  
  
                                      [1m[4m[31mRCWA[0m
  
  
                        [1m[4m[31mResidue Class-Wise Affine Groups[0m
  
  
                               ( Version 1.7.2 )
  
  
                                  May 3, 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
  
  
  
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  [1m[4m[31mAbstract[0m
  The  [1mRCWA[0m  package  provides  methods  for  investigating [22m[36mR[0mesidue [22m[36mC[0mlass-[22m[36mW[0mise
  [22m[36mA[0mffine  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 [22m[36mresidue class-wise affine[0m 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
  [22m[36mRestklassenweise    affine    Gruppen[0m.   This   thesis   is   published   at
  [34mhttp://deposit.ddb.de/cgi-bin/dokserv?idn=977164071[0m  (Archivserver  Deutsche
  Bibliothek)  and  at  [34mhttp://elib.uni-stuttgart.de/opus/volltexte/2005/2448/[0m
  (OPUS-Datenbank Universität Stuttgart). A copy of this thesis and an english
  translation thereof are distributed with this package (see [1mthesis/thesis.pdf[0m
  resp. [1mthesis/thesis_e.pdf[0m).
  
  
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  [1m[4m[31mCopyright[0m
  (C) 2003 - 2006 by Stefan Kohl
  
  This package is distributed under the GNU General Public License.
  
  
  -------------------------------------------------------
  
  
  [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 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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