( Version 1.5.1 )
March 29, 2006
Stefan Kohl
e-mail: kohl@mathematik.uni-stuttgart.de
WWW: http://www.cip.mathematik.uni-stuttgart.de/~kohlsn
Address:
Institut für Geometrie und Topologie
Universität Stuttgart
70550 Stuttgart
Germany
The RCWA package provides methods for investigating Residue Class-Wise Affine groups by means of computation. Residue class-wise affine groups are permutation groups acting on the integers, whose elements are bijective residue class-wise affine mappings. Typically they are infinite.
A mapping f: Z -> Z is called residue class-wise affine provided that there is a positive integer m such that the restrictions of f to the residue classes (mod m) are all affine. This means that for any residue class r(m) in Z/mZ there are coefficients a_r(m), b_r(m), c_r(m) in Z such that the restriction of the mapping f to the set r(m) = {r + km | k in Z} is given by
a_r(m) * n + b_r(m) f|_r(m): r(m) -> Z, n |-> -------------------. c_r(m)
Residue class-wise affine groups are countable. "Many" of them act multiply transitively on Z or on subsets thereof. Only relatively basic facts about their structure are known so far. This package is intended to serve as a tool for obtaining a better understanding of their rich and interesting group theoretical and combinatorial structure.
Residue class-wise affine groups can be generalized in a natural way to euclidean rings other than the ring of integers. While this package undoubtedly provides most functionality for residue class-wise affine groups over the integers, at least rudimentarily it also covers the cases that the underlying ring is a semilocalization of Z or a polynomial ring in one variable over a finite field.
The original motivation for investigating residue class-wise affine groups comes from the famous 3n+1 Conjecture, which is an assertion about a surjective, but not injective residue class-wise affine mapping.
Residue class-wise affine groups are introduced in the author's thesis Restklassenweise affine Gruppen. This thesis is published at http://deposit.ddb.de/cgi-bin/dokserv?idn=977164071 (Archivserver Deutsche Bibliothek) and at http://elib.uni-stuttgart.de/opus/volltexte/2005/2448/ (OPUS-Datenbank Universität Stuttgart). A copy of this thesis and an english translation thereof are distributed with this package (see thesis/thesis.pdf
resp. thesis/thesis_e.pdf
).
(C) 2003 - 2006 by Stefan Kohl
This package is distributed under the GNU General Public License.
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