4,112 research outputs found
The rapid points of a complex oscillation
By considering a counting-type argument on Brownian sample paths, we prove a
result similar to that of Orey and Taylor on the exact Hausdorff dimension of
the rapid points of Brownian motion. Because of the nature of the proof we can
then apply the concepts to so-called complex oscillations (or 'algorithmically
random Brownian motion'), showing that their rapid points have the same
dimension.Comment: 11 page
The Fourier dimension of Brownian limsup fractals
Robert Kaufman's proof that the set of rapid points of Brownian motion has a
Fourier dimension equal to its Hausdorff dimension was first published in 1974.
A study of the proof of the original paper revealed several gaps in the
arguments and a slight inaccuracy in the main theorem. This paper presents a
new version of the construction and incorporates some recent results in order
to establish a corrected version of Kaufman's theorem. The method of proof can
then be extended to show that functionally determined rapid points of Brownian
motion also form Salem sets for absolutely continuous functions of finite
energy.Comment: 11 page
Zeno machines and hypercomputation
This paper reviews the Church-Turing Thesis (or rather, theses) with
reference to their origin and application and considers some models of
"hypercomputation", concentrating on perhaps the most straight-forward option:
Zeno machines (Turing machines with accelerating clock). The halting problem is
briefly discussed in a general context and the suggestion that it is an
inevitable companion of any reasonable computational model is emphasised. It is
hinted that claims to have "broken the Turing barrier" could be toned down and
that the important and well-founded role of Turing computability in the
mathematical sciences stands unchallenged.Comment: 11 pages. First submitted in December 2004, substantially revised in
July and in November 2005. To appear in Theoretical Computer Scienc
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