Complexity of pattern classes and Lipschitz property

Rademacher and Gaussian complexities are successfully used in learning theory for measuring the capacity of the class of functions to be learned. One of the most important properties for these complexities is their Lipschitz property: a composition of a class of functions with a fixed Lipschitz function may increase its complexity by at most twice the Lipschitz constant. The proof of this property is non-trivial (in contrast to the other properties) and it is believed that the proof in the Gaussian case is conceptually more difficult then the one for the Rademacher case. In this paper we give a detailed prove of the Lipschitz property for the Rademacher case and generalize the same idea to an arbitrary complexity (including the Gaussian). We also discuss a related topic about the Rademacher complexity of a class consisting of all the Lipschitz functions with a given Lipschitz constant. We show that the complexity is surprisingly low in the one-dimensional case. The question for higher dimensions remains open

The mass-metallicity relation for high-redshift damped Ly-alpha galaxies

We used our database of ESO VLT-UVES spectra of quasars to build up a sample of 67 Damped Lyman-alpha (DLA) systems with redshifts 1.7<zabs<3.7. For each system, we measured average metallicities relative to Solar, [X/H] (with either X=Zn, S or Si), and the velocity widths of low-ionization line profiles, W1. We find that there is a tight correlation between the two quantities, detected at the 5sigma significance level. The existence of such a correlation, over more than two orders of magnitude spread in metallicity, is likely to be the consequence of an underlying mass-metallicity relation for the galaxies responsible for DLA absorption lines. The best-fit linear relation is [X/H]=1.35(\pm 0.11)\log W1 -3.69(\pm 0.18)$ with W1 expressed in km/s. While the slope of this velocity-metallicity relation is the same within uncertainties between the higher and the lower redshift bins of our sample, there is a hint of an increase of the intercept point of the relation with decreasing redshift. This suggests that galaxy halos of a given mass tend to become more metal-rich with time. Moreover, the slope of this relation is consistent with that of the luminosity-metallicity relation for local galaxies. The DLA systems having the lowest metallicities among the DLA population would therefore, on average, correspond to the galaxies having the lowest masses. In turn, these galaxies should have the lowest luminosities among the DLA galaxy population. This may explain the recent result that the few DLA systems with detected Ly-alpha emission have higher than average metallicities.Comment: proceedings of IAU Colloquium No. 199, 2005, ``Probing Galaxies through Quasar Absorption Lines'', P.R. Williams, C. Shu, B. Menard, ed

Multiphase Plasma in Sub-Damped Lyman Alpha Systems: A Hidden Metal Reservoir

We present a VLT/UVES spectrum of a proximate sub-damped Lyman-alpha (sub-DLA) system at z=2.65618 toward the quasar Q0331-4505 (z_qso=2.6785+/-0.0030). Absorption lines of O I, Si II, Si III, Si IV, C II, C III, C IV, Fe II, Al II, and O VI are seen in the sub-DLA, which has a neutral hydrogen column density log N(H I)=19.82+/-0.05. The absorber is at a velocity of 1820+/-250 km/s from the quasar; however, its low metallicity [O/H]=-1.64+/-0.07, lack of partial coverage, lack of temporal variations between observations taken in 2003 and 2006, and non-detection of N V imply the absorber is not a genuine intrinsic system. By measuring the O VI column density and assuming equal metallicities in the neutral and ionized gas, we determine the column density of hot ionized hydrogen in this sub-DLA, and in two other sub-DLAs with O VI drawn from the literature. Coupling this with determinations of the typical amount of warm ionized hydrogen in sub-DLAs, we confirm that sub-DLAs are a more important metal reservoir than DLAs, in total comprising at least 6-22% of the metal budget at z~2.5.Comment: 5 pages, 3 color figures, accepted for publication in ApJ

Mass-metallicity relation from z=5 to the present: Evidence for a transition in the mode of galaxy growth at z=2.6 due to the end of sustained primordial gas infall

We analyze the redshift evolution of the mass-metallicity relation in a sample of 110 Damped Ly$\alpha$ absorbers spanning the redshift range $z=0.11-5.06$ and find that the zero-point of the correlation changes significantly with redshift. The evolution is such that the zero-point is constant at the early phases of galaxy growth (i.e. no evolution) but then features a sharp break at $z=2.6\pm 0.2$ with a rapid incline towards lower redshifts such that damped absorbers of identical masses are more metal rich at later times than earlier. The slope of this mass metallicity correlation evolution is $0.35 \pm 0.07$ dex per unit redshift. We compare this result to similar studies of the redshift evolution of emission selected galaxy samples and find a remarkable agreement with the slope of the evolution of galaxies of stellar mass log$(M_{*}/M_\odot) \approx 8.5$. This allows us to form an observational tie between damped absorbers and galaxies seen in emission. We use results from simulations to infer the virial mass of the dark matter halo of a typical DLA galaxy and find a ratio $(M_{vir}/M_{*}) \approx 30$. We compare our results to those of several other studies that have reported strong transition-like events at redshifts around $z=2.5-2.6$ and argue that all those observations can be understood as the consequence of a transition from a situation where galaxies were fed more unprocessed infalling gas than they could easily consume to one where they suddenly become infall starved and turn to mainly processing, or re-processing, of previously acquired gas.Comment: 8 pages, 5 figures, accepted for publication in MNRA

