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Paper on representation of the Reals (and others from Darmstadt)
The following paper is now available by anonymous ftp from the site at
Imperial College.
A Faithful Computational Model of the Real Numbers
Philipp S{\"u}nderhauf
ABSTRACT. We investigate the representation of real numbers by
sequences of digits, thought of as radix expansions. ``Faithful''
refers to the fact that we overcome the classical problem of
multiple representations for certain numbers. This is established by
employing a suitable quasi-uniform structure on the set of finite
sequences. (The paper contains a motivating introduction to
quasi-uniformities.) The completion of this space adds exactly one
representative for each real number. Moreover, the quasi-uniformity
induced on the set of total elements is exactly the usual uniformity
on the reals. Hence we do also give a faithful representation of
the topological structure of the real numbers.
The quasi-uniformity on our model may be described in a finitary
fashion: There is a base consisting of relations U_n such that in
order to determine whether x U_n y holds, one needs to know only
the first n digits of the sequences x and y.
Among the continuous endofunctions on our model, the uniformly
continuous ones turn out to play a prominent role: They
correspond to continuous endofunctions on the reals.
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ftp instructions
$ftp theory.doc.ic.ac.uk
Name: anonymous
Password: <your e-mail address>
ftp> cd papers/Jung
ftp> binary
ftp> get reals.ps.Z
ftp> bye
$uncompress reals.ps.Z
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This is the place to draw your attention to the directory papers/Jung
at theory.doc.ic.ac.uk which contains several papers from members of the
Darmstadt "AG Domains".
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Philipp S"underhauf
Fachbereich Mathematik (AG1)
Technische Hochschule Darmstadt
Schlo{\ss}gartenstr. 7 fax: x49-6151-164011
D-64289 Darmstadt email: sunderhauf@mathematik.th-darmstadt.de