Probing the basins of attraction of a recurrent neural network

A recurrent neural network is considered that can retrieve a collection of patterns, as well as slightly perturbed versions of this `pure' set of patterns via fixed points of its dynamics. By replacing the set of dynamical constraints, i.e., the fixed point equations, by an extended collection of fixed-point-like equations, analytical expressions are found for the weights w_ij(b) of the net, which depend on a certain parameter b. This so-called basin parameter b is such that for b=0 there are, a priori, no perturbed patterns to be recognized by the net. It is shown by a numerical study, via probing sets, that a net constructed to recognize perturbed patterns, i.e., with values of the connections w_ij(b) with b unequal zero, possesses larger basins of attraction than a net made with the help of a pure set of patterns, i.e., with connections w_ij(b=0). The mathematical results obtained can, in principle, be realized by an actual, biological neural net.Comment: 17 pages, LaTeX, 2 figure

Tumbling of a rigid rod in a shear flow

The tumbling of a rigid rod in a shear flow is analyzed in the high viscosity limit. Following Burgers, the Master Equation is derived for the probability distribution of the orientation of the rod. The equation contains one dimensionless number, the Weissenberg number, which is the ratio of the shear rate and the orientational diffusion constant. The equation is solved for the stationary state distribution for arbitrary Weissenberg numbers, in particular for the limit of high Weissenberg numbers. The stationary state gives an interesting flow pattern for the orientation of the rod, showing the interplay between flow due to the driving shear force and diffusion due to the random thermal forces of the fluid. The average tumbling time and tumbling frequency are calculated as a function of the Weissenberg number. A simple cross-over function is proposed which covers the whole regime from small to large Weissenberg numbers.Comment: 22 pages, 9 figure

Conserving Approximations in Time-Dependent Density Functional Theory

In the present work we propose a theory for obtaining successively better approximations to the linear response functions of time-dependent density or current-density functional theory. The new technique is based on the variational approach to many-body perturbation theory (MBPT) as developed during the sixties and later expanded by us in the mid nineties. Due to this feature the resulting response functions obey a large number of conservation laws such as particle and momentum conservation and sum rules. The quality of the obtained results is governed by the physical processes built in through MBPT but also by the choice of variational expressions. We here present several conserving response functions of different sophistication to be used in the calculation of the optical response of solids and nano-scale systems.Comment: 11 pages, 4 figures, revised versio

Atomic quasi-Bragg diffraction in a magnetic field

We report on a new technique to split an atomic beam coherently with an easily adjustable splitting angle. In our experiment metastable helium atoms in the |{1s2s}^3S_1 M=1> state diffract from a polarization gradient light field formed by counterpropagating \sigma^+ and \sigma^- polarized laser beams in the presence of a homogeneous magnetic field. In the near-adiabatic regime, energy conservation allows the resonant exchange between magnetic energy and kinetic energy. As a consequence, symmetric diffraction of |M=0> or |M=-1> atoms in a single order is achieved, where the order can be chosen freely by tuning the magnetic field. We present experimental results up to 6th order diffraction (24 \hbar k momentum splitting, i.e., 2.21 m/s in transverse velocity) and present a simple theoretical model that stresses the similarity with conventional Bragg scattering. The resulting device constitutes a flexible, adjustable, large-angle, three-way coherent atomic beam splitter with many potential applications in atom optics and atom interferometry.Comment: 4 pages, 5 figure

Age, Metallicity, and the Distance to the Magellanic Clouds From Red Clump Stars

We show that the luminosity dependence of the red clump stars on age and metallicity can cause a difference of up to < ~0.6 mag in the mean absolute I magnitude of the red clump between different stellar populations. We show that this effect may resolve the apparent ~0.4 mag discrepancy between red clump-derived distance moduli to the Magellanic Clouds and those from, e.g., Cepheid variables. Taking into account the population effects on red clump luminosity, we determine a distance modulus to the LMC of 18.36 +/- 0.17 mag, and to the SMC of 18.82 +/- 0.20 mag. Our alternate red clump LMC distance is consistent with the value (m-M){LMC} = 18.50 +/- 0.10 adopted by the HST Cepheid Key Project. We briefly examine model predictions of red clump luminosity, and find that variations in helium abundance and core mass could bring the Clouds closer by some 0.10--0.15 mag, but not by the ~0.4 mag that would result from setting the mean absolute I-magnitude of the Cloud red clumps equal to the that of the Solar neighborhood red clump.Comment: Accepted for publication in The Astrophysical Journal Letters, AASTeX 4.0, 10 pages, 1 postscript figur

Spin-polarized stable phases of the 2-D electron fluid at finite temperatures

The Helmholtz free energy F of the interacting 2-D electron fluid is calculated nonperturbatively using a mapping of the quantum fluid to a classical Coulomb fluid [Phys. Rev. Letters, vol. 87, 206404 (2001)]. For density parameters rs such that rs<~25, the fluid is unpolarized at all temperatures t=T/EF where EF is the Fermi energy. For lower densities, the system becomes fully spin polarized for t<~0.35, and partially polarized for 0.35<t< 2, depending on the density. At rs ~25-30, and t ~0.35, an ''ambispin'' phase where F is almost independent of the spin polarization is found. These results support recent claims, based on quantum Monte Carlo results, for a stable, fully spin-polarized fluid phase at T = 0 for rs larger than about 25-26.Comment: Latex manuscript (4-5 pages) and two postscript figures; see also http://nrcphy1.phy.nrc.ca/ims/qp/chandre/chnc

Waiting and Residence Times of Brownian Interface Fluctuations

We report on the residence times of capillary waves above a given height $h$ and on the typical waiting time in between such fluctuations. The measurements were made on phase separated colloid-polymer systems by laser scanning confocal microscopy. Due to the Brownian character of the process, the stochastics vary with the chosen measurement interval $\Delta t$. In experiments, the discrete scanning times are a practical cutoff and we are able to measure the waiting time as a function of this cutoff. The measurement interval dependence of the observed waiting and residence times turns out to be solely determined by the time dependent height-height correlation function $g(t)$. We find excellent agreement with the theory presented here along with the experiments.Comment: 5 figure