164,937 research outputs found
The number of generalized balanced lines
Let be a set of red points and blue points in general
position in the plane, with . A line determined by them is said
to be balanced if in each open half-plane bounded by the difference
between the number of red points and blue points is . We show that every set
as above has at least balanced lines. The main techniques in the proof
are rotations and a generalization, sliding rotations, introduced here.Comment: 6 pages, 3 figures, several typos fixed, reference adde
Decomposition tables for experiments I. A chain of randomizations
One aspect of evaluating the design for an experiment is the discovery of the
relationships between subspaces of the data space. Initially we establish the
notation and methods for evaluating an experiment with a single randomization.
Starting with two structures, or orthogonal decompositions of the data space,
we describe how to combine them to form the overall decomposition for a
single-randomization experiment that is ``structure balanced.'' The
relationships between the two structures are characterized using efficiency
factors. The decomposition is encapsulated in a decomposition table. Then, for
experiments that involve multiple randomizations forming a chain, we take
several structures that pairwise are structure balanced and combine them to
establish the form of the orthogonal decomposition for the experiment. In
particular, it is proven that the properties of the design for such an
experiment are derived in a straightforward manner from those of the individual
designs. We show how to formulate an extended decomposition table giving the
sources of variation, their relationships and their degrees of freedom, so that
competing designs can be evaluated.Comment: Published in at http://dx.doi.org/10.1214/09-AOS717 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Balanced partitions of 3-colored geometric sets in the plane
Let SS be a finite set of geometric objects partitioned into classes or colors . A subset S'¿SS'¿S is said to be balanced if S'S' contains the same amount of elements of SS from each of the colors. We study several problems on partitioning 33-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m2m lines of each color, there is a segment intercepting mm lines of each color. (b) Given nn red points, nn blue points and nn green points on any closed Jordan curve ¿¿, we show that for every integer kk with 0=k=n0=k=n there is a pair of disjoint intervals on ¿¿ whose union contains exactly kk points of each color. (c) Given a set SS of nn red points, nn blue points and nn green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of SS.Peer ReviewedPostprint (published version
Effect of Loss on Multiplexed Single-Photon Sources
An on-demand single-photon source is a key requirement for scaling many
optical quantum technologies. A promising approach to realize an on-demand
single-photon source is to multiplex an array of heralded single-photon sources
using an active optical switching network. However, the performance of
multiplexed sources is degraded by photon loss in the optical components and
the non-unit detection efficiency of the heralding detectors. We provide a
theoretical description of a general multiplexed single-photon source with
lossy components and derive expressions for the output probabilities of
single-photon emission and multi-photon contamination. We apply these
expressions to three specific multiplexing source architectures and consider
their tradeoffs in design and performance. To assess the effect of lossy
components on near- and long-term experimental goals, we simulate the
multiplexed sources when used for many-photon state generation under various
amounts of component loss. We find that with a multiplexed source composed of
switches with ~0.2-0.4 dB loss and high efficiency number-resolving detectors,
a single-photon source capable of efficiently producing 20-40 photon states
with low multi-photon contamination is possible, offering the possibility of
unlocking new classes of experiments and technologies.Comment: Journal versio
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