5,332 research outputs found
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Stable phantom-divide crossing in two scalar models with matter
We construct cosmological models with two scalar fields, which has the
structure as in the ghost condensation model or k-essence model. The models can
describe the stable phantom crossing, which should be contrasted with one
scalar tensor models, where the infinite instability occurs at the crossing the
phantom divide. We give a general formulation of the reconstruction in terms of
the e-foldings N by including the matter although in the previous two scalar
models, which are extensions of the scalar tensor model, it was difficult to
give a formulation of the reconstruction when we include matters. In the
formulation of the reconstruction, we start with a model with some arbitrary
functions, and find the functions which generates the history in the expansion
of the universe. We also give general arguments for the stabilities of the
models and the reconstructed solution. The viability of a model is also
investigated by comparing the observational data.Comment: 12 pages, 1 figur
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