On the leading coefficient of polynomials orthogonal over domains with corners

Let $G$ be the interior domain of a piecewise analytic Jordan curve without cusps. Let $\{p_n\}_{n=0}^\infty$ be the sequence of polynomials that are orthonormal over $G$ with respect to the area measure, with each $p_n$ having leading coefficient $\lambda_n>0$. N. Stylianopoulos has recently proven that the asymptotic behavior of $\lambda_n$ as $n\to\infty$ is given by $$\frac{n+1}{\pi}\frac{\gamma^{2n+2}}{ \lambda_n^{2}}=1-\alpha_n,$$ where $\alpha_n=O(1/n)$ as $n\to\infty$ and $\gamma$ is the reciprocal of the logarithmic capacity of the boundary $\partial G$. In this paper, we prove that the $O(1/n)$ estimate for the error term $\alpha_n$ is, in general, best possible, by exhibiting an example for which $$\liminf_{n\to\infty}\,n\alpha_n>0.$$ The proof makes use of the Faber polynomials, about which a conjecture is formulated.Comment: 7 page

The Coexistence of Classical Bulges, Pseudobulges, and Supermassive Black Holes

Some S0 and early-type spiral galaxies possess "composite bulges"; in these galaxies, the photometric bulge -- the central stellar light in excess of the disk light -- is composed of both a "(disky) pseudobulge", with a flattened, disklike morphology and relatively cool stellar kinematics, and a rounder, kinematically hot "classical bulge" embedded within. I speculate that supermassive black holes (SMBH) in such galaxies may correlate with the classical-bulge component only, and not with the pseudobulge component; preliminary comparisons with SMBH masses appear to support this hypothesis.Comment: LaTeX, 4 pages, 2 PDF figures. To appear in the proceedings of "The Monster's Fiery Breath: Feedback in Galaxies, Groups, and Clusters", eds. Sebastian Heinz and Eric Wilcots (AIP conference series

Can D-Branes Wrap Nonrepresentable Cycles?

Sometimes a homology cycle of a nonsingular compactification manifold cannot be represented by a nonsingular submanifold. We want to know whether such nonrepresentable cycles can be wrapped by D-branes. A brane wrapping a representable cycle carries a K-theory charge if and only if its Freed-Witten anomaly vanishes. However some K-theory charges are only carried by branes that wrap nonrepresentable cycles. We provide two examples of Freed-Witten anomaly-free D6-branes wrapping nonrepresentable cycles in the presence of a trivial NS 3-form flux. The first occurs in type IIA string theory compactified on the Sp(2) group manifold and the second in IIA on a product of lens spaces. We find that the first D6-brane carries a K-theory charge while the second does not.Comment: 11 pages, no figure