Expected Supremum of a Random Linear Combination of Shifted Kernels

We address the expected supremum of a linear combination of shifts of the sinc kernel with random coefficients. When the coefficients are Gaussian, the expected supremum is of order \sqrt{\log n}, where n is the number of shifts. When the coefficients are uniformly bounded, the expected supremum is of order \log\log n. This is a noteworthy difference to orthonormal functions on the unit interval, where the expected supremum is of order \sqrt{n\log n} for all reasonable coefficient statistics.Comment: To appear in the Journal of Fourier Analysis and Application

Typical entanglement of stabilizer states

How entangled is a randomly chosen bipartite stabilizer state? We show that if the number of qubits each party holds is large the state will be close to maximally entangled with probability exponentially close to one. We provide a similar tight characterization of the entanglement present in the maximally mixed state of a randomly chosen stabilizer code. Finally, we show that typically very few GHZ states can be extracted from a random multipartite stabilizer state via local unitary operations. Our main tool is a new concentration inequality which bounds deviations from the mean of random variables which are naturally defined on the Clifford group.Comment: Final version, to appear in PRA. 11 pages, 1 figur

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

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

Average output entropy for quantum channels

We study the regularized average Renyi output entropy \bar{S}_{r}^{\reg} of quantum channels. This quantity gives information about the average noisiness of the channel output arising from a typical, highly entangled input state in the limit of infinite dimensions. We find a closed expression for \beta_{r}^{\reg}, a quantity which we conjecture to be equal to \Srreg. We find an explicit form for \beta_{r}^{\reg} for some entanglement-breaking channels, and also for the qubit depolarizing channel $\Delta_{\lambda}$ as a function of the parameter $\lambda$. We prove equality of the two quantities in some cases, in particular we conclude that for $\Delta_{\lambda}$ both are non-analytic functions of the variable $\lambda$.Comment: 32 pages, several plots and figures; positivity condition added for Theorem on entanglement breaking channels; new result for entrywise positive channel

The SDSS Damped Lya Survey: Data Release 1

We present the results from an automated search for damped Lya (DLA) systems in the quasar spectra of Data Release 1 from the Sloan Digital Sky Survey (SDSS-DR1). At z~2.5, this homogeneous dataset has greater statistical significance than the previous two decades of research. We derive a statistical sample of 71 damped Lya systems (>50 previously unpublished) at z>2.1 and measure HI column densities directly from the SDSS spectra. The number of DLA systems per unit redshift is consistent with previous measurements and we expect our survey has >95% completeness. We examine the cosmological baryonic mass density of neutral gas Omega_g inferred from the damped Lya systems from the SDSS-DR1 survey and a combined sample drawn from the literature. Contrary to previous results, the Omega_g values do not require a significant correction from Lyman limit systems at any redshift. We also find that the Omega_g values for the SDSS-DR1 sample do not decline at high redshift and the combined sample shows a (statistically insignificant) decrease only at z>4. Future data releases from SDSS will provide the definitive survey of DLA systems at z~2.5 and will significantly reduce the uncertainty in Omega_g at higher redshift.Comment: 12 pages, includes color figures. Accepted to PASP, April 20 200

Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics

Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss some techniques that yield uniform approximation over the whole eigenvalue spectrum and can take large steps even for high eigenvalues. In particular, we will focus on methods based on coefficient approximation which replace the coefficient functions of the Sturm-Liouville problem by simpler approximations and then solve the approximating problem. The use of (modified) Magnus or Neumann integrators allows to extend the coefficient approximation idea to higher order methods

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