51,992 research outputs found
Theoretical investigation of the quantum noise in ghost imaging
Ghost imaging is a method to nonlocally image an object by transmitting pairs
of entangled photons through the object and a reference optical system
respectively. We present a theoretical analysis of the quantum noise in this
imaging technique. The dependence of the noise on the properties of the
apertures in the imaging system are discussed and demonstrated with a numerical
example. For a given source, the resolution and the signal-to-noise ratio
cannot be improved at the same time
Exotic orbits due to spin-spin coupling around Kerr black holes
We report exotic orbital phenomena of spinning test particles orbiting around
a Kerr black hole, i.e., some orbits of spinning particles are asymmetrical
about the equatorial plane. When a nonspinning test particle orbits around a
Kerr black hole in a strong field region, due to relativistic orbital
precessions, the pattern of trajectories is symmetrical about the equatorial
plane of the Kerr black hole. However, the patterns of the spinning particles'
orbit are no longer symmetrical about the equatorial plane for some orbital
configurations and large spins. We argue that these asymmetrical patterns come
from the spin-spin interactions between spinning particles and Kerr black
holes, because the directions of spin-spin forces can be arbitrary, and
distribute asymmetrically about the equatorial plane.Comment: 15 pages, 20 figure
Equivalence of weak and strong modes of measures on topological vector spaces
A strong mode of a probability measure on a normed space can be defined
as a point such that the mass of the ball centred at uniformly
dominates the mass of all other balls in the small-radius limit. Helin and
Burger weakened this definition by considering only pairwise comparisons with
balls whose centres differ by vectors in a dense, proper linear subspace of
, and posed the question of when these two types of modes coincide. We show
that, in a more general setting of metrisable vector spaces equipped with
measures that are finite on bounded sets, the density of and a uniformity
condition suffice for the equivalence of these two types of modes. We
accomplish this by introducing a new, intermediate type of mode. We also show
that these modes can be inequivalent if the uniformity condition fails. Our
results shed light on the relationships between among various notions of
maximum a posteriori estimator in non-parametric Bayesian inference.Comment: 22 pages, 3 figure
A Partially Linear Framework for Massive Heterogeneous Data
We consider a partially linear framework for modelling massive heterogeneous
data. The major goal is to extract common features across all sub-populations
while exploring heterogeneity of each sub-population. In particular, we propose
an aggregation type estimator for the commonality parameter that possesses the
(non-asymptotic) minimax optimal bound and asymptotic distribution as if there
were no heterogeneity. This oracular result holds when the number of
sub-populations does not grow too fast. A plug-in estimator for the
heterogeneity parameter is further constructed, and shown to possess the
asymptotic distribution as if the commonality information were available. We
also test the heterogeneity among a large number of sub-populations. All the
above results require to regularize each sub-estimation as though it had the
entire sample size. Our general theory applies to the divide-and-conquer
approach that is often used to deal with massive homogeneous data. A technical
by-product of this paper is the statistical inferences for the general kernel
ridge regression. Thorough numerical results are also provided to back up our
theory.Comment: 40 pages main text + 40 pages suppl, To appear in Annals of
Statistic
Quasi-invariance of countable products of Cauchy measures under non-unitary dilations
Consider an infinite sequence (Un)n∈N of independent Cauchy random variables, defined by a sequence (δn)n∈N of location parameters and a sequence (γn)n∈N of scale parameters. Let (Wn)n∈N be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence (σnγn)n∈N of scale parameters, with σn≠0 for all n∈N. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of (Un)n∈N and (Wn)n∈N are equivalent if and only if the sequence (|σn|−1)n∈N is square-summable
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