Error bounds on block Gauss Seidel solutions of coupled\ud multiphysics problems

Mathematical models in many fields often consist of coupled sub–models, each of which describe a different physical process. For many applications, the quantity of interest from these models may be written as a linear functional of the solution to the governing equations. Mature numerical solution techniques for the individual sub–models often exist. Rather than derive a numerical solution technique for the full coupled model, it is therefore natural to investigate whether these techniques may be used by coupling in a block Gauss–Seidel fashion. In this study, we derive two a posteriori bounds for such linear functionals. These bounds may be used on each Gauss–Seidel iteration to estimate the error in the linear functional computed using the single physics solvers, without actually solving the full, coupled problem. We demonstrate the use of the bound first by using a model problem from linear algebra, and then a linear ordinary differential equation example. We then investigate the effectiveness of the bound using a non–linear coupled fluid–temperature problem. One of the bounds derived is very sharp for most linear functionals considered, allowing us to predict very accurately when to terminate our block Gauss–Seidel iteration.\ud \ud Copyright c 2000 John Wiley & Sons, Ltd

Intra-day Variability of Sagittarius A* at 3 Millimeters

We report observations and analysis of flux monitoring of Sagittarius A* at 3-mm wavelength using the OVRO millimeter interferometer over a period of eight days (2002 May 23-30). Frequent phase and flux referencing (every 5 minutes) with the nearby calibrator source J1744-312 was employed to control for instrumental and atmospheric effects. Time variations are sought by computing and subtracting, from each visibility in the database, an average visibility obtained from all the data acquired in our monitoring program having similar uv spacings. This removes the confusing effects of baseline-dependent, correlated flux interference caused by the static, thermal emission from the extended source Sgr A West. Few-day variations up to ~20% and intra-day variability of \~20% and in some cases up to ~40% on few-hour time scales emerge from the differenced data on SgrA*. Power spectra of the residuals indicate the presence of hourly variations on all but two of the eight days. Monte Carlo simulation of red-noise light curves indicates that the hourly variations are well described by a red-noise power spectrum with P(f) ~ f^(-1). Of particular interest is a ~2.5 hour variation seen prominently on two consecutive days. An average power spectrum from all eight days of data reveals noteworthy power on this time scale. There is some indication that few-hour variations are more pronounced on days when the average daily flux is highest. We briefly discuss the possibility that these few-hour variations are due to the dynamical modulation of accreting gas around the central supermassive black hole, as well as the implications for the structure of the SgrA* photosphere at 3 mm. Finally, these data have enabled us to produce a high sensitivity 3-mm map of the extended thermal emission surrounding SgrA*.Comment: Accepted for publication in The Astrophysical Journal Letters, 8 pages, 4 figure

Violating conformal invariance: Two-dimensional clusters grafted to wedges, cones, and branch points of Riemann surfaces

We present simulations of 2-d site animals on square and triangular lattices in non-trivial geomeLattice animals are one of the few critical models in statistical mechanics violating conformal invariance. We present here simulations of 2-d site animals on square and triangular lattices in non-trivial geometries. The simulations are done with the newly developed PERM algorithm which gives very precise estimates of the partition sum, yielding precise values for the entropic exponent $\theta$ ($Z_N \sim \mu^N N^{-\theta}$). In particular, we studied animals grafted to the tips of wedges with a wide range of angles $\alpha$, to the tips of cones (wedges with the sides glued together), and to branching points of Riemann surfaces. The latter can either have $k$ sheets and no boundary, generalizing in this way cones to angles $\alpha > 360$ degrees, or can have boundaries, generalizing wedges. We find conformal invariance behavior, $\theta \sim 1/\alpha$, only for small angles ($\alpha \ll 2\pi$), while $\theta \approx const -\alpha/2\pi$ for $\alpha \gg 2\pi$. These scalings hold both for wedges and cones. A heuristic (non-conformal) argument for the behavior at large $\alpha$ is given, and comparison is made with critical percolation.Comment: 4 pages, includes 3 figure

