250,038 research outputs found
Bounds on three- and higher-distance sets
A finite set X in a metric space M is called an s-distance set if the set of
distances between any two distinct points of X has size s. The main problem for
s-distance sets is to determine the maximum cardinality of s-distance sets for
fixed s and M. In this paper, we improve the known upper bound for s-distance
sets in n-sphere for s=3,4. In particular, we determine the maximum
cardinalities of three-distance sets for n=7 and 21. We also give the maximum
cardinalities of s-distance sets in the Hamming space and the Johnson space for
several s and dimensions.Comment: 12 page
Ptolemaic Indexing
This paper discusses a new family of bounds for use in similarity search,
related to those used in metric indexing, but based on Ptolemy's inequality,
rather than the metric axioms. Ptolemy's inequality holds for the well-known
Euclidean distance, but is also shown here to hold for quadratic form metrics
in general, with Mahalanobis distance as an important special case. The
inequality is examined empirically on both synthetic and real-world data sets
and is also found to hold approximately, with a very low degree of error, for
important distances such as the angular pseudometric and several Lp norms.
Indexing experiments demonstrate a highly increased filtering power compared to
existing, triangular methods. It is also shown that combining the Ptolemaic and
triangular filtering can lead to better results than using either approach on
its own
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
Curvature dark energy reconstruction through different cosmographic distance definitions
In the context of gravity, dark energy is a geometrical
fluid with negative equation of state. Since the function is
not known \emph{a priori}, the need of a model independent reconstruction of
its shape represents a relevant technique to determine which
model is really favored with respect to others. To this aim, we relate
cosmography to a generic and its derivatives in order to
provide a model independent investigation at redshift . Our analysis
is based on the use of three different cosmological distance definitions, in
order to alleviate the duality problem, i.e. the problem of which cosmological
distance to use with specific cosmic data sets. We therefore consider the
luminosity, , flux, , and angular, , distances and we find
numerical constraints by the Union 2.1 supernovae compilation and measurement
of baryonic acoustic oscillations, at . We notice that all
distances reduce to the same expression, i.e. , at first order. Thus, to fix the cosmographic series of observables, we
impose the initial value of by fitting through supernovae
only, in the redshift regime . We find that the pressure of curvature
dark energy fluid is slightly lower than the one related to the cosmological
constant. This indicates that a possible evolving curvature dark energy
realistically fills the current universe. Moreover, the combined use of
and shows that the sign of the acceleration parameter agrees
with theoretical bounds, while its variation, namely the jerk parameter, is
compatible with . Finally, we infer the functional form of
by means of a truncated polynomial approximation, in terms of
fourth order scale factor .Comment: 10 pages, 4 figures, to appear in Annalen der Physi
Higher Distance Energies and Expanders with Structure
We adapt the idea of higher moment energies, originally used in Additive
Combinatorics, so that it would apply to problems in Discrete Geometry. This
new approach leads to a variety of new results, such as
(i) Improved bounds for the problem of distinct distances with local
properties.
(ii) Improved bounds for problems involving expanding polynomials in
(Elekes-Ronyai type bounds) when one or two of the sets have
structure.
Higher moment energies seem to be related to additional problems in Discrete
Geometry, to lead to new elegant theory, and to raise new questions
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