75,661 research outputs found
On the minimum diameter of plane integral point sets
Since ancient times mathematicians consider geometrical objects with integral
side lengths. We consider plane integral point sets , which are
sets of points in the plane with pairwise integral distances where not all
the points are collinear.
The largest occurring distance is called its diameter. Naturally the question
about the minimum possible diameter of a plane integral point set
consisting of points arises. We give some new exact values and describe
state-of-the-art algorithms to obtain them. It turns out that plane integral
point sets with minimum diameter consist very likely of subsets with many
collinear points. For this special kind of point sets we prove a lower bound
for achieving the known upper bound up to a
constant in the exponent.
A famous question of Erd\H{o}s asks for plane integral point sets with no 3
points on a line and no 4 points on a circle. Here, we talk of point sets in
general position and denote the corresponding minimum diameter by
. Recently could be determined via an
exhaustive search.Comment: 12 pages, 5 figure
The Shape of the Level Sets of the First Eigenfunction of a Class of Two Dimensional Schr\"odinger Operators
We study the first Dirichlet eigenfunction of a class of Schr\"odinger
operators with a convex potential V on a domain . We find two length
scales and , and an orientation of the domain , which
determine the shape of the level sets of the eigenfunction. As an intermediate
step, we also establish bounds on the first eigenvalue in terms of the first
eigenvalue of an associated ordinary differential operator.Comment: 56 pages, 3 figure
Enumeration of integral tetrahedra
We determine the numbers of integral tetrahedra with diameter up to
isomorphism for all via computer enumeration. Therefore we give an
algorithm that enumerates the integral tetrahedra with diameter at most in
time and an algorithm that can check the canonicity of a given
integral tetrahedron with at most 6 integer comparisons. For the number of
isomorphism classes of integral matrices with diameter
fulfilling the triangle inequalities we derive an exact formula.Comment: 10 pages, 1 figur
There are integral heptagons, no three points on a line, no four on a circle
We give two configurations of seven points in the plane, no three points in a
line, no four points on a circle with pairwise integral distances. This answers
a famous question of Paul Erd\H{o}s.Comment: 4 pages, 1 figur
Constructing -clusters
A set of -lattice points in the plane, no three on a line and no four on a
circle, such that all pairwise distances and all coordinates are integral is
called an -cluster (in ). We determine the smallest existent
-cluster with respect to its diameter. Additionally we provide a toolbox of
algorithms which allowed us to computationally locate over 1000 different
-clusters, some of them having huge integer edge lengths. On the way, we
exhaustively determined all Heronian triangles with largest edge length up to
.Comment: 18 pages, 2 figures, 2 table
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