From continua to R-trees

We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X.Comment: This is the version published by Algebraic & Geometric Topology on 1 November 200

Synchronization in disordered Josephson junction arrays: Small-world connections and the Kuramoto model

We study synchronization in disordered arrays of Josephson junctions. In the first half of the paper, we consider the relation between the coupled resistively- and capacitively shunted junction (RCSJ) equations for such arrays and effective phase models of the Winfree type. We describe a multiple-time scale analysis of the RCSJ equations for a ladder array of junctions \textit{with non-negligible capacitance} in which we arrive at a second order phase model that captures well the synchronization physics of the RCSJ equations for that geometry. In the second half of the paper, motivated by recent work on small world networks, we study the effect on synchronization of random, long-range connections between pairs of junctions. We consider the effects of such shortcuts on ladder arrays, finding that the shortcuts make it easier for the array of junctions in the nonzero voltage state to synchronize. In 2D arrays we find that the additional shortcut junctions are only marginally effective at inducing synchronization of the active junctions. The differences in the effects of shortcut junctions in 1D and 2D can be partly understood in terms of an effective phase model.Comment: 31 pages, 21 figure

Character varieties and harmonic maps to R-trees

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible $SL_2({\mathbb C})$ representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an ${\mathbb R}$-tree which is minimal and whose length function is projectively equivalent to the Morgan-Shalen limit of the sequence of representations. We then examine the implications of the existence of a harmonic map when the action on the tree fixes an end.Comment: 12 pages. Latex. to appear in Math. Res. Let

Efficient Processing of Spatial Joins Using R-Trees

Abstract: In this paper, we show that spatial joins are very suitable to be processed on a parallel hardware platform. The parallel system is equipped with a so-called shared virtual memory which is well-suited for the design and implementation of parallel spatial join algorithms. We start with an algorithm that consists of three phases: task creation, task assignment and parallel task execu-tion. In order to reduce CPU- and I/O-cost, the three phases are processed in a fashion that pre-serves spatial locality. Dynamic load balancing is achieved by splitting tasks into smaller ones and reassigning some of the smaller tasks to idle processors. In an experimental performance compar-ison, we identify the advantages and disadvantages of several variants of our algorithm. The most efficient one shows an almost optimal speed-up under the assumption that the number of disks is sufficiently large. Topics: spatial database systems, parallel database systems