The Development of GRAPE, a Gamma Ray Polarimeter Experiment

The measurement of hard X‐ray polarization in γ‐ray bursts (GRBs) would add yet another piece of information in our effort to resolve the true nature of these enigmatic objects. Here we report on the development of a dedicated polarimeter design with a relatively large FoV that is capable of studying hard X‐ray polarization (50–300 keV) from GRBs. This compact design, based on the use of a large area position‐sensitive PMT (PSPMT), is referred to as GRAPE (Gamma‐RAy Polarimeter Experiment). The feature of GRAPE that is especially attractive for studies of GRBs is the significant off‐axis polarization response (at angles greater than 60°). For an array of GRAPE modules, current sensitivity estimates give minimum detectable polarization (MDP) levels of a few percent for the brightest GRBs

Hard x-ray polarimeter for gamma-ray bursts and solar flares

We report on the development of a dedicated polarimeter design that is capable of studying the linear polarization of hard X-rays (50-300 keV) from gamma-ray bursts and solar flares. This compact design, based on the use of a large area position-sensitive PMT (PSPMT), is referred to as GRAPE (Gamma-RAy Polarimeter Experiment). The PSPMT is used to determine the Compton interaction location within an array of small plastic scintillator elements. Some of the photons that scatter within the plastic scintillator array are subsequently absorbed by a small centrally-located array of CsI(Tl) crystals that is read out by an independent multi-anode PMT. One feature of GRAPE that is especially attractive for studies of gamma-ray bursts is the significant off-axis response (at angles \u3e 60 degrees). The modular nature of this design lends itself toward its accomodation on a balloon or spacecraft platform. For an array of GRAPE modules, sensitivity levels below a few percent can be achieved for both gamma-ray bursts and solar flares. Here we report on the latest results from the testing of a laboratory science model

Dedicated polarimeter design for hard x-ray and soft gamma-ray astronomy

We have developed a modular design for a hard X-ray and soft gamma-ray polrimeter that we call GRAPE (Gamma RAy Polarimeter Experiment). Optimized for the energy range of 50-300 keV, the GRAPE design is a Compton polarimeter based on the use of an array of plastic scintillator scattering elements in conjunction with a centrally positioned high-Z calorimeter detector. Here we shall review the results from a laboratory model of the baseline GRAPE design. The baseline design uses a 5-inch diameter position sensitive PMT (PSPMT) for readout of the plastic scintillator array and a small array of CsI detectors for measurement of the scattered photon. An improved design, based on the use of large area multi-anode PMTs (MAPMTs), is also discussed along with plans for laboratory testing of a prototype. An array of GRAPE modules could be used as the basis for a dedicated science mission, either on a long duration balloon or on an orbital mission. With a large effective FoV, a non-imaging GRAPE mission would be ideal for studying polarization in transient sources (gamma ray bursts and solar flares). It may also prove useful for studying periodically varying sources, such as pulsars. An imaging system would improve the sensitivity of the polarization measurements for transient and periodic sources and may also permit the measurement of polarization in steady-state sources

Velocity-Metallicity Correlation for high-z DLA Galaxies: Evidence for a Mass-Metallicity Relation?

We used our database of VLT-UVES quasar spectra to build up a sample of 70 Damped Lyman-alpha (DLA) or strong sub-DLA systems with total neutral hydrogen column densities of log N(HI)>20 and redshifts in the range 1.7<z_abs<4.3. For each of the systems, we measured in an homogeneous manner the metallicities relative to Solar, [X/H] (with X=Zn, or S or Si), and the velocity widths of low-ionization line profiles, Delta V. We provide for the first time evidence for a correlation between DLA metallicity and line profile velocity width, which is detected at the 6.1sigma significance level. This confirms the trend previously observed in a much smaller sample by Wolfe & Prochaska (1998). The best-fit linear relation is [X/H]=1.55(\pm 0.12) log Delta V -4.33(\pm 0.23) with Delta V expressed in km/s. The slope of the DLA velocity-metallicity relation is the same within uncertainties between the higher (z_abs>2.43) and the lower (z_abs<2.43) redshift halves of our sample. However, the two populations of systems are statistically different. There is a strong redshift evolution in the sense that the mean metallicity and mean velocity width increase with decreasing redshift. We argue that the existence of a DLA velocity-metallicity correlation, over more than a factor of 100 spread in metallicity, is probably the consequence of an underlying mass-metallicity relation for the galaxies responsible for DLA absorption lines. Assuming a simple linear scaling of the galaxy luminosity with the mass of the dark-matter halo, we find that the slope of the DLA velocity-metallicity relation is consistent with that of the luminosity-metallicity relation derived for local galaxies. [...] abridged.Comment: 10 pages, 4 figures, A&A in pres

Eigenvalue variance bounds for Wigner and covariance random matrices

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd\"os, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices