RCWA

Residue Class-Wise Affine Groups

( Version 1.7.2 )

May 3, 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

Abstract

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).

Copyright

(C) 2003 - 2006 by Stefan Kohl

This package is distributed under the GNU General Public License.

Contents

1. Preface
   1.1 Motivation
   1.2 Purpose of this package
   1.3 Scope of this package
   1.4 Acknowledgements
2. Residue Class-Wise Affine Mappings
   2.1 Basic definitions
   2.2 Entering residue class-wise affine mappings
      2.2-1 ClassShift
      2.2-2 ClassReflection
      2.2-3 ClassTransposition
      2.2-4 PrimeSwitch
      2.2-5 RcwaMapping
      2.2-6 LaTeXObj
   2.3 Basic functionality for rcwa mappings
   2.4 Factoring rcwa mappings
      2.4-1 FactorizationIntoCSCRCT
      2.4-2 mKnot
   2.5 Determinant and sign
      2.5-1 Determinant
      2.5-2 Sign
   2.6 Attributes and properties derived from the coefficients
      2.6-1 Multiplier
      2.6-2 Divisor
      2.6-3 PrimeSet
      2.6-4 IsIntegral
      2.6-5 IsClassWiseOrderPreserving
   2.7 Functionality related to the affine partial mappings
      2.7-1 LargestSourcesOfAffineMappings
      2.7-2 Multpk
      2.7-3 SetOnWhichMappingIsClassWiseOrderPreserving
      2.7-4 FixedPointsOfAffinePartialMappings
   2.8 Transition graphs and transition matrices
      2.8-1 TransitionGraph
      2.8-2 OrbitsModulo
      2.8-3 FactorizationOnConnectedComponents
      2.8-4 TransitionMatrix
      2.8-5 Sources
      2.8-6 Sinks
      2.8-7 Loops
   2.9 Trajectories
      2.9-1 Trajectory
      2.9-2 Trajectory
      2.9-3 IncreasingOn
   2.10 Localizations of rcwa mappings of the integers
      2.10-1 LocalizedRcwaMapping
   2.11 Extracting roots of rcwa mappings
      2.11-1 Root
   2.12 Special functions for non-bijective mappings
      2.12-1 RightInverse
      2.12-2 CommonRightInverse
      2.12-3 ImageDensity
   2.13 Probabilistic guesses on the behaviour of trajectories
      2.13-1 LikelyContractionCentre
      2.13-2 GuessedDivergence
   2.14 The categories and families of rcwa mappings
      2.14-1 IsRcwaMapping
      2.14-2 RcwaMappingsFamily
3. Residue Class-Wise Affine Groups
   3.1 Constructing residue class-wise affine groups
      3.1-1 RCWA
      3.1-2 Random
      3.1-3 IsomorphismRcwaGroupOverZ
   3.2 Attributes and properties of rcwa groups
      3.2-1 Modulus
      3.2-2 IsTame
      3.2-3 PrimeSet
   3.3 Membership testing, order computation, permutation- / matrix representations
      3.3-1 \in
      3.3-2 Size
      3.3-3 IsomorphismPermGroup
      3.3-4 IsomorphismMatrixGroup
   3.4 Factoring elements into generators
      3.4-1 PreImagesRepresentative
      3.4-2 PreImagesRepresentatives
   3.5 The action of an rcwa group on the underlying ring R
      3.5-1 IsTransitive
      3.5-2 RepresentativeAction
      3.5-3 RepresentativeActionPreImage
      3.5-4 RepresentativeAction
      3.5-5 ShortOrbits
      3.5-6 OrbitsModulo
      3.5-7 Ball
   3.6 Conjugacy in RCWA(R)
      3.6-1 IsConjugate
      3.6-2 RepresentativeAction
      3.6-3 ShortCycles
      3.6-4 NrConjugacyClassesOfRCWAZOfOrder
   3.7 Restriction and induction
      3.7-1 Restriction
      3.7-2 Restriction
      3.7-3 Induction
      3.7-4 DirectProduct
      3.7-5 WreathProduct
   3.8 Special attributes for tame rcwa groups
      3.8-1 RespectedPartition
      3.8-2 ActionOnRespectedPartition
      3.8-3 IntegralConjugate
      3.8-4 IntegralizingConjugator
   3.9 The categories of rcwa groups
      3.9-1 IsRcwaGroup
4. Examples
   4.1 Factoring Collatz' permutation of the integers
   4.2 An rcwa mapping which seems to be contracting, but very slow
   4.3 Checking a result by P. Andaloro
   4.4 Two examples by Matthews and Leigh
   4.5 Exploring the structure of a wild rcwa group
   4.6 A wild rcwa mapping which has only finite cycles
   4.7 An abelian rcwa group over a polynomial ring
   4.8 An rcwa representation of a small group
   4.9 An rcwa representation of the symmetric group on 10 points
   4.10 Checking for solvability
   4.11 Some examples over (semi)localizations of the integers
   4.12 Twisting 257-cycles into an rcwa mapping with modulus 32
   4.13 The behaviour of the moduli of powers
   4.14 Images and preimages under the Collatz mapping
   4.15 A group which acts 4-transitively on the positive integers
   4.16 A group which acts 3-transitively, but not 4-transitively on Z
   4.17 Grigorchuk groups
   4.18 Forward orbits of a monoid with 2 generators
   4.19 Representations of the free group of rank 2
   4.20 Representations of the modular group PSL(2,Z)
5. The Algorithms Implemented in RCWA
6. Installation and auxiliary functions
   6.1 Requirements
   6.2 Installation
   6.3 The Info class of the package
      6.3-1 InfoRCWA
   6.4 The testing routine
      6.4-1 RCWATest
   6.5 Building the manual
      6.5-1 RCWABuildManual




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