41,546 research outputs found
Quasi-periodic motions in dynamical systems. Review of a renormalisation group approach
Power series expansions naturally arise whenever solutions of ordinary
differential equations are studied in the regime of perturbation theory. In the
case of quasi-periodic solutions the issue of convergence of the series is
plagued of the so-called small divisor problem. In this paper we review a
method recently introduced to deal with such a problem, based on
renormalisation group ideas and multiscale techniques. Applications to both
quasi-integrable Hamiltonian systems (KAM theory) and non-Hamiltonian
dissipative systems are discussed. The method is also suited to situations in
which the perturbation series diverges and a resummation procedure can be
envisaged, leading to a solution which is not analytic in the perturbation
parameter: we consider explicitly examples of solutions which are only
infinitely differentiable in the perturbation parameter, or even defined on a
Cantor set.Comment: 36 pages, 8 figures, review articl
The partition function zeroes of quantum critical points
The Lee–Yang theorem for the zeroes of the partition function is not strictly applicable to quantum systems because the zeroes are defined in units of the fugacity ehΔτ, and the Euclidean-time lattice spacing Δτ can be divergent in the infrared (IR). We recently presented analytic arguments describing how a new space-Euclidean time zeroes expansion can be defined, which reproduces Lee and Yang's scaling but avoids the unresolved branch points associated with the breaking of nonlocal symmetries such as Parity. We now present a first numerical analysis for this new zeroes approach for a quantum spin chain system. We use our scheme to quantify the renormalization group flow of the physical lattice couplings to the IR fixed point of this system. We argue that the generic Finite-Size Scaling (FSS) function of our scheme is identically the entanglement entropy of the lattice partition function and, therefore, that we are able to directly extract the central charge, c, of the quantum spin chain system using conformal predictions for the scaling of the entanglement entropy
Linear stability in networks of pulse-coupled neurons
In a first step towards the comprehension of neural activity, one should
focus on the stability of the various dynamical states. Even the
characterization of idealized regimes, such as a perfectly periodic spiking
activity, reveals unexpected difficulties. In this paper we discuss a general
approach to linear stability of pulse-coupled neural networks for generic
phase-response curves and post-synaptic response functions. In particular, we
present: (i) a mean-field approach developed under the hypothesis of an
infinite network and small synaptic conductances; (ii) a "microscopic" approach
which applies to finite but large networks. As a result, we find that no matter
how large is a neural network, its response to most of the perturbations
depends on the system size. There exists, however, also a second class of
perturbations, whose evolution typically covers an increasingly wide range of
time scales. The analysis of perfectly regular, asynchronous, states reveals
that their stability depends crucially on the smoothness of both the
phase-response curve and the transmitted post-synaptic pulse. The general
validity of this scenarion is confirmed by numerical simulations of systems
that are not amenable to a perturbative approach.Comment: 13 pages, 7 figures, submitted to Frontiers in Computational
Neuroscienc
Superheat: An R package for creating beautiful and extendable heatmaps for visualizing complex data
The technological advancements of the modern era have enabled the collection
of huge amounts of data in science and beyond. Extracting useful information
from such massive datasets is an ongoing challenge as traditional data
visualization tools typically do not scale well in high-dimensional settings.
An existing visualization technique that is particularly well suited to
visualizing large datasets is the heatmap. Although heatmaps are extremely
popular in fields such as bioinformatics for visualizing large gene expression
datasets, they remain a severely underutilized visualization tool in modern
data analysis. In this paper we introduce superheat, a new R package that
provides an extremely flexible and customizable platform for visualizing large
datasets using extendable heatmaps. Superheat enhances the traditional heatmap
by providing a platform to visualize a wide range of data types simultaneously,
adding to the heatmap a response variable as a scatterplot, model results as
boxplots, correlation information as barplots, text information, and more.
