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<Abstract>

The &RCWA; package provides methods for computations with
<E>R</E>esidue <E>C</E>lass-<E>W</E>ise <E>A</E>ffine mappings
(<E>rcwa</E> mappings for short) and the groups generated by bijective
mappings of this type. <P/>

The rcwa mappings are a type of mappings of certain euclidean rings
<M>R</M> into themselves. A standard choice is <M>R = \mathbb{Z}</M>.
Similar classes of mappings are known as <E>Generalized Collatz Mappings</E>,
<E>Periodically Linear Functions</E> or <E>Number Theoretic Functions with
Periodic Rational Coefficients</E>. In fact, the development of this package
has partly been motivated by the <M>3n+1</M> - conjecture.
A regularly updated comprehensive annotated bibliography for this conjecture
which has been compiled by Jeffrey&nbsp;C. Lagarias can be found at
<URL>http://arxiv.org/abs/math.NT/0309224</URL>. <P/>

The bijective rcwa mappings of <M>R</M> form a proper subgroup of
Sym(<M>R</M>), called RCWA(<M>R</M>). In general, computing with arbitrary
mappings from <M>R</M> into <M>R</M> 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. <P/>

An object of particular interest is the group RCWA(<Alt Only="LaTeX">
<M>\mathbb{Z}</M></Alt><Alt Not="LaTeX"><M>Z</M></Alt>) of all residue
class-wise affine bijections of the integers. <P/>

Residue class-wise affine groups are introduced in the author's thesis
<E>Restklassenweise affine Gruppen</E>. This thesis is available at
<URL>http://elib.uni-stuttgart.de/opus/volltexte/2005/2448/</URL>.
A copy of it as well as an english translation is distributed with this
package (see <F>thesis/thesis.pdf</F> resp.
<F>thesis/thesis&uscore;e.pdf</F>).

</Abstract>

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