157,562 research outputs found
Maximal integral point sets over Z^2
Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P = {p1,..., pn} ⊂ Z² a maximal integral point set over Z 2 if all pairwise distances are integral and every additional point pn+1 destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle
Maximal integral point sets in affine planes over finite fields
Motivated by integral point sets in the Euclidean plane, we consider integral
point sets in affine planes over finite fields. An integral point set is a set
of points in the affine plane over a finite field
, where the formally defined squared Euclidean distance of every
pair of points is a square in . It turns out that integral point
sets over can also be characterized as affine point sets
determining certain prescribed directions, which gives a relation to the work
of Blokhuis. Furthermore, in one important sub-case integral point sets can be
restated as cliques in Paley graphs of square order. In this article we give
new results on the automorphisms of integral point sets and classify maximal
integral point sets over for . Furthermore, we give two
series of maximal integral point sets and prove their maximality.Comment: 18 pages, 3 figures, 2 table
Integral point sets over finite fields
We consider point sets in the affine plane where each
Euclidean distance of two points is an element of . These sets
are called integral point sets and were originally defined in -dimensional
Euclidean spaces . We determine their maximal cardinality
. For arbitrary commutative rings
instead of or for further restrictions as no three points on a
line or no four points on a circle we give partial results. Additionally we
study the geometric structure of the examples with maximum cardinality.Comment: 22 pages, 4 figure
Volume computation for polytopes and partition functions for classical root systems
This paper presents an algorithm to compute the value of the inverse Laplace
transforms of rational functions with poles on arrangements of hyperplanes. As
an application, we present an efficient computation of the partition function
for classical root systems.Comment: 55 pages, 14 figures. Maple programs available at
http://www.math.polytechnique.fr/~vergne/work/IntegralPoints.htm
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