493 research outputs found
On Arnold's Problem on the Classifications of Convex Lattice Polytopes
In 1980, V.I. Arnold studied the classification problem for convex lattice
polygons of given area. Since then this problem and its analogues have been
studied by B'ar'any, Pach, Vershik, Liu, Zong and others. Upper bounds for the
numbers of non-equivalent ddimensional convex lattice polytopes of given volume
or cardinality have been achieved. In this paper, by introducing and studying
the unimodular groups acting on convex lattice polytopes, we obtain lower
bounds for the number of non-equivalent d-dimensional convex lattice polytopes
of bounded volume or given cardinality, which are essentially tight.Comment: 15 pages, 3 figure
Minkowski Bisectors, Minkowski Cells, and Lattice Coverings
This article introduces and studies Minkowski Bisectors, Minkowski Cells, and
Lattice Coverings.Comment: 9 figure
On the translative packing densities of tetrahedra and cubooctahedra
In this paper, upper bounds for the densities of the densest translative
tetrahedron packings and the densest translative cubooctahedron packings are
obtained.Comment: 37 pages, 9 figure
Characterization of the Two-Dimensional Six-Fold Lattice Tiles
This paper characterizes all the convex domains which can form six-fold
lattice tilings of the Euclidean plane. They are parallelograms, centrally
symmetric hexagons, one type of centrally symmetric octagons and two types of
decagons.Comment: 20 pages, 14 figures. arXiv admin note: substantial text overlap with
arXiv:1712.0112
A Mathematical Theory for Random Solid Packings
Packings of identical objects have fascinated both scientists and laymen
alike for centuries, in particular the sphere packings and the packings of
identical regular tetrahedra. Mathematicians have tried for centuries to
determine the densest packings; Crystallographers and chemists have been
fascinated by the lattice packings for centuries as well. On the other hand,
physicists, geologists, material scientists and engineers have been challenged
by the mysterious random packings for decades. Experiments have shown the
existence of the dense random sphere packings and the loose random sphere
packings for more than half a century. However, a rigorous definition for them
is still missing. The purpose of this paper is to review the random solid
packings and to create a mathematical theory for it.Comment: 6 pages, 1 figur
Classification of the sublattices of a lattice
In 1945-46, C. L. Siegel proved that an -dimensional lattice of
determinant has at most different sublattices
of determinant . In 1997, the exact number of the
different sublattices of index was determined by Baake. This paper presents
a systematic treatment for counting the sublattices and deduces a formula for
the number of the sublattice classes of determinant .Comment: 8 page
Characterization of the Two-Dimensional Five-Fold Lattice Tiles
In 1885, Fedorov discovered that a convex domain can form a lattice tiling of
the Euclidean plane if and only if it is a parallelogram or a centrally
symmetric hexagon. It is known that there is no other convex domain which can
form a two-, three- or four-fold lattice tiling in the Euclidean plane, but
there is a centrally symmetric convex decagon which can form a five-fold
lattice tiling. This paper characterizes all the convex domains which can form
a five-fold lattice tiling of the Euclidean plane.Comment: 20 pages, 14 figures. arXiv admin note: text overlap with
arXiv:1711.02514, arXiv:1710.0550
A quantitative program for Hadwiger's covering conjecture and Borsuk's partition conjecture
In this article we encode Hadwiger's covering conjecture and Borsuk's
partition conjecture into continuous functions defined on the spaces of convex
bodies, propose a four-step program to approach them, and obtain some partial
results.Comment: 17 pages, four figure
Classification of Convex lattice polytopes
In this paper we study the classification problem of convex lattice ploytopes
with respect to given volume or given cardinality
A note on lattice coverings
This note presents an interesting counterexample to a basic covering problem.Comment: 3 pages, 3 figure
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