• Deutsch
Login

OPUS

  • Home
  • Search
  • Browse
  • Publish
  • FAQ
Search Fields

Refine

Author

  • Sascha Kurz (3)
  • Axel Kohnert (1)
  • Christian Haase (1)

Year of publication

  • 2005 (2)
  • 2007 (1)

Language

  • English (3) (remove)

Keywords

  • integral point sets (3) (remove)

3 search hits

search hits 1 to 3

Sort by

  • Year
  • Year
  • Title
  • Title
  • Author
  • Author
Show/Hide Abstract A bijection between the d-dimensional simplices with distances in {1,2} and the partitions of d+1 (2005)
Christian Haase Sascha Kurz
We give a construction for the d-dimensional simplices with all distances in {1,2} from the set of partitions of d+1.
Show/Hide Abstract A note on Erdös-Diophantine graphs and Diophantine carpets (2005)
Axel Kohnert Sascha Kurz
A Diophantine figure is a set of points on the integer grid $\mathbb{Z}^{2}$ where all mutual Euclidean distances are integers. We also speak of Diophantine graphs. The vertices are points in $\mathbb{Z}^{2}$ (the coordinates)and the edges are labeled with the distance between the two adjacent vertices, which is integral. In this language a Diophantine figure is a complete Diophantine graph. Two Diophantine graphs are equivalent if they only differ by translation or rotation of vertices. Due to a famous theorem of Erdös and Anning there are complete Diophantine graphs which are not contained in larger ones. We call them Erdös-Diophantine graphs. A special class of Diophantine graphs are Diophantine carpets. These are planar triangulations of a subset of the integer grid. We give an effective construction for Erdös-Diophantine graphs and characterize the chromatic number of Diophantine carpets.
Show/Hide Abstract Integral point sets over finite fields (2007)
Sascha Kurz
We consider point sets in the affine plane GF(q)^2 where each Euclidean distance of two points is an element of GF(q). These sets are called integral point sets and were originally defined in m-dimensional Euclidean spaces. We determine their maximal cardinality I(GF(q),2). For arbitrary commutative rings R instead of GF(q) 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.

search hits 1 to 3

OPUS4 Logo

  • Contact
  • Imprint
  • Sitelinks