20,358 research outputs found
Nonlinear approximation with nonstationary Gabor frames
We consider sparseness properties of adaptive time-frequency representations
obtained using nonstationary Gabor frames (NSGFs). NSGFs generalize classical
Gabor frames by allowing for adaptivity in either time or frequency. It is
known that the concept of painless nonorthogonal expansions generalizes to the
nonstationary case, providing perfect reconstruction and an FFT based
implementation for compactly supported window functions sampled at a certain
density. It is also known that for some signal classes, NSGFs with flexible
time resolution tend to provide sparser expansions than can be obtained with
classical Gabor frames. In this article we show, for the continuous case, that
sparseness of a nonstationary Gabor expansion is equivalent to smoothness in an
associated decomposition space. In this way we characterize signals with sparse
expansions relative to NSGFs with flexible time resolution. Based on this
characterization we prove an upper bound on the approximation error occurring
when thresholding the coefficients of the corresponding frame expansions. We
complement the theoretical results with numerical experiments, estimating the
rate of approximation obtained from thresholding the coefficients of both
stationary and nonstationary Gabor expansions.Comment: 19 pages, 2 figure
Kinks in the Presence of Rapidly Varying Perturbations
Dynamics of sine-Gordon kinks in the presence of rapidly varying periodic
perturbations of different physical origins is described analytically and
numerically. The analytical approach is based on asymptotic expansions, and it
allows to derive, in a rigorous way, an effective nonlinear equation for the
slowly varying field component in any order of the asymptotic procedure as
expansions in the small parameter , being the frequency
of the rapidly varying ac driving force. Three physically important examples of
such a dynamics, {\em i.e.}, kinks driven by a direct or parametric ac force,
and kinks on rotating and oscillating background, are analysed in detail. It is
shown that in the main order of the asymptotic procedure the effective equation
for the slowly varying field component is {\em a renormalized sine-Gordon
equation} in the case of the direct driving force or rotating (but phase-locked
to an external ac force) background, and it is {\em the double sine-Gordon
equation} for the parametric driving force. The properties of the kinks
described by the renormalized nonlinear equations are analysed, and it is
demonstrated analytically and numerically which kinds of physical phenomena may
be expected in dealing with the renormalized, rather than the unrenormalized,
nonlinear dynamics. In particular, we predict several qualitatively new effects
which include, {\em e.g.}, the perturbation-inducedComment: New copy of the paper of the above title to replace the previous one,
lost in the midst of the bulletin board. RevTeX 3.
Combining perturbation theories with halo models
We investigate the building of unified models that can predict the
matter-density power spectrum and the two-point correlation function from very
large to small scales, being consistent with perturbation theory at low and
with halo models at high . We use a Lagrangian framework to re-interpret the
halo model and to decompose the power spectrum into "2-halo" and "1-halo"
contributions, related to "perturbative" and "non-perturbative" terms. We
describe a simple implementation of this model and present a detailed
comparison with numerical simulations, from up to Mpc, and from up to Mpc. We show that the
1-halo contribution contains a counterterm that ensures a tail at low
and is important not to spoil the predictions on the scales probed by baryon
acoustic oscillations, to Mpc. On the other hand,
we show that standard perturbation theory is inadequate for the 2-halo
contribution, because higher order terms grow too fast at high , so that
resummation schemes must be used. We describe a simple implementation, based on
a 1-loop "direct steepest-descent" resummation for the 2-halo contribution that
allows fast numerical computations, and we check that we obtain a good match to
simulations at low and high . Our simple implementation already fares better
than standard 1-loop perturbation theory on large scales and simple fits to the
power spectrum at high , with a typical accuracy of 1% on large scales and
10% on small scales. We obtain similar results for the two-point correlation
function. However, there remains room for improvement on the transition scale
between the 2-halo and 1-halo contributions, which may be the most difficult
regime to describe.Comment: 29 page
Consistent Modeling of Velocity Statistics and Redshift-Space Distortions in One-Loop Perturbation Theory
The peculiar velocities of biased tracers of the cosmic density field contain
important information about the growth of large scale structure and generate
anisotropy in the observed clustering of galaxies. Using N-body data, we show
that velocity expansions for halo redshift-space power spectra are converged at
the percent-level at perturbative scales for most line-of-sight angles
when the first three pairwise velocity moments are included, and that the third
moment is well-approximated by a counterterm-like contribution. We compute
these pairwise-velocity statistics in Fourier space using both Eulerian and
Lagrangian one-loop perturbation theory using a cubic bias scheme and a
complete set of counterterms and stochastic contributions. We compare the
models and show that our models fit both real-space velocity statistics and
redshift-space power spectra for both halos and a mock sample of galaxies at
sub-percent level on perturbative scales using consistent sets of parameters,
making them appealing choices for the upcoming era of spectroscopic,
peculiar-velocity and kSZ surveys.Comment: 63 pages, 11 figures, updated to match version accepted by JCA
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