Superheat allows the user to explore their data to greater depths and to take
advantage of the heterogeneity present in the data to inform analysis
decisions. The goal of this paper is two-fold: (1) to demonstrate the potential
of the heatmap as a default visualization method for a wide range of data types
using reproducible examples, and (2) to highlight the customizability and ease
of implementation of the superheat package in R for creating beautiful and
extendable heatmaps. The capabilities and fundamental applicability of the
superheat package will be explored via three case studies, each based on
publicly available data sources and accompanied by a file outlining the
step-by-step analytic pipeline (with code).Comment: 26 pages, 10 figure
Power spectrum of the maxBCG sample: detection of acoustic oscillations using galaxy clusters
We use the direct Fourier method to calculate the redshift-space power
spectrum of the maxBCG cluster catalog -- currently by far the largest existing
galaxy cluster sample. The total number of clusters used in our analysis is
12,616. After accounting for the radial smearing effect caused by photometric
redshift errors and also introducing a simple treatment for the nonlinear
effects, we show that currently favored low matter density "concordance" LCDM
cosmology provides a very good fit to the estimated power. Thanks to the large
volume (~0.4 h^{-3}Gpc^{3}), high clustering amplitude (linear effective bias
parameter b_{eff} ~3x(0.85/sigma_8)), and sufficiently high sampling density
(~3x10^{-5} h^{3}Mpc^{-3}) the recovered power spectrum has high enough signal
to noise to allow us to find evidence (~2 sigma CL) for the baryonic acoustic
oscillations (BAO). In case the clusters are additionally weighted by their
richness the resulting power spectrum has slightly higher large-scale amplitude
and smaller damping on small scales. As a result the confidence level for the
BAO detection is somewhat increased: ~2.5 sigma. The ability to detect BAO with
relatively small number of clusters is encouraging in the light of several
proposed large cluster surveys.Comment: MNRAS accepted, extended analysis of arXiv:0705.1843, 15 page
Periodicity Manifestations in the Turbulent Regime of Globally Coupled Map Lattice
We revisit the globally coupled map lattice (GCML). We show that in the so
called turbulent regime various periodic cluster attractor states are formed
even though the coupling between the maps are very small relative to the
non-linearity in the element maps.
Most outstanding is a maximally symmetric three cluster attractor in period
three motion (MSCA) due to the foliation of the period three window of the
element logistic maps. An analytic approach is proposed which explains
successfully the systematics of various periodicity manifestations in the
turbulent regime. The linear stability of the period three cluster attractors
is investigated.Comment: 34 pages, 8 Postscript figures, all in GCML-MSCA.Zi
Convergence results for projected line-search methods on varieties of low-rank matrices via \L{}ojasiewicz inequality
The aim of this paper is to derive convergence results for projected
line-search methods on the real-algebraic variety of real
matrices of rank at most . Such methods extend Riemannian
optimization methods, which are successfully used on the smooth manifold
of rank- matrices, to its closure by taking steps along
gradient-related directions in the tangent cone, and afterwards projecting back
to . Considering such a method circumvents the
difficulties which arise from the nonclosedness and the unbounded curvature of
. The pointwise convergence is obtained for real-analytic
functions on the basis of a \L{}ojasiewicz inequality for the projection of the
antigradient to the tangent cone. If the derived limit point lies on the smooth
part of , i.e. in , this boils down to more
or less known results, but with the benefit that asymptotic convergence rate
estimates (for specific step-sizes) can be obtained without an a priori
curvature bound, simply from the fact that the limit lies on a smooth manifold.
At the same time, one can give a convincing justification for assuming critical
points to lie in : if is a critical point of on
, then either has rank , or
Steady-state stabilization due to random delays in maps with self-feedback loops and in globally delayed-coupled maps
We study the stability of the fixed-point solution of an array of mutually
coupled logistic maps, focusing on the influence of the delay times,
, of the interaction between the th and th maps. Two of us
recently reported [Phys. Rev. Lett. {\bf 94}, 134102 (2005)] that if
are random enough the array synchronizes in a spatially homogeneous
steady state. Here we study this behavior by comparing the dynamics of a map of
an array of delayed-coupled maps with the dynamics of a map with
self-feedback delayed loops. If is sufficiently large, the dynamics of a
map of the array is similar to the dynamics of a map with self-feedback loops
with the same delay times. Several delayed loops stabilize the fixed point,
when the delays are not the same; however, the distribution of delays plays a
key role: if the delays are all odd a periodic orbit (and not the fixed point)
is stabilized. We present a linear stability analysis and apply some
mathematical theorems that explain the numerical results.Comment: 14 pages, 13 figures, important changes (title changed, discussion,
figures, and references added
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