scholarly journals Parallel Enumeration of Triangulations

10.37236/7318 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Charles Jordan ◽  
Michael Joswig ◽  
Lars Kastner
Keyword(s):  
Euclidean Space ◽  
Tropical Geometry ◽  
Point Sets ◽  
Massive Parallelization

We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing full triangulations only; this case is particularly relevant for tropical geometry. Most importantly, down-flip reverse search allows for massive parallelization, i.e., it scales well even for many cores. Our implementation allows to compute the triangulations of much larger point sets than before.

Geometrical Bijections in Discrete Lattices

1999 ◽  
Vol 8 (1-2) ◽  
pp. 109-129 ◽  
Author(s):  
HANS-GEORG CARSTENS ◽  
WALTER A. DEUBER ◽  
WOLFGANG THUMSER ◽  
ELKE KOPPENRADE
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Point Sets ◽  
Discrete Lattices ◽  
Standard Lattice

We define uniformly spread sets as point sets in d-dimensional Euclidean space that are wobbling equivalent to the standard lattice ℤd. A linear image ϕ(ℤd) of ℤd is shown to be uniformly spread if and only if det(ϕ) = 1. Explicit geometrical and number-theoretical constructions are given. In 2-dimensional Euclidean space we obtain bounds for the wobbling distance for rotations, shearings and stretchings that are close to optimal. Our methods also allow us to analyse the discrepancy of certain billiards. Finally, we take a look at paradoxical situations and exhibit recursive point sets that are wobbling equivalent, but not recursively so.

scholarly journals Higher horospherical limit sets for G-modules over CAT(0)-spaces

2021 ◽  
pp. 1-31
Author(s):  
ROBERT BIERI ◽  
ROSS GEOGHEGAN
Keyword(s):  
Group Theory ◽  
Euclidean Space ◽  
Discrete Group ◽  
Tropical Geometry ◽  
Dimensional Euclidean Space ◽  
Limit Sets ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Suitable Space

Abstract The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

scholarly journals Fixed point sets of group actions on euclidean space

Topology ◽  
1975 ◽  
Vol 14 (4) ◽  
pp. 339-345 ◽  
Author(s):  
Allan L. Edmonds ◽  
Ronnie Lee
Keyword(s):  
Fixed Point ◽  
Euclidean Space ◽  
Group Actions ◽  
Point Sets ◽  
Fixed Point Sets

A Note on Convexly Independent Subsets of an Infinite Set of Points

2002 ◽  
Vol 9 (2) ◽  
pp. 303-307
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Point Sets ◽  
Set Of Points ◽  
Infinite Set

Abstract We consider convexly independent subsets of a given infinite set of points in the plane (Euclidean space) and evaluate the cardinality of such subsets. It is demonstrated, in particular, that situations are essentially different for countable and uncountable point sets.

On Multiple Integral Geometric Integrals and Their Applications to Probability Theory

1970 ◽  
Vol 22 (1) ◽  
pp. 151-163 ◽  
Author(s):  
Franz Streit
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Integral Geometry ◽  
Mathematical Theory ◽  
Probability Space ◽  
Convex Bodies ◽  
Multiple Integral ◽  
Mean Value ◽  
Dimensional Euclidean Space ◽  
Point Sets

It has been pointed out repeatedly in the literature that the methods of integral geometry (a mathematical theory founded by Wilhelm Blaschke and considerably extended by several mathematicians) provide highly suitable means for the solution of problems concerning “geometrical probabilities“ [2; 6; 12; 15]. The possibilities for the application of these integral geometric results to the evaluation of probabilities, satisfying certain conditions of invariance with respect to a group of transformations which acts on the probability space, are obviously not yet exhausted. In this article, such applications are presented. First, some concepts and notation are introduced (§1). In the next section we derive some integral geometric relations (§ 2). These results are generalizations of known systems of formulae and they are valid in the k-dimensional Euclidean space. In § 3, we determine mean-value formulae for the fundamental characteristics of point-sets, generated by randomly placed convex bodies.

The number of non-homogeneous lattice points in n-dimensional point sets

1953 ◽  
Vol 49 (1) ◽  
pp. 156-157 ◽  
Author(s):  
D. B. Sawyer
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Lattice Points ◽  
Dimensional Euclidean Space ◽  
Infinite Volume ◽  
Point Sets ◽  
Homogeneous Lattice ◽  
Set Of Points

Let R be a set of points in n-dimensional Euclidean space, and let Δ′(R) denote the lower bound of the determinants of non-homogeneous lattices which have no point in R. For Δ′(R) to be non-zero it is necessary, as Macbeath has shown (2), that R should have infinite volume.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

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