In search of a Hagedorn transition in SU(N) lattice gauge theories at large-N

We investigate on the lattice the metastable confined phase above Tc in SU(N) gauge theories, for N=8,10, and 12. In particular we focus on the decrease with the temperature of the mass of the lightest state that couples to Polyakov loops. We find that at T=Tc the corresponding effective string tension \sigma_{eff}(T) is approximately half its value at T=0, and that as we increase T beyond Tc, while remaining in the confined phase, \sigma_{eff}(T) continues to decrease. We extrapolate \sigma_{eff}(T) to even higher temperatures, and interpret the temperature where it vanishes as the Hagedorn temperature T_H. For SU(12) we find that T_H/Tc=1.116(9), when we use the exponent of the three-dimensional XY model for the extrapolation, which seems to be slightly preferred over a mean-field exponent by our data.Comment: 20 pages, 12 figures. New version includes: a more extensive error analysis, a discussion on the behavior of masses near T_H, and additional acknowledgements and references. Results and conclusions do not chang

How to COAAD Images. II. A Coaddition Image that is Optimal for Any Purpose in the Background-dominated Noise Limit

Image coaddition is one of the most basic operations that astronomers perform. In Paper I, we presented the optimal ways to coadd images in order to detect faint sources and to perform flux measurements under the assumption that the noise is approximately Gaussian. Here, we build on these results and derive from first principles a coaddition technique that is optimal for any hypothesis testing and measurement (e.g., source detection, flux or shape measurements, and star/galaxy separation), in the background-noise-dominated case. This method has several important properties. The pixels of the resulting coadded image are uncorrelated. This image preserves all the information (from the original individual images) on all spatial frequencies. Any hypothesis testing or measurement that can be done on all the individual images simultaneously, can be done on the coadded image without any loss of information. The PSF of this image is typically as narrow, or narrower than the PSF of the best image in the ensemble. Moreover, this image is practically indistinguishable from a regular single image, meaning that any code that measures any property on a regular astronomical image can be applied to it unchanged. In particular, the optimal source detection statistic derived in Paper I is reproduced by matched filtering this image with its own PSF. This coaddition process, which we call proper coaddition, can be understood as the maximum signal-to-noise ratio measurement of the Fourier transform of the image, weighted in such a way that the noise in the entire Fourier domain is of equal variance. This method has important implications for multi-epoch seeing-limited deep surveys, weak lensing galaxy shape measurements, and diffraction-limited imaging via speckle observations. The last topic will be covered in depth in future papers. We provide an implementation of this algorithm in MATLAB

How to COAAD Images. I. Optimal Source Detection and Photometry of Point Sources Using Ensembles of Images

Stacks of digital astronomical images are combined in order to increase image depth. The variable seeing conditions, sky background, and transparency of ground-based observations make the coaddition process nontrivial. We present image coaddition methods that maximize the signal-to-noise ratio (S/N) and optimized for source detection and flux measurement. We show that for these purposes, the best way to combine images is to apply a matched filter to each image using its own point-spread function (PSF) and only then to sum the images with the appropriate weights. Methods that either match the filter after coaddition or perform PSF homogenization prior to coaddition will result in loss of sensitivity. We argue that our method provides an increase of between a few and 25% in the survey speed of deep ground-based imaging surveys compared with weighted coaddition techniques. We demonstrate this claim using simulated data as well as data from the Palomar Transient Factory data release 2. We present a variant of this coaddition method, which is optimal for PSF or aperture photometry. We also provide an analytic formula for calculating the S/N for PSF photometry on single or multiple observations. In the next paper in this series, we present a method for image coaddition in the limit of background-dominated noise, which is optimal for any statistical test or measurement on the constant-in-time image (e.g., source detection, shape or flux measurement, or star–galaxy separation), making the original data redundant. We provide an implementation of these algorithms in MATLAB

Quality of internal representation shapes learning performance in feedback neural networks

A fundamental feature of complex biological systems is the ability to form feedback interactions with their environment. A prominent model for studying such interactions is reservoir computing, where learning acts on low-dimensional bottlenecks. Despite the simplicity of this learning scheme, the factors contributing to or hindering the success of training in reservoir networks are in general not well understood. In this work, we study non-linear feedback networks trained to generate a sinusoidal signal, and analyze how learning performance is shaped by the interplay between internal network dynamics and target properties. By performing exact mathematical analysis of linearized networks, we predict that learning performance is maximized when the target is characterized by an optimal, intermediate frequency which monotonically decreases with the strength of the internal reservoir connectivity. At the optimal frequency, the reservoir representation of the target signal is high-dimensional, de-synchronized, and thus maximally robust to noise. We show that our predictions successfully capture the qualitative behaviour of performance in non-linear networks. Moreover, we find that the relationship between internal representations and performance can be further exploited in trained non-linear networks to explain behaviours which do not have a linear counterpart. Our results indicate that a major determinant of learning success is the quality of the internal representation of the target, which in turn is shaped by an interplay between parameters controlling the internal network and those defining the task

Chiral crystals in strong-coupling lattice QCD at nonzero chemical potential

We study the effective action for strong-coupling lattice QCD with one-component staggered fermions in the case of nonzero chemical potential and zero temperature. The structure of this action suggests that at large chemical potentials its ground state is a crystalline `chiral density wave' that spontaneously breaks chiral symmetry and translation invariance. In mean-field theory, on the other hand, we find that this state is unstable. We show that lattice artifacts are partly responsible for this, and suggest that if this phase exists in QCD, then finding it in Monte-Carlo simulations would require simulating on relatively fine lattices. In particular, the baryon mass in lattice units, m_B, should be considerably smaller than its strong-coupling limit of m_B~3.Comment: 33 pages, 8 figure