Inverse Depolarization: A Potential Probe of Internal Faraday Rotation and Helical Magnetic Fields in Extragalactic Radio Jets

Motivated by recent observations that show increasing fractional linear polarization with increasing wavelength in a small number of optically thin jet features, i.e. "inverse depolarization", we present a physical model that can explain this effect and may provide a new and complementary probe of the low energy particle population and possible helical magnetic fields in extragalactic radio jets. In our model, structural inhomogeneities in the jet magnetic field create cancellation of polarization along the line of sight. Internal Faraday rotation, which increases like wavelength squared, acts to align the polarization from the far and near sides of the jet, leading to increased polarization at longer wavelengths. Structural inhomogeneities of the right type are naturally produced in helical magnetic fields and will also appear in randomly tangled magnetic fields. We explore both alternatives and find that, for random fields, the length scale for tangling cannot be too small a fraction of the jet diameter and still be consistent with the relatively high levels of fractional polarization observed in these features. We also find that helical magnetic fields naturally produce transverse structure for inverse depolarization which may be observable even in partially resolved jets.Comment: 12 pages, 4 figures, accepted for publication in ApJ

Spectral Properties of Continuum Fibonacci Schr\"odinger Operators

We study continuum Schr\"odinger operators on the real line whose potentials are comprised of two compactly supported square-integrable functions concatenated according to an element of the Fibonacci substitution subshift over two letters. We show that the Hausdorff dimension of the spectrum tends to one in the small-coupling and high-energy regimes, regardless of the shape of the potential pieces

Noncommutative topology and Jordan operator algebras

Jordan operator algebras are norm-closed spaces of operators on a Hilbert space with $a^2 \in A$ for all $a \in A$. We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a `full' noncommutative topology in the C*-algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of C*-algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about $L+L^*$ for a left ideal $L$ in a C*-algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in $C^*$-algebras and various subspaces of $C^*$-algebras, related to open and closed projections, and technical $C^*$-algebraic results of Brown.Comment: Revision, many typos corrected and exposition improved in places. Section 2 expanded with some applications of the main theorem of that sectio

Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schoedinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schroedinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference

A Project Based Approach to Statistics and Data Science

In an increasingly data-driven world, facility with statistics is more important than ever for our students. At institutions without a statistician, it often falls to the mathematics faculty to teach statistics courses. This paper presents a model that a mathematician asked to teach statistics can follow. This model entails connecting with faculty from numerous departments on campus to develop a list of topics, building a repository of real-world datasets from these faculty, and creating projects where students interface with these datasets to write lab reports aimed at consumers of statistics in other disciplines. The end result is students who are well prepared for interdisciplinary research, who are accustomed to coping with the idiosyncrasies of real data, and who have sharpened their technical writing and speaking skills

Direct Distance Measurements to Superluminal Radio Sources

We present a new technique for directly measuring the distances to superluminal radio sources. By comparing the observed proper motions of components in a parsec scale radio jet to their measured Doppler factors, we can deduce the distance to the radio source independent of the standard rungs in the cosmological distance ladder. This technique requires that the jet angle to the line of sight and the ratio of pattern to flow velocities are sufficiently constrained. We evaluate a number of possibilities for constraining these parameters and demonstrate the technique on a well defined component in the parsec scale jet of the quasar 3C279 (z = 0.536). We find an angular size distance to 3C279 of greater than 1.8 (+0.5,-0.3) n^{1/8} Gpc, where n is the ratio of the energy density in the magnetic field to the energy density in the radiating particles in that jet component. For an Einstein-de Sitter Universe, this measurement would constrain the Hubble constant to be H < 65 n^{-1/8} km/s/Mpc at the two sigma level. Similar measurements on higher redshift sources may help discriminate between cosmological models.Comment: 18 pages, 8 figures, to be published in The Astrophysical Journa

Chemical Modelling of Young Stellar Objects, I. Method and Benchmarks

Upcoming facilities such as the Herschel Space Observatory or ALMA will deliver a wealth of molecular line observations of young stellar objects (YSOs). Based on line fluxes, chemical abundances can then be estimated by radiative transfer calculations. To derive physical properties from abundances, the chemical network needs to be modeled and fitted to the observations. This modeling process is however computationally exceedingly demanding, particularly if in addition to density and temperature, far UV (FUV) irradiation, X-rays, and multi-dimensional geometry have to be considered. We develop a fast tool, suitable for various applications of chemical modeling in YSOs. A grid of the chemical composition of the gas having a density, temperature, FUV irradiation and X-ray flux is pre-calculated as a function of time. A specific interpolation approach is developed to reduce the database to a feasible size. Published models of AFGL 2591 are used to verify the accuracy of the method. A second benchmark test is carried out for FUV sensitive molecules. The novel method for chemical modeling is more than 250,000 times faster than direct modeling and agrees within a mean factor of 1.35. The tool is distributed for public use. In the course of devloping the method, the chemical evolution is explored: We find that X-ray chemistry in envelopes of YSOs can be reproduced by means of an enhanced cosmic-ray ionization rate. We further find that the abundance of CH+ in low-density gas with high ionization can be enhanced by the recombination of doubly ionized carbon (C++) and suggest a new value for the initial abundance of the main sulphur carrier in the hot-core.Comment: Accepted by ApJS. 24 pages, 15 figures. A version with higher resolution images is available from http://www.astro.phys.ethz.ch/staff/simonbr/papgridI.pdf . Online data available at http://www.astro.phys.ethz.ch/chemgrid.html . Second paper of this series of papers available at arXiv:0906.058

Completely contractive projections on operator algebras

The main goal of this paper is to find operator algebra variants of certain deep results of Stormer, Friedman and Russo, Choi and Effros, Effros and Stormer, Robertson and Youngson, Youngson, and others, concerning projections on C*-algebras and their ranges. (See papers of these authors referenced in the bibliography.) In particular we investigate the `bicontractive projection problem' and related questions in the category of operator algebras. To do this, we will add the ingredient of `real positivity' from recent papers of the first author with Read.Comment: To appear Pacific J Math; several corrections and small improvements. Keywords Operator algebra, completely contractive map, projection, conditional expectation, bicontractive projection, real positive, noncommutative Banach-Stone theore

Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II

We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast Fourier Transform algorithms %\cite{sovi}, we make explicit use of the path algebra connection to the construction of Gel'fand-Tsetlin bases and work in the setting of quivers. We relate this framework to the construction of a {\em configuration space} derived from a Bratteli diagram. In this setting the complexity of an algorithm for computing a Fourier transform reduces to the calculation of the dimension of the associated configuration space. Our methods give improved upper bounds for computing the Fourier transform for the general linear groups over finite fields, the classical Weyl groups, and homogeneous spaces of finite groups, while also recovering the best known algorithms for the symmetric group and compact Lie groups.Comment: 53 pages, 5 appendices, 34 figure

Contractive projections and operator spaces

Parallel to the study of finite dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces H_n^k,0< k < n+1, generalizing the row and column Hilbert spaces R_n,C_n and show that an atomic subspace X of B(H) which is the range of a contractive projection on B(H) is isometrically completely contractive to a direct sum of the H_n^k and Cartan factors of types 1 to 4. In particular, for finite dimensional X, this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w*-closed JW*-triples without an infinite dimensional rank 1 w^*-closed idealComment: 40 pages, latex, the paper was submitted in October of 2000 and an announcement with the same title appeared in C. R. Acad. Sci. Paris 331 (2000), 873-87