Entropy and Poincar\'e recurrence from a geometrical viewpoint

We study Poincar\'e recurrence from a purely geometrical viewpoint. We prove that the metric entropy is given by the exponential growth rate of return times to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss theorem. Moreover, we show that minimal return times to dynamical balls grow linearly with respect to its length. Finally, some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures are given.Comment: 11 pages, revised versio

Cosmological Model Predictions for Weak Lensing: Linear and Nonlinear Regimes

Weak lensing by large scale structure induces correlated ellipticities in the images of distant galaxies. The two-point correlation is determined by the matter power spectrum along the line of sight. We use the fully nonlinear evolution of the power spectrum to compute the predicted ellipticity correlation. We present results for different measures of the second moment for angular scales \theta \simeq 1'-3 degrees and for alternative normalizations of the power spectrum, in order to explore the best strategy for constraining the cosmological parameters. Normalizing to observed cluster abundance the rms amplitude of ellipticity within a 15' radius is \simeq 0.01 z_s^{0.6}, almost independent of the cosmological model, with z_s being the median redshift of background galaxies. Nonlinear effects in the evolution of the power spectrum significantly enhance the ellipticity for \theta < 10' -- on 1' the rms ellipticity is \simeq 0.05, which is nearly twice the linear prediction. This enhancement means that the signal to noise for the ellipticity is only weakly increasing with angle for 2'< \theta < 2 degrees, unlike the expectation from linear theory that it is strongly peaked on degree scales. The scaling with cosmological parameters also changes due to nonlinear effects. By measuring the correlations on small (nonlinear) and large (linear) angular scales, different cosmological parameters can be independently constrained to obtain a model independent estimate of both power spectrum amplitude and matter density \Omega_m. Nonlinear effects also modify the probability distribution of the ellipticity. Using second order perturbation theory we find that over most of the range of interest there are significant deviations from a normal distribution.Comment: 38 pages, 11 figures included. Extended discussion of observational prospects, matches accepted version to appear in Ap

The Little-Hopfield model on a Random Graph

We study the Hopfield model on a random graph in scaling regimes where the average number of connections per neuron is a finite number and where the spin dynamics is governed by a synchronous execution of the microscopic update rule (Little-Hopfield model).We solve this model within replica symmetry and by using bifurcation analysis we prove that the spin-glass/paramagnetic and the retrieval/paramagnetictransition lines of our phase diagram are identical to those of sequential dynamics.The first-order retrieval/spin-glass transition line follows by direct evaluation of our observables using population dynamics. Within the accuracy of numerical precision and for sufficiently small values of the connectivity parameter we find that this line coincides with the corresponding sequential one. Comparison with simulation experiments shows excellent agreement.Comment: 14 pages, 4 figure

Atributos químicos das fases sólida e aquosa do solo de sítios de restinga sob diferentes coberturas vegetais no estado de Sergipe.

bitstream/CPATC/19773/1/f_11_2007.pd