25,104 research outputs found
The horizon problem for prevalent surfaces
We investigate the box dimensions of the horizon of a fractal surface defined
by a function . In particular we show that a prevalent surface
satisfies the `horizon property', namely that the box dimension of the horizon
is one less than that of the surface. Since a prevalent surface has box
dimension 3, this does not give us any information about the horizon of
surfaces of dimension strictly less than 3. To examine this situation we
introduce spaces of functions with surfaces of upper box dimension at most
\alpha, for \alpha [2,3). In this setting the behaviour of the horizon is
more subtle. We construct a prevalent subset of these spaces where the lower
box dimension of the horizon lies between the dimension of the surface minus
one and 2. We show that in the sense of prevalence these bounds are as tight as
possible if the spaces are defined purely in terms of dimension. However, if we
work in Lipschitz spaces, the horizon property does indeed hold for prevalent
functions. Along the way, we obtain a range of properties of box dimensions of
sums of functions
Label-free, single molecule detection of cytokines using optical microcavities
Interleukin-2 (IL2) is a cytokine that regulates T-cell growth and is used in cancer therapies. By
sensitizing a microcavity sensor surface with anti-IL2 and monitoring the resonant frequency,
single molecules of IL2 can be detected
On the Assouad dimension of self-similar sets with overlaps
It is known that, unlike the Hausdorff dimension, the Assouad dimension of a
self-similar set can exceed the similarity dimension if there are overlaps in
the construction. Our main result is the following precise dichotomy for
self-similar sets in the line: either the \emph{weak separation property} is
satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the
\emph{weak separation property} is not satisfied, in which case the Assouad
dimension is maximal (equal to one).
In the first case we prove that the self-similar set is Ahlfors regular, and
in the second case we use the fact that if the \emph{weak separation property}
is not satisfied, one can approximate the identity arbitrarily well in the
group generated by the similarity mappings, and this allows us to build a
\emph{weak tangent} that contains an interval. We also obtain results in higher
dimensions and provide illustrative examples showing that the
`equality/maximal' dichotomy does not extend to this setting.Comment: 24 pages, 2 figure
Endogenous protease(s) in extracts of Neurospora mycella activate the exonuclease associated with a putative Rec-nuclease
Activation of exonuclease by endogenous protease
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