14,123 research outputs found
Minkowski Tensors of Anisotropic Spatial Structure
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences
The singular set of mean curvature flow with generic singularities
A mean curvature flow starting from a closed embedded hypersurface in
must develop singularities. We show that if the flow has only generic
singularities, then the space-time singular set is contained in finitely many
compact embedded -dimensional Lipschitz submanifolds plus a set of
dimension at most . If the initial hypersurface is mean convex, then all
singularities are generic and the results apply.
In and , we show that for almost all times the evolving
hypersurface is completely smooth and any connected component of the singular
set is entirely contained in a time-slice. For or -convex hypersurfaces
in all dimensions, the same arguments lead to the same conclusion: the flow is
completely smooth at almost all times and connected components of the singular
set are contained in time-slices. A key technical point is a strong
{\emph{parabolic}} Reifenberg property that we show in all dimensions and for
all flows with only generic singularities. We also show that the entire flow
clears out very rapidly after a generic singularity.
These results are essentially optimal
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