Multi-Wavelength Observations of Short-Duration Gamma-Ray Bursts: Recent Results

The number of detections as well as significantly deep non-detections of optical/NIR afterglows of Type I (short-duration population) Gamma-Ray Bursts (GRBs) has become large enough that statistically meaningful samples can now be constructed. I present within some recent results on the luminosity distribution of Type I GRB afterglows in comparison to those of Type II GRBs (collapsar population), the issue of the existence of jet breaks in Type I GRB afterglows, and the discovery of dark Type I GRBs.Comment: 10 pages, 3 figures, based on an invited talk, to appear in the proceedings of the Gamma-Ray Burst Symposium 2012- IAA-CSIC - Marbella, editors: Castro-Tirado, A. J., Gorosabel, J. and Park, I. H; v2: accepted, slightly expanded, minor changes after referee repor

On Finite Rank Deformations of Wigner Matrices II: Delocalized Perturbations

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices. We assume that the matrix entries have finite fourth moment and extend the results by Capitaine, Donati-Martin, and F\'eral for perturbations whose eigenvectors are delocalized.Comment: We explained some proofs in greater detail, corrected several small misprints, and updated the bibliograph

A Noninformative Prior on a Space of Distribution Functions

In a given problem, the Bayesian statistical paradigm requires the specification of a prior distribution that quantifies relevant information about the unknowns of main interest external to the data. In cases where little such information is available, the problem under study may possess an invariance under a transformation group that encodes a lack of information, leading to a unique prior---this idea was explored at length by E.T. Jaynes. Previous successful examples have included location-scale invariance under linear transformation, multiplicative invariance of the rate at which events in a counting process are observed, and the derivation of the Haldane prior for a Bernoulli success probability. In this paper we show that this method can be extended, by generalizing Jaynes, in two ways: (1) to yield families of approximately invariant priors, and (2) to the infinite-dimensional setting, yielding families of priors on spaces of distribution functions. Our results can be used to describe conditions under which a particular Dirichlet Process posterior arises from an optimal Bayesian analysis, in the sense that invariances in the prior and likelihood lead to one and only one posterior distribution

Schematic Cut elimination and the Ordered Pigeonhole Principle [Extended Version]

In previous work, an attempt was made to apply the schematic CERES method [8] to a formal proof with an arbitrary number of {\Pi} 2 cuts (a recursive proof encapsulating the infinitary pigeonhole principle) [5]. However the derived schematic refutation for the characteristic clause set of the proof could not be expressed in the formal language provided in [8]. Without this formalization a Herbrand system cannot be algorithmically extracted. In this work, we provide a restriction of the proof found in [5], the ECA-schema (Eventually Constant Assertion), or ordered infinitary pigeonhole principle, whose analysis can be completely carried out in the framework of [8], this is the first time the framework is used for proof analysis. From the refutation of the clause set and a substitution schema we construct a Herbrand system.Comment: Submitted to IJCAR 2016. Will be a reference for Appendix material in that paper. arXiv admin note: substantial text overlap with arXiv:1503.0855

Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime. Asymptotics of the stationary distribution

We consider a heterogeneous queueing system consisting of one large pool of $O(r)$ identical servers, where $r\to\infty$ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the distributional sense. The system is heavily loaded in the Halfin-Whitt sense, namely the nominal utilization is $1-a/\sqrt{r}$ where $a>0$ is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue $Q^r(\infty)$. While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form $O(\sqrt{r})$ on both $Q^r(\infty)$ and the number of idle servers. The bounds are uniform w.r.t. parameter $r$ and the service policy. In particular, we show that $\limsup_r E \exp(\theta r^{-1/2}Q^r(\infty))<\infty$. Therefore, the sequence $r^{-1/2}Q^r(\infty)$ is tight and has a uniform exponential tail bound. We further consider the system with strictly positive abandonment rates, and show that in this case every weak limit $\hat{Q}(\infty)$ of $r^{-1/2}Q^r(\infty)$ has a sub-Gaussian tail. Namely $E[\exp(\theta (\hat{Q}(\infty))^2)]0$.Comment: 21 page