23,067 research outputs found
Wavefront sets in algebraic quantum field theory
The investigation of wavefront sets of n-point distributions in quantum field
theory has recently acquired some attention stimulated by results obtained with
the help of concepts from microlocal analysis in quantum field theory in curved
spacetime. In the present paper, the notion of wavefront set of a distribution
is generalized so as to be applicable to states and linear functionals on nets
of operator algebras carrying a covariant action of the translation group in
arbitrary dimension. In the case where one is given a quantum field theory in
the operator algebraic framework, this generalized notion of wavefront set,
called "asymptotic correlation spectrum", is further investigated and several
of its properties for physical states are derived. We also investigate the
connection between the asymptotic correlation spectrum of a physical state and
the wavefront sets of the corresponding Wightman distributions if there is a
Wightman field affiliated to the local operator algebras. Finally we present a
new result (generalizing known facts) which shows that certain spacetime points
must be contained in the singular supports of the 2n-point distributions of a
non-trivial Wightman field.Comment: 34 pages, LaTex2
Partially Symmetric Functions are Efficiently Isomorphism-Testable
Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem
requires a randomized algorithm to distinguish functions that are identical to
f up to relabeling of the input variables from functions that are far from
being so. An important open question in property testing is to determine for
which functions f we can test f-isomorphism with a constant number of queries.
Despite much recent attention to this question, essentially only two classes of
functions were known to be efficiently isomorphism testable: symmetric
functions and juntas.
We unify and extend these results by showing that all partially symmetric
functions---functions invariant to the reordering of all but a constant number
of their variables---are efficiently isomorphism-testable. This class of
functions, first introduced by Shannon, includes symmetric functions, juntas,
and many other functions as well. We conjecture that these functions are
essentially the only functions efficiently isomorphism-testable.
To prove our main result, we also show that partial symmetry is efficiently
testable. In turn, to prove this result we had to revisit the junta testing
problem. We provide a new proof of correctness of the nearly-optimal junta
tester. Our new proof replaces the Fourier machinery of the original proof with
a purely combinatorial argument that exploits the connection between sets of
variables with low influence and intersecting families.
Another important ingredient in our proofs is a new notion of symmetric
influence. We use this measure of influence to prove that partial symmetry is
efficiently testable and also to construct an efficient sample extractor for
partially symmetric functions. We then combine the sample extractor with the
testing-by-implicit-learning approach to complete the proof that partially
symmetric functions are efficiently isomorphism-testable.Comment: 22 page
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
New insight on galaxy structure from GALPHAT I. Motivation, methodology, and benchmarks for Sersic models
We introduce a new galaxy image decomposition tool, GALPHAT (GALaxy
PHotometric ATtributes), to provide full posterior probability distributions
and reliable confidence intervals for all model parameters. GALPHAT is designed
to yield a high speed and accurate likelihood computation, using grid
interpolation and Fourier rotation. We benchmark this approach using an
ensemble of simulated Sersic model galaxies over a wide range of observational
conditions: the signal-to-noise ratio S/N, the ratio of galaxy size to the PSF
and the image size, and errors in the assumed PSF; and a range of structural
parameters: the half-light radius and the Sersic index . We
characterise the strength of parameter covariance in Sersic model, which
increases with S/N and , and the results strongly motivate the need for the
full posterior probability distribution in galaxy morphology analyses and later
inferences.
The test results for simulated galaxies successfully demonstrate that, with a
careful choice of Markov chain Monte Carlo algorithms and fast model image
generation, GALPHAT is a powerful analysis tool for reliably inferring
morphological parameters from a large ensemble of galaxies over a wide range of
different observational conditions. (abridged)Comment: Submitted to MNRAS. The submitted version with high resolution
figures can be downloaded from
http://www.astro.umass.edu/~iyoon/GALPHAT/galphat1.pd
Passivity and microlocal spectrum condition
In the setting of vector-valued quantum fields obeying a linear wave-equation
in a globally hyperbolic, stationary spacetime, it is shown that the two-point
functions of passive quantum states (mixtures of ground- or KMS-states) fulfill
the microlocal spectrum condition (which in the case of the canonically
quantized scalar field is equivalent to saying that the two-point function is
of Hadamard form). The fields can be of bosonic or fermionic character. We also
give an abstract version of this result by showing that passive states of a
topological *-dynamical system have an asymptotic pair correlation spectrum of
a specific type.Comment: latex2e, 29 pages. Change in references, typos remove
Goodness-of-fit testing and quadratic functional estimation from indirect observations
We consider the convolution model where i.i.d. random variables having
unknown density are observed with additive i.i.d. noise, independent of the
's. We assume that the density belongs to either a Sobolev class or a
class of supersmooth functions. The noise distribution is known and its
characteristic function decays either polynomially or exponentially
asymptotically. We consider the problem of goodness-of-fit testing in the
convolution model. We prove upper bounds for the risk of a test statistic
derived from a kernel estimator of the quadratic functional based on
indirect observations. When the unknown density is smoother enough than the
noise density, we prove that this estimator is consistent,
asymptotically normal and efficient (for the variance we compute). Otherwise,
we give nonparametric upper bounds for the risk of the same estimator. We give
an approach unifying the proof of nonparametric minimax lower bounds for both
problems. We establish them for Sobolev densities and for supersmooth densities
less smooth than exponential noise. In the two setups we obtain exact testing
constants associated with the asymptotic minimax rates.Comment: Published in at http://dx.doi.org/10.1214/009053607000000118 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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