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