17,229 research outputs found
Interpolation inequalities in pattern formation
We prove some interpolation inequalities which arise in the analysis of
pattern formation in physics. They are the strong version of some already known
estimates in weak form that are used to give a lower bound of the energy in
many contexts (coarsening and branching in micromagnetics and superconductors).
The main ingredient in the proof of our inequalities is a geometric
construction which was first used by Choksi, Conti, Kohn, and one of the
authors in the study of branching in superconductors
Stability of Curvature Measures
We address the problem of curvature estimation from sampled compact sets. The
main contribution is a stability result: we show that the gaussian, mean or
anisotropic curvature measures of the offset of a compact set K with positive
-reach can be estimated by the same curvature measures of the offset of a
compact set K' close to K in the Hausdorff sense. We show how these curvature
measures can be computed for finite unions of balls. The curvature measures of
the offset of a compact set with positive -reach can thus be approximated
by the curvature measures of the offset of a point-cloud sample. These results
can also be interpreted as a framework for an effective and robust notion of
curvature
Birthday Inequalities, Repulsion, and Hard Spheres
We study a birthday inequality in random geometric graphs: the probability of
the empty graph is upper bounded by the product of the probabilities that each
edge is absent. We show the birthday inequality holds at low densities, but
does not hold in general. We give three different applications of the birthday
inequality in statistical physics and combinatorics: we prove lower bounds on
the free energy of the hard sphere model and upper bounds on the number of
independent sets and matchings of a given size in d-regular graphs.
The birthday inequality is implied by a repulsion inequality: the expected
volume of the union of spheres of radius r around n randomly placed centers
increases if we condition on the event that the centers are at pairwise
distance greater than r. Surprisingly we show that the repulsion inequality is
not true in general, and in particular that it fails in 24-dimensional
Euclidean space: conditioning on the pairwise repulsion of centers of
24-dimensional spheres can decrease the expected volume of their union
Subsampling in Smoothed Range Spaces
We consider smoothed versions of geometric range spaces, so an element of the
ground set (e.g. a point) can be contained in a range with a non-binary value
in . Similar notions have been considered for kernels; we extend them to
more general types of ranges. We then consider approximations of these range
spaces through -nets and -samples (aka
-approximations). We characterize when size bounds for
-samples on kernels can be extended to these more general
smoothed range spaces. We also describe new generalizations for -nets to these range spaces and show when results from binary range spaces can
carry over to these smoothed ones.Comment: This is the full version of the paper which appeared in ALT 2015. 16
pages, 3 figures. In Algorithmic Learning Theory, pp. 224-238. Springer
International Publishing, 201
Applications of Minkowski Functionals to the Statistical Analysis of Dark Matter Models
A new method for the statistical analysis of 3D point processes, based on the
family of Minkowski functionals, is explained and applied to modelled galaxy
distributions generated by a toy-model and cosmological simulations of the
large-scale structure in the Universe. These measures are sensitive to both,
geometrical and topological properties of spatial patterns and appear to be
very effective in discriminating different point processes. Moreover by the
means of conditional subsampling, different building blocks of large-scale
structures like sheets, filaments and clusters can be detected and extracted
from a given distribution.Comment: 13 pages, Latex, 2 gzipped tar-files, to appear in: Proc. ``1st SFB
workshop on Astro-particle physics'', Ringberg, Tegernsee, 199
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