Stochastic switching in infinite dimensions with applications to random parabolic PDEs

We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides.Comment: 30 pages. Published version containing some minor corrections and improvement

Smooth invariant densities for random switching on the torus

We consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2d-torus, with the random switchings happening according to a Poisson process. Assuming that the driving vector fields are transversal to each other at all points of the torus and that each of them allows for a smooth invariant density and no periodic orbits, we prove that the switched system also has a smooth invariant density, for every switching rate. Our approach is based on an integration by parts formula inspired by techniques from Malliavin calculus.Comment: 19 page

Sticky central limit theorems on open books

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fr\'{e}chet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension $1$ and hence measure $0$) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).Comment: Published in at http://dx.doi.org/10.1214/12-AAP899 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Testing Lorentz Invariance with Neutrinos from Ultrahigh Energy Cosmic Ray Interactions

We have previously shown that a very small amount of Lorentz invariance violation (LIV), which suppresses photomeson interactions of ultrahigh energy cosmic rays (UHECRs) with cosmic background radiation (CBR) photons, can produce a spectrum of cosmic rays that is consistent with that currently observed by the Pierre Auger Observatory (PAO) and HiRes experiments. Here, we calculate the corresponding flux of high energy neutrinos generated by the propagation of UHECR protons through the CBR in the presence of LIV. We find that LIV produces a reduction in the flux of the highest energy neutrinos and a reduction in the energy of the peak of the neutrino energy flux spectrum, both depending on the strength of the LIV. Thus, observations of the UHE neutrino spectrum provide a clear test for the existence and amount of LIV at the highest energies. We further discuss the ability of current and future proposed detectors make such observations.Comment: final version to appear in Astroparticle Physic

The Palomar Transient Factory Orion Project: Eclipsing Binaries and Young Stellar Objects

The Palomar Transient Factory (PTF) Orion project is an experiment within the broader PTF survey, a systematic automated exploration of the sky for optical transients. Taking advantage of the wide field of view available using the PTF camera at the Palomar 48" telescope, 40 nights were dedicated in December 2009-January 2010 to perform continuous high-cadence differential photometry on a single field containing the young (7-10Myr) 25 Ori association. The primary motivation for the project is to search for planets around young stars in this region. The unique data set also provides for much ancillary science. In this first paper we describe the survey and data reduction pipeline, and present initial results from an inspection of the most clearly varying stars relating to two of the ancillary science objectives: detection of eclipsing binaries and young stellar objects. We find 82 new eclipsing binary systems, 9 of which we are candidate 25 Ori- or Orion OB1a-association members. Of these, 2 are potential young W UMa type systems. We report on the possible low-mass (M-dwarf primary) eclipsing systems in the sample, which include 6 of the candidate young systems. 45 of the binary systems are close (mainly contact) systems; one shows an orbital period among the shortest known for W UMa binaries, at 0.2156509 \pm 0.0000071d, with flat-bottomed primary eclipses, and a derived distance consistent with membership in the general Orion association. One of the candidate young systems presents an unusual light curve, perhaps representing a semi-detached binary system with an inflated low-mass primary or a star with a warped disk, and may represent an additional young Orion member. Finally, we identify 14 probable new classical T-Tauri stars in our data, along with one previously known (CVSO 35) and one previously reported as a candidate weak-line T-Tauri star (SDSS J052700.12+010136.8).Comment: 66 pages, 27 figures, accepted to Astronomical Journal. Minor typographical corrections and update to author affiliation

IPAC Image Processing and Data Archiving for the Palomar Transient Factory

The Palomar Transient Factory (PTF) is a multiepochal robotic survey of the northern sky that acquires data for the scientific study of transient and variable astrophysical phenomena. The camera and telescope provide for wide-field imaging in optical bands. In the five years of operation since first light on 2008 December 13, images taken with Mould-R and SDSS-g′ camera filters have been routinely acquired on a nightly basis (weather permitting), and two different Hα filters were installed in 2011 May (656 and 663 nm). The PTF image-processing and data-archival program at the Infrared Processing and Analysis Center (IPAC) is tailored to receive and reduce the data, and, from it, generate and preserve astrometrically and photometrically calibrated images, extracted source catalogs, and co-added reference images. Relational databases have been deployed to track these products in operations and the data archive. The fully automated system has benefited by lessons learned from past IPAC projects and comprises advantageous features that are potentially incorporable into other ground-based observatories. Both off-the-shelf and in-house software have been utilized for economy and rapid development. The PTF data archive is curated by the NASA/IPAC Infrared Science Archive (IRSA). A state-of-the-art custom Web interface has been deployed for downloading the raw images, processed images, and source catalogs from IRSA. Access to PTF data products is currently limited to an initial public data release (M81, M44, M42, SDSS Stripe 82, and the Kepler Survey Field). It is the intent of the PTF collaboration to release the full PTF data archive when sufficient funding becomes available

Tracking Student Propositions in an Inquiry System

We built software to support student reasoning about a phenomenon and development of hypotheses to explain it. The goal is to engage students in asking questions, generating hypotheses and testing predictions. Rashi, an intelligent tutor, tracks students’ investigations (e.g., hypotheses, questions, data collection, and inferences) and helps articulate how evidence and theories are related. The tutor provides advice, such as recognizing when data does not support a hypothesis Cases are presented in geology, biology or engineering, and students are scaffolded to use an inquiry-based approach to posit a theory to explain the situation. Generic and reusable structured tools guide students through exploration of ill-structured problem spaces, supporting student knowledge and scaffolding reasoning and diagnostic skills