454 research outputs found
Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions
Let be a set of points and a convex -gon in .
We analyze in detail the topological (or discrete) changes in the structure of
the Voronoi diagram and the Delaunay triangulation of , under the convex
distance function defined by , as the points of move along prespecified
continuous trajectories. Assuming that each point of moves along an
algebraic trajectory of bounded degree, we establish an upper bound of
on the number of topological changes experienced by the
diagrams throughout the motion; here is the maximum length of an
-Davenport-Schinzel sequence, and is a constant depending on the
algebraic degree of the motion of the points. Finally, we describe an algorithm
for efficiently maintaining the above structures, using the kinetic data
structure (KDS) framework
Approximate Nearest Neighbor Search Amid Higher-Dimensional Flats
We consider the Approximate Nearest Neighbor (ANN) problem where the input set consists of n k-flats in the Euclidean Rd, for any fixed parameters k 0 is another prespecified parameter. We present an algorithm that achieves this task with n^{k+1}(log(n)/epsilon)^O(1) storage and preprocessing (where the constant of proportionality in the big-O notation depends on d), and can answer a query in O(polylog(n)) time (where the power of the logarithm depends on d and k). In particular, we need only near-quadratic storage to answer ANN queries amidst a set of n lines in any fixed-dimensional Euclidean space. As a by-product, our approach also yields an algorithm, with similar performance bounds, for answering exact nearest neighbor queries amidst k-flats with respect to any polyhedral distance function. Our results are more general, in that they also
provide a tradeoff between storage and query time
A TQFT associated to the LMO invariant of three-dimensional manifolds
We construct a Topological Quantum Field Theory (in the sense of Atiyah)
associated to the universal finite-type invariant of 3-dimensional manifolds,
as a functor from the category of 3-dimensional manifolds with parametrized
boundary, satisfying some additional conditions, to an algebraic-combinatorial
category. It is built together with its truncations with respect to a natural
grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The
TQFT(s) induce(s) a (series of) representation(s) of a subgroup of
the Mapping Class Group that contains the Torelli group. The N=1 truncation
produces a TQFT for the Casson-Walker-Lescop invariant.Comment: 28 pages, 13 postscript figures. Version 2 (Section 1 has been
considerably shorten, and section 3 has been slightly shorten, since they
will constitute a separate paper. Section 4, which contained only announce of
results, has been suprimated; it will appear in detail elsewhere.
Consequently some statements have been re-numbered. No mathematical changes
have been made.
Theory of inelastic scattering from quantum impurities
We use the framework set up recently to compute non-perturbatively inelastic
scattering from quantum impurities [G. Zar\'and {\it et al.}, Phys. Rev. Lett.
{\bf 93}, 107204 (2004)] to study the the energy dependence of the single
particle -matrix and the inelastic scattering cross section for a number of
quantum impurity models. We study the case of the spin two-channel
Kondo model, the Anderson model, and the usual single-channel Kondo
model. We discuss the difference between non-Fermi liquid and Fermi liquid
models and study how a cross-over between the non-Fermi liquid and Fermi liquid
regimes appears in case of channel anisotropy for the two-channel Kondo
model. We show that for the most elementary non-Fermi liquid system, the
two-channel Kondo model, half of the scattering remains inelastic even at the
Fermi energy. Details of the derivation of the reduction formulas and a simple
path integral approach to connect the -matrix to local correlation functions
are also presented.Comment: published versio
Quantum state measurements using multi-pixel photon detectors
The characterization and conditional preparation of multi-photon quantum
states requires the use of photon number resolving detectors. We study the use
of detectors based on multiple avalanche photodiode pixels in this context. We
develop a general model that provides the positive operator value measures for
these detectors. The model incorporates the effect of cross-talk between pixels
which is unique to these devices. We validate the model by measuring coherent
state photon number distributions and reconstructing them with high precision.
Finally, we evaluate the suitability of such detectors for quantum state
tomography and entanglement-based quantum state preparation, highlighting the
effects of dark counts and cross-talk between pixels.Comment: 7 pages, 6 figure
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