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

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

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

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

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