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

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

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

Perceptions of a Dating Couple Conflict Resolution Interaction and Relationship Quality as Predictors of Depressive Symptoms in a College Student Sample

This study examines how perceptions of a conflict resolution interaction are related to measures of relationship quality and adjustment in a college student sample. Participants included 152 college students involved in a romantic relationship. All participants completed questionnaires to assess features of their romantic relationship and to measure depression. Couples participated in a recorded conflict resolution discussion, and used a video-recall procedure to assess their subjective perceptions of the interaction. Analyses revealed that depressive symptoms were significantly correlated with both low levels of positivity and high levels of negativity during the interaction and in the relationship generally. A stepwise multiple regression analysis revealed an association between perception of the interaction and depression in males, and an association between interaction in the relationship generally and depression in females. Results indicate the importance of socially supportive interaction and conflict resolution skills in college-aged couples to establish high-quality relationships and prevent the onset of depressive symptoms

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

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