15,227 research outputs found
Geometry of canonical self-similar tilings
We give several different geometric characterizations of the situation in
which the parallel set of a self-similar set can be described
by the inner -parallel set of the associated
canonical tiling , in the sense of \cite{SST}. For example,
if and only if the boundary of the
convex hull of is a subset of , or if the boundary of , the
unbounded portion of the complement of , is the boundary of a convex set. In
the characterized situation, the tiling allows one to obtain a tube formula for
, i.e., an expression for the volume of as a function of
. On the way, we clarify some geometric properties of canonical
tilings.
Motivated by the search for tube formulas, we give a generalization of the
tiling construction which applies to all self-affine sets having empty
interior and satisfying the open set condition. We also characterize the
relation between the parallel sets of and these tilings.Comment: 20 pages, 6 figure
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EARLY HISTOCHEMICAL CHANGES IN IRRADIATED SALIVARY GLANDS AND LYMPH-NODES OF THE RAT.
Tube formulas and complex dimensions of self-similar tilings
We use the self-similar tilings constructed by the second author in
"Canonical self-affine tilings by iterated function systems" to define a
generating function for the geometry of a self-similar set in Euclidean space.
This tubular zeta function encodes scaling and curvature properties related to
the complement of the fractal set, and the associated system of mappings. This
allows one to obtain the complex dimensions of the self-similar tiling as the
poles of the tubular zeta function and hence develop a tube formula for
self-similar tilings in \. The resulting power series in
is a fractal extension of Steiner's classical tube formula for
convex bodies K \ci \bRd. Our sum has coefficients related to the curvatures
of the tiling, and contains terms for each integer , just as
Steiner's does. However, our formula also contains terms for each complex
dimension. This provides further justification for the term "complex
dimension". It also extends several aspects of the theory of fractal strings to
higher dimensions and sheds new light on the tube formula for fractals strings
obtained in "Fractal Geometry and Complex Dimensions" by the first author and
Machiel van Frankenhuijsen.Comment: 41 pages, 6 figures, incorporates referee comments and references to
new result
Catching crabs: a case study in local-scale English conservation
Wells-next-the-Sea and Cromer in Norfolk (England) both rely upon their local
crab populations, since crabbing (gillying) is a major part of their tourist
industry. Compared to a control site with no crabbing, crabs from Wells harbour
and Cromer pier were found to have nearly six times the amount of limb damage.
Crabs caught by the general public had more injuries than crabs caught in
controlled conditions, suggesting the buckets in which the crabs were kept were
to blame. Since there is much evidence that such injuries have negative impacts
on the survival and reproductive success of the shore crab, this is taken as
evidence of non-lethal injury from humans having a population-level effect on
these animals. Questionnaire data demonstrated a public lack of awareness and
want for information, which was then used to obtain funding to produce a
leaflet campaign informing the public of how to crab responsibly. All data
collected is available online at http://dx.doi.org/10.6084/m9.figshare.979288
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