28,597 research outputs found
On the Complexity of Polytope Isomorphism Problems
We show that the problem to decide whether two (convex) polytopes, given by
their vertex-facet incidences, are combinatorially isomorphic is graph
isomorphism complete, even for simple or simplicial polytopes. On the other
hand, we give a polynomial time algorithm for the combinatorial polytope
isomorphism problem in bounded dimensions. Furthermore, we derive that the
problems to decide whether two polytopes, given either by vertex or by facet
descriptions, are projectively or affinely isomorphic are graph isomorphism
hard.
The original version of the paper (June 2001, 11 pages) had the title ``On
the Complexity of Isomorphism Problems Related to Polytopes''. The main
difference between the current and the former version is a new polynomial time
algorithm for polytope isomorphism in bounded dimension that does not rely on
Luks polynomial time algorithm for checking two graphs of bounded valence for
isomorphism. Furthermore, the treatment of geometric isomorphism problems was
extended.Comment: 16 pages; to appear in: Graphs and Comb.; replaces our paper ``On the
Complexity of Isomorphism Problems Related to Polytopes'' (June 2001
Intrinsic data depth for Hermitian positive definite matrices
Nondegenerate covariance, correlation and spectral density matrices are
necessarily symmetric or Hermitian and positive definite. The main contribution
of this paper is the development of statistical data depths for collections of
Hermitian positive definite matrices by exploiting the geometric structure of
the space as a Riemannian manifold. The depth functions allow one to naturally
characterize most central or outlying matrices, but also provide a practical
framework for inference in the context of samples of positive definite
matrices. First, the desired properties of an intrinsic data depth function
acting on the space of Hermitian positive definite matrices are presented.
Second, we propose two computationally fast pointwise and integrated data depth
functions that satisfy each of these requirements and investigate several
robustness and efficiency aspects. As an application, we construct depth-based
confidence regions for the intrinsic mean of a sample of positive definite
matrices, which is applied to the exploratory analysis of a collection of
covariance matrices associated to a multicenter research trial
Schwarzschild Tests of the Wahlquist-Estabrook-Buchman-Bardeen Tetrad Formulation for Numerical Relativity
A first order symmetric hyperbolic tetrad formulation of the Einstein
equations developed by Estabrook and Wahlquist and put into a form suitable for
numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted
to explicit spherical symmetry and tested for accuracy and stability in the
evolution of spherically symmetric black holes (the Schwarzschild geometry).
The lapse and shift which specify the evolution of the coordinates relative to
the tetrad congruence are reset at frequent time intervals to keep the
constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the
spatial coordinate satisfying a kind of minimal rate of strain condition. By
arranging through initial conditions that the constant-time hypersurfaces are
asymptotically hyperbolic, we simplify the boundary value problem and improve
stability of the evolution. Results are obtained for both tetrad gauges
(``Nester'' and ``Lorentz'') of the WEBB formalism using finite difference
numerical methods. We are able to obtain stable unconstrained evolution with
the Nester gauge for certain initial conditions, but not with the Lorentz
gauge.Comment: (accepted by Phys. Rev. D) minor changes; typos correcte
Ramanujan Complexes and bounded degree topological expanders
Expander graphs have been a focus of attention in computer science in the
last four decades. In recent years a high dimensional theory of expanders is
emerging. There are several possible generalizations of the theory of expansion
to simplicial complexes, among them stand out coboundary expansion and
topological expanders. It is known that for every d there are unbounded degree
simplicial complexes of dimension d with these properties. However, a major
open problem, formulated by Gromov, is whether bounded degree high dimensional
expanders, according to these definitions, exist for d >= 2. We present an
explicit construction of bounded degree complexes of dimension d = 2 which are
high dimensional expanders. More precisely, our main result says that the
2-skeletons of the 3-dimensional Ramanujan complexes are topological expanders.
Assuming a conjecture of Serre on the congruence subgroup property, infinitely
many of them are also coboundary expanders.Comment: To appear in FOCS 201
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