On Vanishing of {K}ronecker Coefficients

It is shown that: (1) The problem of deciding positivity of Kronecker coefficients is NP-hard. (2) There exists a positive ($\# P$)-formula for a subclass of Kronecker coefficients whose positivity is NP-hard to decide. (3) For any $0 < \epsilon \le 1$, there exists $0<a<1$ such that, for all $m$, there exist $\Omega(2^{m^a})$ partition triples $(\lambda,\mu,\mu)$ in the Kronecker cone such that: (a) the Kronecker coefficient $k^\lambda_{\mu,\mu}$ is zero, (b) the height of $\mu$ is $m$, (c) the height of $\lambda$ is $\le m^\epsilon$, and (d) $|\lambda|= |\mu| \le m^3$. The last result takes a step towards proving the existence of occurrence-based representation-theoretic obstructions in the context of the GCT approach to the permanent vs. determinant problem. Its proof also illustrates the effectiveness of the explicit proof strategy of GCT

Mineralogical Composition of Diabase and Altered Dolostone from the St. Francois Mountains near Annapolis, Missouri, USA

Dolostones of Cambrian and Ordovician age are found in the St. Francois Mountains on the Ozark Plateau in southeastern Missouri and often lie unconformably over Precambrian basement rocks due to the paleo-topography. In some areas, there is distinct evidence of alteration and mineralization within the dolostones. One such location is found near Annapolis, Missouri where nearly all of the original dolomitic material is replaced with quartz. A diabase dike is in close proximity to the silicified dolostone and may have provided the conduit for fluids during alteration. The purpose of this study was to examine samples from the relatively unaltered dolostone, the silicified dolostone, and the diabase in order to establish the mineral changes associated with the alteration. Several samples were collected and analyzed in hand sample, in thin section, and using X-Ray Diffraction. In the diabase, plagioclase was abundant in both hand sample and thin section. The XRD results confirmed the presence of plagioclase as labradorite and supported evidence of magnetite as well. The highly altered dolostone exhibited original sedimentary laminations and varied in color. In thin section, the altered dolostone was very fine grained and the minerals that appeared to be present were quartz and an opaque mineral. After XRD analysis it was concluded that quartz was present and a pyrite structured sulfide mineral was present as well. The other four samples were dolomitic with laminations that varied in color from dark red to cream. XRD results indicated that dolomite, ankerite, and minrecordite were present in the dolostone

A Bose gas in a single-beam optical dipole trap

We study an ultracold Bose gas in an optical dipole trap consisting of one single focused laser beam. An analytical expression for the corresponding density of states beyond the usual harmonic approximation is obtained. We are thus able to discuss the existence of a critical temperature for Bose-Einstein condensation and find that the phase transition must be enabled by a cutoff near the threshold. Moreover, we study the dynamics of evaporative cooling and observe significant deviations from the findings for the well-established harmonic approximation. Furthermore, we investigate Bose-Einstein condensates in such a trap in Thomas-Fermi approximation and determine analytical expressions for chemical potential, internal energy and Thomas-Fermi radii beyond the usual harmonic approximation

Radio observations of the cool gas, dust, and star formation in the first galaxies

We summarize cm through submm observations of the host galaxies of z ~ 6 quasars. These observations reveal the cool molecular gas (the fuel for star formation), the warm dust (heated by star formation), the fine structure line emission (tracing the CNM and PDRs), and the synchrotron emission. Our results imply active star formation in ~ 30% of the host galaxies, with star formation rates ~ 10^3 M_sun/year, and molecular gas masses ~ 10^10 M_sun. Imaging of the [CII] emission from the most distant quasar reveals a 'maximal starburst disk' on a scale ~ 1.5 kpc. Gas dynamical studies suggest a departure of these galaxies from the low-z M_{BH} -- M_{bulge} relation, with the black holes being, on average, 15 times more massive than expected. Overall, we are witnessing the co-eval formation of massive galaxies and supermassive black holes within 1 Gyr of the Big Bang.Comment: First Stars and Galaxies: Challenges in the Next Decade, AIP, 2010; Austin TX (eds Whelan, Bromm, Yoshida); 7 page