20,543 research outputs found
Separating invariants for the basic G_a-actions
We explicitly construct a finite set of separating invariants for the basic
\Ga-actions. These are the finite dimensional indecomposable rational linear
representations of the additive group \Ga of a field of characteristic zero,
and their invariants are the kernel of the Weitzenb\"ock derivation
.Comment: 10 page
The Brownian fan
We provide a mathematical study of the modified Diffusion Monte Carlo (DMC)
algorithm introduced in the companion article \cite{DMC}. DMC is a simulation
technique that uses branching particle systems to represent expectations
associated with Feynman-Kac formulae. We provide a detailed heuristic
explanation of why, in cases in which a stochastic integral appears in the
Feynman-Kac formula (e.g. in rare event simulation, continuous time filtering,
and other settings), the new algorithm is expected to converge in a suitable
sense to a limiting process as the time interval between branching steps goes
to 0. The situation studied here stands in stark contrast to the "na\"ive"
generalisation of the DMC algorithm which would lead to an exponential
explosion of the number of particles, thus precluding the existence of any
finite limiting object. Convergence is shown rigorously in the simplest
possible situation of a random walk, biased by a linear potential. The
resulting limiting object, which we call the "Brownian fan", is a very natural
new mathematical object of independent interest.Comment: 53 pages, 2 figures. Formerly 2nd part of arXiv:1207.286
Probing correlations of early magnetic fields using mu-distortion
The damping of a non-uniform magnetic field between the redshifts of about
and injects energy into the photon-baryon plasma and causes the
CMB to deviate from a perfect blackbody spectrum, producing a so-called
-distortion. We can calculate the correlation of
this distortion with the temperature anisotropy of the CMB to search for a
correlation between the magnetic field and the
curvature perturbation ; knowing the
correlation would help us distinguish between different models of
magnetogenesis. Since the perturbations which produce the -distortion will
be much smaller scale than the relevant density perturbations, the observation
of this correlation is sensitive to the squeezed limit of , which is naturally parameterized by (a
parameter defined analogously to ). We find that a PIXIE-like
CMB experiments has a signal to noise , where is the magnetic field's
strength on -distortion scales normalized to today's redshift; thus, a 10
nG field would be detectable with . However, if
the field is of inflationary origin, we generically expect it to be accompanied
by a curvature bispectrum induced by the magnetic
field. For sufficiently small magnetic fields, the signal will dominate, but for nG, one would have
to consider the specifics of the inflationary magnetogenesis model.
We also discuss the potential post-magnetogenesis sources of a correlation and explain why there will be no contribution from
the evolution of the magnetic field in response to the curvature perturbation.Comment: 23 pages, 1 figure. v2: Noted that a competing effect could
potentially be smaller than originally stated. Fixed references. Matches JCAP
versio
An introduction to moduli stacks, with a view towards Higgs bundles on algebraic curves
This article is based in part on lecture notes prepared for the summer school
"The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles" at the
Institute for Mathematical Sciences at the National University of Singapore in
July of 2014. The aim is to provide a brief introduction to algebraic stacks,
and then to give several constructions of the moduli stack of Higgs bundles on
algebraic curves. The first construction is via a "bootstrap" method from the
algebraic stack of vector bundles on an algebraic curve. This construction is
motivated in part by Nitsure's GIT construction of a projective moduli space of
semi-stable Higgs bundles, and we describe the relationship between Nitsure's
moduli space and the algebraic stacks constructed here. The third approach is
via deformation theory, where we directly construct the stack of Higgs bundles
using Artin's criterion.Comment: 145 pages, AMS LaTeX, to appear in the NUS IMS Lecture Note Series on
The Geometry, Topology, and Physics of Moduli Spaces of Higgs Bundle
Theoretical Estimates of Intrinsic Galaxy Alignment
It has recently been argued that the observed ellipticities of galaxies may
be determined at least in part by the primordial tidal gravitational field in
which the galaxy formed. Long-range correlations in the tidal field could thus
lead to an ellipticity-ellipticity correlation for widely separated galaxies.
We present a new model relating ellipticity to angular momentum, which can be
calculated in linear theory. We use this model to calculate the angular power
spectrum of intrinsic galaxy shape correlations. We show that for low redshift
galaxy surveys, our model predicts that intrinsic correlations will dominate
correlations induced by weak lensing, in good agreement with previous
theoretical work and observations. We find that our model produces `E-mode'
correlations enhanced by a factor of 3.5 over B-modes on small scales, making
it harder to disentangle intrinsic correlations from those induced by weak
gravitational lensing.Comment: 14 pages, 2 figures, MNRAS in press. Error corrected in lensing
calculation; revised versio
Experimental Evaluation of Book Drawing Algorithms
A -page book drawing of a graph consists of a linear ordering of
its vertices along a spine and an assignment of each edge to one of the
pages, which are half-planes bounded by the spine. In a book drawing, two edges
cross if and only if they are assigned to the same page and their vertices
alternate along the spine. Crossing minimization in a -page book drawing is
NP-hard, yet book drawings have multiple applications in visualization and
beyond. Therefore several heuristic book drawing algorithms exist, but there is
no broader comparative study on their relative performance. In this paper, we
propose a comprehensive benchmark set of challenging graph classes for book
drawing algorithms and provide an extensive experimental study of the
performance of existing book drawing algorithms.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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