1,591 research outputs found
Surface tension and interfacial fluctuations in d-dimensional Ising model
The surface tension of rough interfaces between coexisting phases in 2D and
3D Ising models are discussed in view of the known results and some original
calculations presented in this paper. The results are summarised in a formula,
which allows to interpolate the corrections to finite-size scaling between two
and three dimensions. The physical meaning of an analytic continuation to
noninteger values of the spatial dimensionality d is discussed. Lattices and
interfaces with properly defined fractal dimensions should fulfil certain
requirements to possibly have properties of an analytic continuation from
d-dimensional hypercubes. Here 2 appears as the marginal value of d below which
the (d-1)-dimensional interface splits in disconnected pieces. Some
phenomenological arguments are proposed to describe such interfaces. They show
that the character of the interfacial fluctuations at d<2 is not the same as
provided by a formal analytic continuation from d-dimensional hypercubes with d
>= 2. It, probably, is true also for the related critical exponents.Comment: 10 pages, no figures. In the second version changes are made to make
it consistent with the published paper (Sec.2 is completed
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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