6,443 research outputs found
Gaussian Bounds for Noise Correlation of Functions
In this paper we derive tight bounds on the expected value of products of
{\em low influence} functions defined on correlated probability spaces. The
proofs are based on extending Fourier theory to an arbitrary number of
correlated probability spaces, on a generalization of an invariance principle
recently obtained with O'Donnell and Oleszkiewicz for multilinear polynomials
with low influences and bounded degree and on properties of multi-dimensional
Gaussian distributions. The results derived here have a number of applications
to the theory of social choice in economics, to hardness of approximation in
computer science and to additive combinatorics problems.Comment: Typos and references correcte
Avalanche dynamics of radio pulsar glitches
We test statistically the hypothesis that radio pulsar glitches result from
an avalanche process, in which angular momentum is transferred erratically from
the flywheel-like superfluid in the star to the slowly decelerating, solid
crust via spatially connected chains of local, impulsive, threshold-activated
events, so that the system fluctuates around a self-organised critical state.
Analysis of the glitch population (currently 285 events from 101 pulsars)
demonstrates that the size distribution in individual pulsars is consistent
with being scale invariant, as expected for an avalanche process. The
waiting-time distribution is consistent with being exponential in seven out of
nine pulsars where it can be measured reliably, after adjusting for
observational limits on the minimum waiting time, as for a constant-rate
Poisson process. PSR J05376910 and PSR J08354510 are the exceptions;
their waiting-time distributions show evidence of quasiperiodicity. In each
object, stationarity requires that the rate equals , where is the angular acceleration of the
crust, is the mean glitch size, and is the
relative angular acceleration of the crust and superfluid. There is no evidence
that changes monotonically with spin-down age. The rate distribution
itself is fitted reasonably well by an exponential for . For , its exact form is unknown; the
exponential overestimates the number of glitching pulsars observed at low
, where the limited total observation time exercises a selection bias.Comment: Accepted for publication in the Astrophysical Journa
Stretched Exponential Relaxation in the Biased Random Voter Model
We study the relaxation properties of the voter model with i.i.d. random
bias. We prove under mild condions that the disorder-averaged relaxation of
this biased random voter model is faster than a stretched exponential with
exponent , where depends on the transition rates
of the non-biased voter model. Under an additional assumption, we show that the
above upper bound is optimal. The main ingredient of our proof is a result of
Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe
Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions
Power-law distributions contain precious information about a large variety of
processes in geoscience and elsewhere. Although there are sound theoretical
grounds for these distributions, the empirical evidence in favor of power laws
has been traditionally weak. Recently, Clauset et al. have proposed a
systematic method to find over which range (if any) a certain distribution
behaves as a power law. However, their method has been found to fail, in the
sense that true (simulated) power-law tails are not recognized as such in some
instances, and then the power-law hypothesis is rejected. Moreover, the method
does not work well when extended to power-law distributions with an upper
truncation. We explain in detail a similar but alternative procedure, valid for
truncated as well as for non-truncated power-law distributions, based in
maximum likelihood estimation, the Kolmogorov-Smirnov goodness-of-fit test, and
Monte Carlo simulations. An overview of the main concepts as well as a recipe
for their practical implementation is provided. The performance of our method
is put to test on several empirical data which were previously analyzed with
less systematic approaches. The databases presented here include the half-lives
of the radionuclides, the seismic moment of earthquakes in the whole world and
in Southern California, a proxy for the energy dissipated by tropical cyclones
elsewhere, the area burned by forest fires in Italy, and the waiting times
calculated over different spatial subdivisions of Southern California. We find
the functioning of the method very satisfactory.Comment: 26 pages, 9 figure
General variational many-body theory with complete self-consistency for trapped bosonic systems
In this work we develop a complete variational many-body theory for a system
of trapped bosons interacting via a general two-body potential. In this
theory both the many-body basis functions {\em and} the respective expansion
coefficients are treated as variational parameters. The optimal variational
parameters are obtained {\em self-consistently} by solving a coupled system of
non-eigenvalue -- generally integro-differential -- equations to get the
one-particle functions and by diagonalizing the secular matrix problem to find
the expansion coefficients. We call this theory multi-configurational Hartree
for bosons or MCHB(M), where M specifies explicitly the number of one-particle
functions used to construct the configurations. General rules for evaluating
the matrix elements of one- and two-particle operators are derived and applied
to construct the secular Hamiltonian matrix. We discuss properties of the
derived equations. It is demonstrated that for any practical computation where
the configurational space is restricted, the description of trapped bosonic
systems strongly depends on the choice of the many-body basis set used, i.e.,
self-consistency is of great relevance. As illustrative examples we consider
bosonic systems trapped in one- and two-dimensional symmetric and asymmetric
double-well potentials. We demonstrate that self-consistency has great impact
on the predicted physical properties of the ground and excited states and show
that the lack of self-consistency may lead to physically wrong predictions. The
convergence of the general MCHB(M) scheme with a growing number M is validated
in a specific case of two bosons trapped in a symmetric double-well.Comment: 53 pages, 8 figure
Constraints on cosmic hemispherical power anomalies from quasars
Recent analyses of the cosmic microwave background (CMB) maps from the WMAP
satellite have uncovered evidence for a hemispherical power anomaly, i.e. a
dipole modulation of the CMB power spectrum at large angular scales with an
amplitude of +/-14 percent. Erickcek et al have put forward an inflationary
model to explain this anomaly. Their scenario is a variation on the curvaton
scenario in which the curvaton possesses a large-scale spatial gradient that
modulates the amplitude of CMB fluctuations. We show that this scenario would
also lead to a spatial gradient in the amplitude of perturbations sigma_8, and
hence to a dipole asymmetry in any highly biased tracer of the underlying
density field. Using the high-redshift quasars from the Sloan Digital Sky
Survey, we find an upper limit on such a gradient of |nabla
sigma_8|/sigma_8<0.027/r_{lss} (99% posterior probability), where r_{lss} is
the comoving distance to the last-scattering surface. This rules out the
simplest version of the curvaton spatial gradient scenario.Comment: matches JCAP accepted version (minor revisions
On the Inability of Markov Models to Capture Criticality in Human Mobility
We examine the non-Markovian nature of human mobility by exposing the
inability of Markov models to capture criticality in human mobility. In
particular, the assumed Markovian nature of mobility was used to establish a
theoretical upper bound on the predictability of human mobility (expressed as a
minimum error probability limit), based on temporally correlated entropy. Since
its inception, this bound has been widely used and empirically validated using
Markov chains. We show that recurrent-neural architectures can achieve
significantly higher predictability, surpassing this widely used upper bound.
In order to explain this anomaly, we shed light on several underlying
assumptions in previous research works that has resulted in this bias. By
evaluating the mobility predictability on real-world datasets, we show that
human mobility exhibits scale-invariant long-range correlations, bearing
similarity to a power-law decay. This is in contrast to the initial assumption
that human mobility follows an exponential decay. This assumption of
exponential decay coupled with Lempel-Ziv compression in computing Fano's
inequality has led to an inaccurate estimation of the predictability upper
bound. We show that this approach inflates the entropy, consequently lowering
the upper bound on human mobility predictability. We finally highlight that
this approach tends to overlook long-range correlations in human mobility. This
explains why recurrent-neural architectures that are designed to handle
long-range structural correlations surpass the previously computed upper bound
on mobility predictability
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