Digital Repository of Mathematical Formulae

The purpose of the NIST Digital Repository of Mathematical Formulae (DRMF) is to create a digital compendium of mathematical formulae for orthogonal polynomials and special functions (OPSF) and of associated mathematical data. The DRMF addresses needs of working mathematicians, physicists and engineers: providing a platform for publication and interaction with OPSF formulae on the web. Using MediaWiki extensions and other existing technology (such as software and macro collections developed for the NIST Digital Library of Mathematical Functions), the DRMF acts as an interactive web domain for OPSF formulae. Whereas Wikipedia and other web authoring tools manifest notions or descriptions as first class objects, the DRMF does that with mathematical formulae. See http://gw32.iu.xsede.org/index.php/Main_Page

KSZ tomography and the bispectrum

Several statistics have been proposed for measuring the kSZ effect by combining the small-scale CMB with galaxy surveys. We review five such statistics, and show that they are all mathematically equivalent to the optimal bispectrum estimator of type $\langle ggT \rangle$. Reinterpreting these kSZ statistics as special cases of bispectrum estimation makes many aspects transparent, for example optimally weighting the estimator, or incorporating photometric redshift errors. We analyze the information content of the bispectrum and show that there are two observables: the small-scale galaxy-electron power spectrum $P_{ge}(k_S)$, and the large-scale galaxy-velocity power spectrum $P_{gv}(k)$. The cosmological constraining power of the kSZ arises from its sensitivity to fluctuations on large length scales, where its effective noise level can be much better than galaxy surveys.Comment: 39 page

Multi-Satellite Observations of Cygnus X-1 to Study the Focused Wind and Absorption Dips

High-mass X-ray binary systems are powered by the stellar wind of their donor stars. The X-ray state of Cygnus X-1 is correlated with the properties of the wind which defines the environment of mass accretion. Chandra-HETGS observations close to orbital phase 0 allow for an analysis of the photoionzed stellar wind at high resolution, but because of the strong variability due to soft X-ray absorption dips, simultaneous multi-satellite observations are required to track and understand the continuum, too. Besides an earlier joint Chandra and RXTE observation, we present first results from a recent campaign which represents the best broad-band spectrum of Cyg X-1 ever achieved: On 2008 April 18/19 we observed this source with XMM-Newton, Chandra, Suzaku, RXTE, INTEGRAL, Swift, and AGILE in X- and gamma-rays, as well as with VLA in the radio. After superior conjunction of the black hole, we detect soft X-ray absorption dips likely due to clumps in the focused wind covering >95 % of the X-ray source, with column densities likely to be of several 10^23 cm^-2, which also affect photon energies above 20 keV via Compton scattering.Comment: 10 pages, contributed talk at the 7th Microquasar Workshop, Foca, Turkey, Sept. 1-5, 200

Non-equilibrium dynamics of stochastic point processes with refractoriness

Stochastic point processes with refractoriness appear frequently in the quantitative analysis of physical and biological systems, such as the generation of action potentials by nerve cells, the release and reuptake of vesicles at a synapse, and the counting of particles by detector devices. Here we present an extension of renewal theory to describe ensembles of point processes with time varying input. This is made possible by a representation in terms of occupation numbers of two states: Active and refractory. The dynamics of these occupation numbers follows a distributed delay differential equation. In particular, our theory enables us to uncover the effect of refractoriness on the time-dependent rate of an ensemble of encoding point processes in response to modulation of the input. We present exact solutions that demonstrate generic features, such as stochastic transients and oscillations in the step response as well as resonances, phase jumps and frequency doubling in the transfer of periodic signals. We show that a large class of renewal processes can indeed be regarded as special cases of the model we analyze. Hence our approach represents a widely applicable framework to define and analyze non-stationary renewal processes.Comment: 8 pages, 4 figure