Non-Gaussian numerical errors versus mass hierarchy

We probe the numerical errors made in renormalization group calculations by varying slightly the rescaling factor of the fields and rescaling back in order to get the same (if there were no round-off errors) zero momentum 2-point function (magnetic susceptibility). The actual calculations were performed with Dyson's hierarchical model and a simplified version of it. We compare the distributions of numerical values obtained from a large sample of rescaling factors with the (Gaussian by design) distribution of a random number generator and find significant departures from the Gaussian behavior. In addition, the average value differ (robustly) from the exact answer by a quantity which is of the same order as the standard deviation. We provide a simple model in which the errors made at shorter distance have a larger weight than those made at larger distance. This model explains in part the non-Gaussian features and why the central-limit theorem does not apply.Comment: 26 pages, 7 figures, uses Revte

Expressing the entropy of lattice systems as sums of conditional entropies

Whether a system is to be considered complex or not depends on how one searches for correlations. We propose a general scheme for calculation of entropies in lattice systems that has high flexibility in how correlations are successively taken into account. Compared to the traditional approach for estimating the entropy density, in which successive approximations builds on step-wise extensions of blocks of symbols, we show that one can take larger steps when collecting the statistics necessary to calculate the entropy density of the system. In one dimension this means that, instead of a single sweep over the system in which states are read sequentially, one take several sweeps with larger steps so that eventually the whole lattice is covered. This means that the information in correlations is captured in a different way, and in some situations this will lead to a considerably much faster convergence of the entropy density estimate as a function of the size of the configurations used in the estimate. The formalism is exemplified with both an example of a free energy minimisation scheme for the two-dimensional Ising model, and an example of increasingly complex spatial correlations generated by the time evolution of elementary cellular automaton rule 60

Determining Ages of APOGEE Giants with Known Distances

We present a sample of local red giant stars observed using the New Mexico State University 1 m telescope with the APOGEE spectrograph, for which we estimate stellar ages and the age distribution from the high-resolution spectroscopic stellar parameters and accurate distance measurements from Hipparcos. The high-resolution (R ~ 23,000), near infrared (H-band, 1.5-1.7 micron) APOGEE spectra provide measurements of the stellar atmospheric parameters (temperature, surface gravity, [M/H], and [alpha/M]). Due to the smaller uncertainties in surface gravity possible with high-resolution spectra and accurate Hipparcos distance measurements, we are able to calculate the stellar masses to within 40%. For red giants, the relatively rapid evolution of stars up the red giant branch allows the age to be constrained based on the mass. We examine methods of estimating age using both the mass-age relation directly and a Bayesian isochrone matching of measured parameters, assuming a constant star formation history (SFH). To improve the prior on the SFH, we use a hierarchical modeling approach to constrain the parameters of a model SFH from the age probability distribution functions of the data. The results of an alpha dependent Gaussian SFH model shows a clear relation between age and [alpha/M] at all ages. Using this SFH model as the prior for an empirical Bayesian analysis, we construct a full age probability distribution function and determine ages for individual stars. The age-metallicity relation is flat, with a slight decrease in [M/H] at the oldest ages and a ~ 0.5 dex spread in metallicity. For stars with ages < 1 Gyr we find a smaller spread, consistent with radial migration having a smaller effect on these young stars than on the older stars.Comment: 14 page, 18 figures, accepted to ApJ with minor revisions, full electronic table of data available upon publicatio

Testing asteroseismology with Gaia DR2: Hierarchical models of the Red Clump

Asteroseismology provides fundamental stellar parameters independent of distance, but subject to systematics under calibration. Gaia DR2 has provided parallaxes for a billion stars, which are offset by a parallax zero-point. Red Clump (RC) stars have a narrow spread in luminosity, thus functioning as standard candles to calibrate these systematics. This work measures how the magnitude and spread of the RC in the Kepler field are affected by changes to temperature and scaling relations for seismology, and changes to the parallax zero-point for Gaia. We use a sample of 5576 RC stars classified through asteroseismology. We apply hierarchical Bayesian latent variable models, finding the population level properties of the RC with seismology, and use those as priors on Gaia parallaxes to find the parallax zero-point offset. We then find the position of the RC using published values for the zero-point. We find a seismic temperature insensitive spread of the RC of ~0.03 mag in the 2MASS K band and a larger and slightly temperature-dependent spread of ~0.13 mag in the Gaia G band. This intrinsic dispersion in the K band provides a distance precision of ~1% for RC stars. Using Gaia data alone, we find a mean zero-point of -41 $\pm$ 10 $\mu$as. This offset yields RC absolute magnitudes of -1.634 $\pm$ 0.018 in K and 0.546 $\pm$ 0.016 in G. Obtaining these same values through seismology would require a global temperature shift of ~-70 K, which is compatible with known systematics in spectroscopy.Comment: Accepted for publication in MNRA

Zero-Temperature Complex Replica Zeros of the ±J\pm J Ising Spin Glass on Mean-Field Systems and Beyond

Zeros of the moment of the partition function $[Z^n]_{\bm{J}}$ with respect to complex $n$ are investigated in the zero temperature limit $\beta \to \infty$, $n\to 0$ keeping $y=\beta n \approx O(1)$. We numerically investigate the zeros of the $\pm J$ Ising spin glass models on several Cayley trees and hierarchical lattices and compare those results. In both lattices, the calculations are carried out with feasible computational costs by using recursion relations originated from the structures of those lattices. The results for Cayley trees show that a sequence of the zeros approaches the real axis of $y$ implying that a certain type of analyticity breaking actually occurs, although it is irrelevant for any known replica symmetry breaking. The result of hierarchical lattices also shows the presence of analyticity breaking, even in the two dimensional case in which there is no finite-temperature spin-glass transition, which implies the existence of the zero-temperature phase transition in the system. A notable tendency of hierarchical lattices is that the zeros spread in a wide region of the complex $y$ plane in comparison with the case of Cayley trees, which may reflect the difference between the mean-field and finite-dimensional systems.Comment: 4 pages, 4 figure

An immune algorithm based fuzzy predictive modeling mechanism using variable length coding and multi-objective optimization allied to engineering materials processing

In this paper, a systematic multi-objective fuzzy modeling approach is proposed, which can be regarded as a three-stage modeling procedure. In the first stage, an evolutionary based clustering algorithm is developed to extract an initial fuzzy rule base from the data. Based on this model, a back-propagation algorithm with momentum terms is used to refine the initial fuzzy model. The refined model is then used to seed the initial population of an immune inspired multi-objective optimization algorithm in the third stage to obtain a set of fuzzy models with improved transparency. To tackle the problem of simultaneously optimizing the structure and parameters, a variable length coding scheme is adopted to improve the efficiency of the search. The proposed modeling approach is applied to a real data set from the steel industry. Results show that the proposed approach is capable of eliciting not only accurate but also transparent fuzzy models