13,400 research outputs found
How well-proportioned are lens and prism spaces?
The CMB anisotropies in spherical 3-spaces with a non-trivial topology are
analysed with a focus on lens and prism shaped fundamental cells. The
conjecture is tested that well proportioned spaces lead to a suppression of
large-scale anisotropies according to the observed cosmic microwave background
(CMB). The focus is put on lens spaces L(p,q) which are supposed to be oddly
proportioned. However, there are inhomogeneous lens spaces whose shape of the
Voronoi domain depends on the position of the observer within the manifold.
Such manifolds possess no fixed measure of well-proportioned and allow a
predestined test of the well-proportioned conjecture. Topologies having the
same Voronoi domain are shown to possess distinct CMB statistics which thus
provide a counter-example to the well-proportioned conjecture. The CMB
properties are analysed in terms of cyclic subgroups Z_p, and new point of view
for the superior behaviour of the Poincar\'e dodecahedron is found
Homogenization of Ferromagnetic Energies on Poisson Random Sets in the Plane
We prove that by scaling nearest-neighbour ferromagnetic energies
de ned on Poisson random sets in the plane we obtain an isotropic perimeter
energy with a surface tension characterised by an asymptotic formula. The
result relies on proving that cells with `very long' or `very short' edges of the
corresponding Voronoi tessellation can be neglected. In this way we may apply
Geometry Measure Theory tools to de ne a compact convergence, and a characterisation
of metric properties of clusters of Voronoi cells using limit theorems
for subadditive processes
Laws of large numbers in stochastic geometry with statistical applications
Given independent random marked -vectors (points) distributed
with a common density, define the measure , where is
a measure (not necessarily a point measure) which stabilizes; this means that
is determined by the (suitably rescaled) set of points near . For
bounded test functions on , we give weak and strong laws of large
numbers for . The general results are applied to demonstrate that an
unknown set in -space can be consistently estimated, given data on which
of the points lie in , by the corresponding union of Voronoi cells,
answering a question raised by Khmaladze and Toronjadze. Further applications
are given concerning the Gamma statistic for estimating the variance in
nonparametric regression.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ5167 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction
General models of Gibbs Delaunay-Voronoi tessellations, which can be viewed
as extensions of Ord's process, are considered. The interaction may occur on
each cell of the tessellation and between neighbour cells. The tessellation may
also be subjected to a geometric hardcore interaction, forcing the cells not to
be too large, too small, or too flat. This setting, natural for applications,
introduces some theoretical difficulties since the interaction is not
necessarily hereditary. Mathematical results available for studying these
models are reviewed and further outcomes are provided. They concern the
existence, the simulation and the estimation of such tessellations. Based on
these results, tools to handle these objects in practice are presented: how to
simulate them, estimate their parameters and validate the fitted model. Some
examples of simulated tessellations are studied in details
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