107,634 research outputs found
Deterministic Sampling and Range Counting in Geometric Data Streams
We present memory-efficient deterministic algorithms for constructing
epsilon-nets and epsilon-approximations of streams of geometric data. Unlike
probabilistic approaches, these deterministic samples provide guaranteed bounds
on their approximation factors. We show how our deterministic samples can be
used to answer approximate online iceberg geometric queries on data streams. We
use these techniques to approximate several robust statistics of geometric data
streams, including Tukey depth, simplicial depth, regression depth, the
Thiel-Sen estimator, and the least median of squares. Our algorithms use only a
polylogarithmic amount of memory, provided the desired approximation factors
are inverse-polylogarithmic. We also include a lower bound for non-iceberg
geometric queries.Comment: 12 pages, 1 figur
Geometrical effects on energy transfer in disordered open quantum systems
We explore various design principles for efficient excitation energy
transport in complex quantum systems. We investigate energy transfer efficiency
in randomly disordered geometries consisting of up to 20 chromophores to
explore spatial and spectral properties of small natural/artificial
Light-Harvesting Complexes (LHC). We find significant statistical correlations
among highly efficient random structures with respect to ground state
properties, excitonic energy gaps, multichromophoric spatial connectivity, and
path strengths. These correlations can even exist beyond the optimal regime of
environment-assisted quantum transport. For random configurations embedded in
spatial dimensions of 30 A and 50 A, we observe that the transport efficiency
saturates to its maximum value if the systems contain 7 and 14 chromophores
respectively. Remarkably, these optimum values coincide with the number of
chlorophylls in (Fenna-Matthews-Olson) FMO protein complex and LHC II monomers,
respectively, suggesting a potential natural optimization with respect to
chromophoric density.Comment: 11 pages, 10 figures. Expanded from the former appendix to
arXiv:1104.481
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Application of Edwards' statistical mechanics to high dimensional jammed sphere packings
The isostatic jamming limit of frictionless spherical particles from Edwards'
statistical mechanics [Song \emph{et al.}, Nature (London) {\bf 453}, 629
(2008)] is generalized to arbitrary dimension using a liquid-state
description. The asymptotic high-dimensional behavior of the self-consistent
relation is obtained by saddle-point evaluation and checked numerically. The
resulting random close packing density scaling is
consistent with that of other approaches, such as replica theory and density
functional theory. The validity of various structural approximations is
assessed by comparing with three- to six-dimensional isostatic packings
obtained from simulations. These numerical results support a growing accuracy
of the theoretical approach with dimension. The approach could thus serve as a
starting point to obtain a geometrical understanding of the higher-order
correlations present in jammed packings.Comment: 13 pages, 7 figure
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