180 research outputs found
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
represent hyperconvex metric spaces that have the same topology (although not
necessarily the same distance function) as subsets of the plane. We show how to
use this data structure to construct the tight span of a metric space given as
an n x n distance matrix, when the tight span is homeomorphic to a subset of
the plane, in time O(n^2), and to add a single point to a planar tight span in
time O(n). As an application of this construction, we show how to test whether
a given finite metric space embeds isometrically into the Manhattan plane in
time O(n^2), and add a single point to the space and re-test whether it has
such an embedding in time O(n).Comment: 39 pages, 15 figure
Embedding into the rectilinear plane in optimal O*(n^2)
We present an optimal O*(n^2) time algorithm for deciding if a metric space
(X,d) on n points can be isometrically embedded into the plane endowed with the
l_1-metric. It improves the O*(n^2 log^2 n) time algorithm of J. Edmonds
(2008). Together with some ingredients introduced by J. Edmonds, our algorithm
uses the concept of tight span and the injectivity of the l_1-plane. A
different O*(n^2) time algorithm was recently proposed by D. Eppstein (2009).Comment: 12 pages, 13 figure
Shape control of active surfaces inspired by the movement of euglenids
We examine a novel mechanism for active surface morphing inspired by the cell body deformations of euglenids. Actuation is accomplished through in-plane simple shear along prescribed slip lines decorating the surface. Under general non-uniform actuation, such local deformation produces Gaussian curvature, and therefore leads to shape changes. Geometrically, a deformation that realizes the prescribed local shear is an isometric embedding. We explore the possibilities and limitations of this bio- inspired shape morphing mechanism, by first characterizing isometric embeddings un- der axisymmetry, understanding the limits of embeddability, and studying in detail the accessibility of surfaces of zero and constant curvature. Modeling mechanically the active surface as a non-Euclidean plate (NEP), we further examine the mechanism beyond the geometric singularities arising from embeddability, where mechanics and buckling play a decisive role. We also propose a non-axisymmetric actuation strategy to accomplish large amplitude bending and twisting motions of elongated cylindrical surfaces. Besides helping understand how euglenids delicately control their shape, our results may provide the background to engineer soft machine
Distance covariance in metric spaces
We extend the theory of distance (Brownian) covariance from Euclidean spaces,
where it was introduced by Sz\'{e}kely, Rizzo and Bakirov, to general metric
spaces. We show that for testing independence, it is necessary and sufficient
that the metric space be of strong negative type. In particular, we show that
this holds for separable Hilbert spaces, which answers a question of Kosorok.
Instead of the manipulations of Fourier transforms used in the original work,
we use elementary inequalities for metric spaces and embeddings in Hilbert
spaces.Comment: Published in at http://dx.doi.org/10.1214/12-AOP803 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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