Simultaneous Confidence Bands for Functional Data Using the Gaussian Kinematic Formula

This article constructs simultaneous confidence bands (SCBs) for functional parameters using the Gaussian Kinematic formula of $t$-processes (tGKF). Although the tGKF relies on Gaussianity, we show that a central limit theorem (CLT) for the parameter of interest is enough to obtain asymptotically precise covering rates even for non-Gaussian processes. As a proof of concept we study the functional signal-plus-noise model and derive a CLT for an estimator of the Lipschitz-Killing curvatures, the only data dependent quantities in the tGKF SCBs. Extensions to discrete sampling with additive observation noise are discussed using scale space ideas from regression analysis. Here we provide sufficient conditions on the processes and kernels to obtain convergence of the functional scale space surface. The theoretical work is accompanied by a simulation study comparing different methods to construct SCBs for the population mean. We show that the tGKF works well even for small sample sizes and only a Rademacher multiplier-$t$ bootstrap performs similarily well. For larger sample sizes the tGKF often outperforms the bootstrap methods and is computational faster. We apply the method to diffusion tensor imaging (DTI) fibers using a scale space approach for the difference of population means. R code is available in our Rpackage SCBfda

The role of cooling flows in galaxy formation

The present structure of galaxies is governed by the radiative dissipation of the gravitational and supernova energy injected during formation. A crucial aspect of this process is whether the gas cools as fast as it falls into the gravitational potential well. If it does then rapid normal star formation is assumed to ensue. If not, and the gas can still cool by the present time, then the situation resembles that of a cooling flow, such as commonly found in clusters of galaxies. The cooled matter is assumed to accumulate as very cold clouds and/or low mass stars, i.e. as baryonic dark matter. In this paper we investigate the likelihood of a cooling flow phase during the hierarchical formation of galaxies. We concentrate on the behaviour of the gas, using a highly simplified treatment of the evolution of the dark matter potential within which the gas evolves. We assume that normal star formation is limited by how much gas the associated supernovae can unbind and allow the gas profile to flatten as a consequence of supernova energy injection. We find that cooling flows are an important phase in the formation of most galaxies with total (dark plus luminous) masses approxgt 10^12 Msun , creating about 20 per cent of the total dark halo in a galaxy such as our own and a smaller but comparable fraction of an elliptical galaxy of similar mass. The onset of a cooling flow determines the upper mass limit for the formation of a visible spheroid from gas, setting a characteristic mass scale for normal galaxies. We argue that disk formation requires a cooling flow phase and that dissipation in the cooling flow phase is the most important factor in the survival of normal galaxies during subsequent hierarchical mergers.Comment: uuencoded compressed postscript. The preprint is also available at http://www.ast.cam.ac.uk/preprint/PrePrint.htm

Existence of mesons after deconfinement

We investigate the possibility for a quark-antiquark pair to form a bound state at temperatures higher than the critical one ($T>T_c$), thus after deconfinement. Our main goal is to find analytical criteria constraining the existence of such mesons. Our formalism relies on a Schr\"{o}dinger equation for which we study the physical consequences of both using the free energy and the internal energy as potential term, assuming a widely accepted temperature-dependent Yukawa form for the free energy and a recently proposed nonperturbative form for the screening mass. We show that using the free energy only allows for the 1S bottomonium to be bound above $T_c$, with a dissociation temperature around $1.5\times T_c$. The situation is very different with the internal energy, where we show that no bound states at all can exist in the deconfined phase. But, in this last case, quasi-bound states could be present at higher temperatures because of a positive barrier appearing in the potential.Comment: 14 pages, 3 figures; only the case T>T_c is discussed in v

Functional delta residuals and applications to functional effect sizes

Given a functional central limit (fCLT) and a parameter transformation, we use the functional delta method to construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the transformed limit process. Moreover, we prove a multiplier bootstrap fCLT theorem for these transformed residuals and show how this can be used to construct simultaneous confidence bands for transformed functional parameters. As motivation for this methodology, we provide the formal application of these residuals to a functional version of the effect size parameter Cohen's $d$, a problem appearing in current brain imaging applications. The performance and necessity of such residuals is illustrated in a simulation experiment for the covering rate of simultaneous confidence bands for the functional Cohen's $d$ parameter

Characterizing and Improving the Reliability of Broadband Internet Access

In this paper, we empirically demonstrate the growing importance of reliability by measuring its effect on user behavior. We present an approach for broadband reliability characterization using data collected by many emerging national initiatives to study broadband and apply it to the data gathered by the Federal Communications Commission's Measuring Broadband America project. Motivated by our findings, we present the design, implementation, and evaluation of a practical approach for improving the reliability of broadband Internet access with multihoming.Comment: 15 pages, 14 figures, 6 table

Two-temperature coronae in active galactic nuclei

We show that coronal magnetic dissipation in thin active sheets that sandwich standard thin accretion disks in active galactic nuclei may account for canonical electron temperatures of a few $\times 10^9$K if protons acquire most of the dissipated energy. Coulomb collisions transfer energy from the ions to the electrons, which subsequently cool rapidly by inverse-Compton scattering. In equilibrium, the proton energy density likely exceeds that of the magnetic field and both well exceed the electron and photon energy densities. The Coulomb energy transfer from protons to electrons is slow enough to maintain a high proton temperature, but fast enough to explain observed rapid X-ray variabilities in Seyferts. The $\sim 10^9$K electron temperature is insensitive to the proton temperature when the latter is $\ge 10^{12}$K.Comment: 5 pages LaTex, and 2 .ps figures, submitted to MNRAS, 4/9

Revealing the X-ray source in IRAS 13224-3809 through flux-dependent reverberation lags

IRAS 13224-3809 was observed in 2011 for 500 ks with the XMM-Newton observatory. We detect highly significant X-ray lags between soft (0.3 - 1 keV) and hard (1.2 - 5 keV) energies. The hard band lags the soft at low frequencies (i.e. hard lag), while the opposite (i.e. soft lag) is observed at high frequencies. In this paper, we study the lag during flaring and quiescent periods. We find that the frequency and absolute amplitude of the soft lag is different during high-flux and low-flux periods. During the low flux intervals, the soft lag is detected at higher frequencies and with smaller amplitude. Assuming that the soft lag is associated with the light travel time between primary and reprocessed emission, this behaviour suggests that the X-ray source is more compact during low-flux intervals, and irradiates smaller radii of the accretion disc (likely because of light bending effects). We continue with an investigation of the lag dependence on energy, and find that isolating the low-flux periods reveals a strong lag signature at the Fe K line energy, similar to results found using 1.3 Ms of data on another well known Narrow-Line Seyfert I galaxy, 1H0707-495.Comment: 6 pages, 8 figures, accepted for publication in MNRA