The spin contribution to the form factor of quantum graphs

Following the quantisation of a graph with the Dirac operator (spin-1/2) we explain how additional weights in the spectral form factor K(\tau) due to spin propagation around orbits produce higher order terms in the small-\tau asymptotics in agreement with symplectic random matrix ensembles. We determine conditions on the group of spin rotations sufficient to generate CSE statistics.Comment: 9 page

Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph

We study the spectral statistics of the Dirac operator on a rose-shaped graph---a graph with a single vertex and all bonds connected at both ends to the vertex. We formulate a secular equation that generically determines the eigenvalues of the Dirac rose graph, which is seen to generalise the secular equation for a star graph with Neumann boundary conditions. We derive approximations to the spectral pair correlation function at large and small values of spectral spacings, in the limit as the number of bonds approaches infinity, and compare these predictions with results of numerical calculations. Our results represent the first example of intermediate statistics from the symplectic symmetry class.Comment: 26 pages, references adde

Carbon and Strontium Abundances of Metal-Poor Stars

We present carbon and strontium abundances for 100 metal-poor stars measured from R$\sim$7000 spectra obtained with the Echellette Spectrograph and Imager at the Keck Observatory. Using spectral synthesis of the G-band region, we have derived carbon abundances for stars ranging from [Fe/H]$=-1.3$ to [Fe/H]$=-3.8$. The formal errors are $\sim 0.2$ dex in [C/Fe]. The strontium abundance in these stars was measured using spectral synthesis of the resonance line at 4215 {\AA}. Using these two abundance measurments along with the barium abundances from our previous study of these stars, we show it is possible to identify neutron-capture-rich stars with our spectra. We find, as in other studies, a large scatter in [C/Fe] below [Fe/H]$= -2$. Of the stars with [Fe/H]$<-2$, 9$\pm$4% can be classified as carbon-rich metal-poor stars. The Sr and Ba abundances show that three of the carbon-rich stars are neutron-capture-rich, while two have normal Ba and Sr. This fraction of carbon enhanced stars is consistent with other studies that include this metallicity range.Comment: ApJ, Accepte

From error bounds to the complexity of first-order descent methods for convex functions

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with H\"olderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple methodology: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form $O(q^{k})$ where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with $\ell^1$ regularization

NGC 2419, M92, and the Age Gradient in the Galactic Halo

The WFPC2 camera on HST has been used to obtain deep main sequence photometry of the low-metallicity ([Fe/H]=-2.14), outer-halo globular cluster NGC 2419. A differential fit of the NGC 2419 CMD to that of the similarly metal-poor \ standard cluster M92 shows that they have virtually identical principal sequences and thus the same age to well within 1 Gyr. Since other low-metallicity clusters throughout the Milky Way halo have this same age to within the 1-Gyr precision of the differential age technique, we conclude that the earliest star (or globular cluster) formation began at essentially the same time everywhere in the Galactic halo throughout a region now almost 200 kpc in diameter. Thus for the metal-poorest clusters in the halo there is no detectable age gradient with Galactocentric distance. To estimate the absolute age of NGC 2419 and M92, we fit newly computed isochrones transformed through model-atmosphere calculations to the (M_V,V-I) plane, with assumed distance scales that represent the range currently debated in the literature. Unconstrained isochrone fits give M_V(RR) = 0.55 \pm 0.06 and a resulting age of 14 to 15 Gyr. Incorporating the full effects of helium diffusion would further reduce this estimate by about 1 Gyr. A distance scale as bright as M_V(RR) = 0.15 for [Fe/H] = -2, as has recently been reported, would leave several serious problems which have no obvious solution in the context of current stellar models.Comment: 32 pages, aastex, 9 postscript figures; accepted for publication in AJ, September 1997. Also available by e-mail from [email protected]

Beyond the Heisenberg time: Semiclassical treatment of spectral correlations in chaotic systems with spin 1/2

The two-point correlation function of chaotic systems with spin 1/2 is evaluated using periodic orbits. The spectral form factor for all times thus becomes accessible. Equivalence with the predictions of random matrix theory for the Gaussian symplectic ensemble is demonstrated. A duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established