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- Newsgroups: sci.math
- Path: sparky!uunet!snorkelwacker.mit.edu!mintaka.lcs.mit.edu!zurich.ai.mit.edu!ara
- From: ara@zurich.ai.mit.edu (Allan Adler)
- Subject: Re: What a weird group!!!
- In-Reply-To: jbaez@riesz.mit.edu's message of 15 Oct 92 04:53:12 GMT
- Message-ID: <ARA.92Oct15162611@camelot.ai.mit.edu>
- Sender: news@mintaka.lcs.mit.edu
- Organization: M.I.T. Artificial Intelligence Lab.
- References: <1992Oct15.045312.13089@galois.mit.edu>
- Date: Thu, 15 Oct 1992 21:26:11 GMT
- Lines: 40
-
- The Bohr compactification of the reals is the dual group of the group R_d,
- where R_d denotes the additive group of the reals with the discrete
- topology. Since the group R_d is the direct limit of its finitely generated
- subgroups, which are free abelian, the Bohr compactification is the
- inverse limit of tori. Since the group R_d is isomorphic to the direct sum
- of two copies of itself, the Bohr compactification of R is topologically
- isomorphic to the product of two copies of itself. Actually, R_d is
- isomorphic to the direct sum of continuum many copies of itself, so
- the Bohr compactification is topologically isomorphic to the product
- of continuum many copies of itself. The cardinality of the Bohr
- compactification of R is 2^c, where c is the cardinality of R.
-
- To get a little bit of a picture of what the Bohr compactification of R
- might look like, consider something simpler: the dual group of the
- additive group Q_d of rationals (with the discrete topology). It too can
- be described as an inverse limit of tori, actually of circle groups,
- but there is another way to describe it which is quite satisfying:
- the dual of Q_d is topologically isomorphic to the additive group of
- adeles modulo the additive group of rationals. Since the group R_d is
- the direct sum of continuum many copies of Q_d, it follows that the
- Bohr compactification of R is the product of continuum many copies of
- (adeles mod rationals).
-
- The relation to almost period functions is that a function on R is
- almost periodic if it extends continuously to the Bohr compactification
- of R. Since the Bohr compactification is a compact abelian group, one
- can take the Fourier transform there and get a function on the dual
- group of the Bohr compactification, i.e. on R_d. Thus, one represents
- an almost periodic function as a series involving exponentials exp(2pi i r x),
- where the various r are not necessarily commensurable (actually, as an
- integral over R_d, which has counting measure. The support of such an
- integrand must be countable, hence the integral is a series.).
-
- Contributions for periodontal surgery may be mailed to:
- Allan Adler
- 36 Rolens Drive, Apt. C4
- Kingston, RI 02881
-
- Allan Adler
- ara@altdorf.ai.mit.edu
-