Discrete Images, Objects, and Functions in Zn

Frontmatter

1. Neighborhood Structures

Abstract

Nearly 250 years ago, a small paper was published by Leonhard Euler [Eu36]. This paper can be considered the birth-certificate of graph theory. Euler stated in a letter from March 1736:

“Es wurde mir einmal eine Aufgabe über eine Insel vorgelegt, die in der Stadt Königsberg gelegen und von einem Fluß umgeben ist, über welchen sieben Brücken führen, und es wurde gefragt, ob jemand die einzelnen Brücken in einem zusammenhängenden Laufe so durchwandern könne, daß jede Brücke nur einmal überquert wird. Dabei wurde mir auch mitgeteilt, daß bisher weder jemand sich für diese Möglichkeit verbirgt noch jemand bewiesen habe, daß es unmöglich sei, dies zu tun.” [Euler, Ko86] ^T1)

Klaus Voss

2. Incidence Structures

Abstract

Topology is a branch of modern mathematics which deals mainly with infinite (not countable) continuous sets, i.e. with point sets where infinitely many points lie in the surrounding of a point.

Klaus Voss

3. Topological Laws and Properties

Abstract

In section 1.4.2, we have mentioned Clifford’s reflections about the boundary of a point set in a discrete space. Boundaries are very important because they separate different point sets, i.e. the local spatial properties change greatly at boundaries. Further, some topological and geometrical characteristics of objects in discrete spaces can be expressed by sum terms corresponding to single surfaces as the formulas of section 2.4.4 show.

Klaus Voss

4. Geometrical Laws and Properties

Abstract

At the end of the last century, the book “Geometrie der Zahlen” was published by H.Minkowski in Leipzig [Mi96]. Minkowski was mainly interested in number theory. But he had also a deep understanding of geometrical problems. The combination of number theoretical ideas and geometrical conceptions has been proven as a very fruitful approach to a new mathematical discipline namely to the geometry of numbers [Ca59].

Klaus Voss

5. Discrete Functions

Abstract

Let D be any set of elements and let R be any other set (D=R is allowed). If we have a certain law L according to which to each element dϵD an element rϵR is associated, we say that this law is a function defined on D.

Klaus Voss

6. Summary and Symbols

Abstract

In this part of the book, the most important formulas and theorems shall be summarized so that the reader may have a comprehensive survey of the content dealt within the book.

Klaus Voss

7. References

Klaus Voss

Backmatter