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SUMMARY:Kolmogorov complexity and Fourier aspects of Brownian motion - Fou
 che\, W (University of South Africa)
DTSTART:20120705T130000Z
DTEND:20120705T133000Z
UID:TALK38870@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:It is well-known that the notion of randomness\, suitably refi
 ned\, goes a long way in dealing with the tension between the ``incompatab
 ility of shortest descriptions and of effecting the most-economical algori
 thmical processing" Manin(2006). In this work\, we continue to explore thi
 s interplay between short descriptions and randomness in the context of Br
 ownian motion and its associated geometry. In this way one sees how random
  phenomena associated with the geometry of Brownian motion\, are implicitl
 y enfolded in each real number which is complex in the sense of Kolmogorov
 . These random phenomena range from fractal geometry\, Fourier analysis an
 d non-classical noises in quantum physics. In this talk we shall discuss c
 ountable dense random sets as the appear in the theory of Brownian motion 
 in the context of algorithmic randomness. We shall also discuss applicatio
 ns to Fourier analysis. In particular\, we also discuss the images of cert
 ain $Pi_2^0$ perfect sets of Hausdorff dimension zero under a complex osci
 llation (which is also known as an algorithmically random Brownian motion)
 . This opens the way to relate certain non-classical noises to Kolmogorov 
 complexity. For example\, the work of the present work enables one to repr
 esent Warren's splitting noise directly in terms of infinite binary string
 s which are Kolmogorov-Chaitin-Martin-L&ouml\;f random.\n
LOCATION:Seminar Room 1\, Newton Institute
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