1,566 research outputs found
Simplicial Flat Norm with Scale
We study the multiscale simplicial flat norm (MSFN) problem, which computes
flat norm at various scales of sets defined as oriented subcomplexes of finite
simplicial complexes in arbitrary dimensions. We show that the multiscale
simplicial flat norm is NP-complete when homology is defined over integers. We
cast the multiscale simplicial flat norm as an instance of integer linear
optimization. Following recent results on related problems, the multiscale
simplicial flat norm integer program can be solved in polynomial time by
solving its linear programming relaxation, when the simplicial complex
satisfies a simple topological condition (absence of relative torsion). Our
most significant contribution is the simplicial deformation theorem, which
states that one may approximate a general current with a simplicial current
while bounding the expansion of its mass. We present explicit bounds on the
quality of this approximation, which indicate that the simplicial current gets
closer to the original current as we make the simplicial complex finer. The
multiscale simplicial flat norm opens up the possibilities of using flat norm
to denoise or extract scale information of large data sets in arbitrary
dimensions. On the other hand, it allows one to employ the large body of
algorithmic results on simplicial complexes to address more general problems
related to currents.Comment: To appear in the Journal of Computational Geometry. Since the last
version, the section comparing our bounds to Sullivan's has been expanded. In
particular, we show that our bounds are uniformly better in the case of
boundaries and less sensitive to simplicial irregularit
Tits buildings and K-stability
A polarized variety is K-stable if, for any test configuration, the
Donaldson-Futaki invariant is positive. In this paper, inspired by classical
geometric invariant theory, we describe the space of test configurations as a
limit of a direct system of Tits buildings. We show that the Donaldson-Futaki
invariant, conveniently normalized, is a continuous function on this space. We
also introduce a pseudo-metric on the space of test configurations. Recall that
K-stability can be enhanced by requiring that the Donaldson-Futaki invariant is
positive on any admissible filtration of the co-ordinate ring. We show that
admissible filtrations give rise to Cauchy sequences of test configurations
with respect to the above mentioned pseudo-metric.Comment: 16 pages. To appear on the Proceedings of the Edinburgh Mathematical
Societ
Riemannian simplices and triangulations
We study a natural intrinsic definition of geometric simplices in Riemannian
manifolds of arbitrary dimension , and exploit these simplices to obtain
criteria for triangulating compact Riemannian manifolds. These geometric
simplices are defined using Karcher means. Given a finite set of vertices in a
convex set on the manifold, the point that minimises the weighted sum of
squared distances to the vertices is the Karcher mean relative to the weights.
Using barycentric coordinates as the weights, we obtain a smooth map from the
standard Euclidean simplex to the manifold. A Riemannian simplex is defined as
the image of this barycentric coordinate map. In this work we articulate
criteria that guarantee that the barycentric coordinate map is a smooth
embedding. If it is not, we say the Riemannian simplex is degenerate. Quality
measures for the "thickness" or "fatness" of Euclidean simplices can be adapted
to apply to these Riemannian simplices. For manifolds of dimension 2, the
simplex is non-degenerate if it has a positive quality measure, as in the
Euclidean case. However, when the dimension is greater than two, non-degeneracy
can be guaranteed only when the quality exceeds a positive bound that depends
on the size of the simplex and local bounds on the absolute values of the
sectional curvatures of the manifold. An analysis of the geometry of
non-degenerate Riemannian simplices leads to conditions which guarantee that a
simplicial complex is homeomorphic to the manifold
On the convergence of Regge calculus to general relativity
Motivated by a recent study casting doubt on the correspondence between Regge
calculus and general relativity in the continuum limit, we explore a mechanism
by which the simplicial solutions can converge whilst the residual of the Regge
equations evaluated on the continuum solutions does not. By directly
constructing simplicial solutions for the Kasner cosmology we show that the
oscillatory behaviour of the discrepancy between the Einstein and Regge
solutions reconciles the apparent conflict between the results of Brewin and
those of previous studies. We conclude that solutions of Regge calculus are, in
general, expected to be second order accurate approximations to the
corresponding continuum solutions.Comment: Updated to match published version. Details of numerical calculations
added, several sections rewritten. 9 pages, 4 EPS figure
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