6,471 research outputs found
Noncommutative geometry on trees and buildings
We describe the construction of theta summable and finitely summable spectral
triples associated to Mumford curves and some classes of higher dimensional
buildings. The finitely summable case is constructed by considering the
stabilization of the algebra of the dual graph of the special fiber of the
Mumford curve and a variant of the Antonescu-Christensen spectral geometries
for AF algebras. The information on the Schottky uniformization is encoded in
the spectral geometry through the Patterson-Sullivan measure on the limit set.
Some higher rank cases are obtained by adapting the construction for trees.Comment: 23 pages, LaTeX, 2 eps figures, contributed to a proceedings volum
Harmonious Hilbert curves and other extradimensional space-filling curves
This paper introduces a new way of generalizing Hilbert's two-dimensional
space-filling curve to arbitrary dimensions. The new curves, called harmonious
Hilbert curves, have the unique property that for any d' < d, the d-dimensional
curve is compatible with the d'-dimensional curve with respect to the order in
which the curves visit the points of any d'-dimensional axis-parallel space
that contains the origin. Similar generalizations to arbitrary dimensions are
described for several variants of Peano's curve (the original Peano curve, the
coil curve, the half-coil curve, and the Meurthe curve). The d-dimensional
harmonious Hilbert curves and the Meurthe curves have neutral orientation: as
compared to the curve as a whole, arbitrary pieces of the curve have each of d!
possible rotations with equal probability. Thus one could say these curves are
`statistically invariant' under rotation---unlike the Peano curves, the coil
curves, the half-coil curves, and the familiar generalization of Hilbert curves
by Butz and Moore.
In addition, prompted by an application in the construction of R-trees, this
paper shows how to construct a 2d-dimensional generalized Hilbert or Peano
curve that traverses the points of a certain d-dimensional diagonally placed
subspace in the order of a given d-dimensional generalized Hilbert or Peano
curve.
Pseudocode is provided for comparison operators based on the curves presented
in this paper.Comment: 40 pages, 10 figures, pseudocode include
Hyperorthogonal well-folded Hilbert curves
R-trees can be used to store and query sets of point data in two or more
dimensions. An easy way to construct and maintain R-trees for two-dimensional
points, due to Kamel and Faloutsos, is to keep the points in the order in which
they appear along the Hilbert curve. The R-tree will then store bounding boxes
of points along contiguous sections of the curve, and the efficiency of the
R-tree depends on the size of the bounding boxes---smaller is better. Since
there are many different ways to generalize the Hilbert curve to higher
dimensions, this raises the question which generalization results in the
smallest bounding boxes. Familiar methods, such as the one by Butz, can result
in curve sections whose bounding boxes are a factor larger
than the volume traversed by that section of the curve. Most of the volume
bounded by such bounding boxes would not contain any data points. In this paper
we present a new way of generalizing Hilbert's curve to higher dimensions,
which results in much tighter bounding boxes: they have at most 4 times the
volume of the part of the curve covered, independent of the number of
dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.Comment: Manuscript submitted to Journal of Computational Geometry. An
abstract appeared in the 31st Int Symp on Computational Geometry (SoCG 2015),
LIPIcs 34:812-82
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