2,120 research outputs found
Calibrations for minimal networks in a covering space setting
In this paper we define a notion of calibration for an equivalent approach to
the classical Steiner problem in a covering space setting and we give some
explicit examples. Moreover we introduce the notion of calibration in families:
the idea is to divide the set of competitors in a suitable way, defining an
appropriate (and weaker) notion of calibration. Then, calibrating the candidate
minimizers in each family and comparing their perimeter, it is possible to find
the minimizers of the minimization problem. Thanks to this procedure we prove
the minimality of the Steiner configurations spanning the vertices of a regular
hexagon and of a regular pentagon
Variational approximation of functionals defined on 1-dimensional connected sets: the planar case
In this paper we consider variational problems involving 1-dimensional
connected sets in the Euclidean plane, such as the classical Steiner tree
problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal
partition problems and provide a variational approximation through
Modica-Mortola type energies proving a -convergence result. We also
introduce a suitable convex relaxation and develop the corresponding numerical
implementations. The proposed methods are quite general and the results we
obtain can be extended to -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
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
A multi-material transport problem and its convex relaxation via rectifiable -currents
In this paper we study a variant of the branched transportation problem, that
we call multi-material transport problem. This is a transportation problem,
where distinct commodities are transported simultaneously along a network. The
cost of the transportation depends on the network used to move the masses, as
it is common in models studied in branched transportation. The main novelty is
that in our model the cost per unit length of the network does not depend only
on the total flow, but on the actual quantity of each commodity. This allows to
take into account different interactions between the transported goods. We
propose an Eulerian formulation of the discrete problem, describing the flow of
each commodity through every point of the network. We provide minimal
assumptions on the cost, under which existence of solutions can be proved.
Moreover, we prove that, under mild additional assumptions, the problem can be
rephrased as a mass minimization problem in a class of rectifiable currents
with coefficients in a group, allowing to introduce a notion of calibration.
The latter result is new even in the well studied framework of the
"single-material" branched transportation.Comment: Accepted: SIAM J. Math. Ana
Some Further Evidence about Magnification and Shape in Neural Gas
Neural gas (NG) is a robust vector quantization algorithm with a well-known
mathematical model. According to this, the neural gas samples the underlying
data distribution following a power law with a magnification exponent that
depends on data dimensionality only. The effects of shape in the input data
distribution, however, are not entirely covered by the NG model above, due to
the technical difficulties involved. The experimental work described here shows
that shape is indeed relevant in determining the overall NG behavior; in
particular, some experiments reveal richer and complex behaviors induced by
shape that cannot be explained by the power law alone. Although a more
comprehensive analytical model remains to be defined, the evidence collected in
these experiments suggests that the NG algorithm has an interesting potential
for detecting complex shapes in noisy datasets
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