegant geometric proof of this theorem in the special case of real numbers.

math.MG↗