Uncovering Ramanujan's "Lost" Notebook: An Oral History

Here we weave together interviews conducted by the author with three prominent figures in the world of Ramanujan's mathematics, George Andrews, Bruce Berndt and Ken Ono. The article describes Andrews's discovery of the "lost" notebook, Andrews and Berndt's effort of proving and editing Ramanujan's notes, and recent breakthroughs by Ono and others carrying certain important aspects of the Indian mathematician's work into the future. Also presented are historical details related to Ramanujan and his mathematics, perspectives on the impact of his work in contemporary mathematics, and a number of interesting personal anecdotes from Andrews, Berndt and Ono

Colors, magnitudes and velocity dispersions in early-type galaxies: Implications for galaxy ages and metallicities

We present an analysis of the color-magnitude-velocity dispersion relation for a sample of 39320 early-type galaxies within the Sloan Digital Sky Survey. We demonstrate that the color-magnitude relation is entirely a consequence of the fact that both the luminosities and colors of these galaxies are correlated with stellar velocity dispersions. Previous studies of the color-magnitude relation over a range of redshifts suggest that the luminosity of an early-type galaxy is an indicator of its metallicity, whereas residuals in color from the relation are indicators of the luminosity-weighted age of its stars. We show that this, when combined with our finding that velocity dispersion plays a crucial role, has a number of interesting implications. First, galaxies with large velocity dispersions tend to be older (i.e., they scatter redward of the color-magnitude relation). Similarly, galaxies with large dynamical mass estimates also tend to be older. In addition, at fixed luminosity, galaxies which are smaller, or have larger velocity dispersions, or are more massive, tend to be older. Second, models in which galaxies with the largest velocity dispersions are also the most metal poor are difficult to reconcile with our data. However, at fixed velocity dispersion, galaxies have a range of ages and metallicities: the older galaxies have smaller metallicities, and vice-versa. Finally, a plot of velocity dispersion versus luminosity can be used as an age indicator: lines of constant age run parallel to the correlation between velocity dispersion and luminosity.Comment: 12 pages, 9 figures. Accepted by A

An Empirical Calibration of the Completeness of the SDSS Quasar Survey

Spectra of nearly 20000 point-like objects to a Galactic reddening corrected magnitude of i=19.1 have been obtained to test the completeness of the SDSS quasar survey. The spatially-unresolved objects were selected from all regions of color space, sparsely sampled from within a 278 sq. deg. area of sky covered by this study. Only ten quasars were identified that were not targeted as candidates by the SDSS quasar survey (including both color and radio source selection). The inferred density of unresolved quasars on the sky that are missed by the SDSS algorithm is 0.44 per sq. deg, compared to 8.28 per sq. deg. for the selected quasar density, giving a completeness of 94.9(+2.6,-3.8) to the limiting magnitude. Omitting radio selection reduces the color-only selection completeness by about 1%. Of the ten newly identified quasars, three have detected broad absorption line systems, six are significantly redder than other quasars at the same redshift, and four have redshifts between 2.7 and 3.0 (the redshift range where the SDSS colors of quasars intersect the stellar locus). The fraction of quasars missed due to image defects and blends is approximately 4%, but this number varies by a few percent with magnitude. Quasars with extended images comprise about 6% of the SDSS sample, and the completeness of the selection algorithm for extended quasars is approximately 81%, based on the SDSS galaxy survey. The combined end-to-end completeness for the SDSS quasar survey is approximately 89%. The total corrected density of quasars on the sky to i=19.1 is estimated to be 10.2 per sq. deg.Comment: 37 pages, 10 figures, accepted for publication in A

Four problems with global carbon markets: a critical review

This article offers a critique of global carbon markets and trading, with a special focus on the Clean Development Mechanism of the Kyoto Protocol. It explores problems with the use of tradable permits to address climate change revolving around four areas: homogeneity, justice, gaming, and information. Homogeneity problems arise from the non-linear nature of climate change and sensitivity of emissions, which complicate attempts to calculate carbon offsets. Justice problems involve issues of dependency and the concentration of wealth among the rich, meaning carbon trading often counteracts attempts to reduce poverty. Gaming problems include pressures to promote high-volume, least-cost projects and the consequences of emissions leakage. Information problems encompass transaction costs related to carbon trading and market participation and the comparatively weak institutional capacity of project evaluators

Two New Gravitationally Lensed Double Quasars from the Sloan Digital Sky Survey

We report the discoveries of the two-image gravitationally lensed quasars, SDSS J0746+4403 and SDSS J1406+6126, selected from the Sloan Digital Sky Survey (SDSS). SDSS J0746+4403, which will be included in our lens sample for statistics and cosmology, has a source redshift of z_s=2.00, an estimated lens redshift of z_l~0.3, and an image separation of 1.08". SDSS J1406+6126 has a source redshift of z_s=2.13, a spectroscopically measured lens redshift of z_l=0.27, and an image separation of 1.98". We find that the two quasar images of SDSS J1406+6126 have different intervening MgII absorption strengths, which are suggestive of large variations of absorbers on kpc scales. The positions and fluxes of both the lensed quasar systems are easily reproduced by simple mass models with reasonable parameter values. These objects bring to 18 the number of lensed quasars that have been discovered from the SDSS data.Comment: 25 pages, 6 figures, The Astronomical Journal accepte

Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the one-dimensional stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov--Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.Comment: 18 pages, no figure