  
  
                                      [1m[4m[31mRCWA[0m
  
  
                        [1m[4m[31mResidue Class-Wise Affine Groups[0m
  
  
                               ( Version 1.3.5 )
  
  
                                January 9, 2006
  
  
                                  Stefan Kohl
  
  
  
  Stefan Kohl
      Email:    [34mmailto:kohl@mathematik.uni-stuttgart.de[0m
      Homepage: [34mhttp://www.cip.mathematik.uni-stuttgart.de/~kohlsn[0m
      Address:  Institut für Geometrie und Topologie
                Universität Stuttgart
                70550 Stuttgart
                Germany
  
  
  
  -------------------------------------------------------
  [1m[4m[31mAbstract[0m
  The  [1mRCWA[0m  package provides methods for computations with [22m[36mR[0mesidue [22m[36mC[0mlass-[22m[36mW[0mise
  [22m[36mA[0mffine  mappings  ([22m[36mrcwa[0m  mappings  for  short)  and  the groups generated by
  bijective mappings of this type.
  
  The  rcwa  mappings are a type of mappings of certain euclidean rings R into
  themselves.  A  standard  choice  is  R = Z. Similar classes of mappings are
  known  as  [22m[36mGeneralized  Collatz  Mappings[0m,  [22m[36mPeriodically Linear Functions[0m or
  [22m[36mNumber Theoretic Functions with Periodic Rational Coefficients[0m. In fact, the
  development  of  this  package  has  partly  been  motivated  by  the 3n+1 -
  conjecture.  A  regularly  updated  comprehensive annotated bibliography for
  this  conjecture which has been compiled by Jeffrey C. Lagarias can be found
  at [34mhttp://arxiv.org/abs/math.NT/0309224[0m.
  
  The  bijective  rcwa  mappings of R form a proper subgroup of Sym(R), called
  RCWA(R).  In  general, computing with arbitrary mappings from R into R is an
  algorithmically  difficult  task.  The  rcwa  mappings  provide  a  class of
  mappings  which  are  accessible  to computations. The investigation of rcwa
  mappings and groups generated by them is the central aim of this package.
  
  An  object  of  particular  interest  is  the  group  RCWA(Z) of all residue
  class-wise affine bijections of the integers.
  
  Residue  class-wise  affine  groups  are  introduced  in the author's thesis
  [22m[36mRestklassenweise    affine    Gruppen[0m.   This   thesis   is   available   at
  [34mhttp://elib.uni-stuttgart.de/opus/volltexte/2005/2448/[0m. A copy of it as well
  as   an   english   translation   is  distributed  with  this  package  (see
  [1mthesis/thesis.pdf[0m resp. [1mthesis/thesis_e.pdf[0m).
  
  
  -------------------------------------------------------
  [1m[4m[31mCopyright[0m
  (C) 2003 - 2005 by Stefan Kohl
  
  This package is distributed under the same conditions as the [1mGAP[0m system.
  
  
  -------------------------------------------------------
  
  
  [1m[4m[31mContent (RCWA)[0m
  
  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 rcwa mappings
      2.2-1 RcwaMapping
      2.2-2 ClassShift
      2.2-3 ClassReflection
      2.2-4 ClassTransposition
      2.2-5 PrimeSwitch
      2.2-6 LaTeXObj
    2.3 Basic functionality for rcwa mappings
    2.4 Factoring rcwa mappings
      2.4-1 FactorizationIntoGenerators
    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
    2.8 Transition graphs and transition matrices
      2.8-1 TransitionGraph
      2.8-2 OrbitsModulo
      2.8-3 FactorizationOnConnectedComponents
      2.8-4 TransitionMatrix
    2.9 Trajectories
      2.9-1 Trajectory
      2.9-2 TrajectoryModulo
      2.9-3 CoefficientsOnTrajectory
      2.9-4 IncreasingOn
    2.10 Special functions for non-bijective mappings and miscellanea
      2.10-1 LikelyContractionCentre
      2.10-2 GuessedDivergence
      2.10-3 ImageDensity
      2.10-4 RightInverse
      2.10-5 CommonRightInverse
    2.11 The categories and families of rcwa mappings
      2.11-1 IsRcwaMapping
      2.11-2 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 IsomorphismRcwaGroup
    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
    3.6 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
    3.8 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 and families of rcwa groups
      3.9-1 IsRcwaGroup
      3.9-2 IntegralRcwaGroupsFamily
  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
  5. The Algorithms Implemented in RCWA
  6. Installation and auxiliary functions
    6.1 Requirements
    6.2 Installation
    6.3 Building the manual
      6.3-1 RCWABuildManual
    6.4 The Info class of the package
      6.4-1 InfoRCWA
    6.5 The testing routine
      6.5-1 RCWATest
  
  
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