Commuting self-adjoint extensions of symmetric operators defined from the partial derivatives

We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(\partial/\partial x_j):j=1,...,d} with domain C_c^\infty(\Omega) where the self-adjointness is defined relative to L^2(\Omega), and \Omega is a given open subset of R^d. The measure on \Omega is Lebesgue measure on R^d restricted to \Omega. The problem originates with I.E. Segal and B. Fuglede, and is difficult in general. In this paper, we provide a representation-theoretic answer in the special case when \Omega=I\times\Omega_2 and I is an open interval. We then apply the results to the case when \Omega is a d-cube, I^d, and we describe possible subsets \Lambda of R^d such that {e^(i2\pi\lambda \dot x) restricted to I^d:\lambda\in\Lambda} is an orthonormal basis in L^2(I^d).Comment: LaTeX2e amsart class, 18 pages, 2 figures; PACS numbers 02.20.Km, 02.30.Nw, 02.30.Tb, 02.60.-x, 03.65.-w, 03.65.Bz, 03.65.Db, 61.12.Bt, 61.44.B

The Sloan-Lens ACS Survey II: stellar populations and internal structure of early-type lens galaxies

We derive Fundamental Plane parameters of 15 early-type lens galaxies identified by the Sloan Lens ACS (SLACS) Survey. The size of the sample allows us to investigate for the first time the distribution of lens galaxies in the FP space. After correcting for evolution, we find that lens galaxies occupy a subset of the local FP. The edge-on projection (approximately M vs M/L) is indistinguishable from that of normal early-type galaxies. However -- within the fundamental plane -- the lens galaxies appear to concentrate at the edge of the region populated by normal early-type galaxies. We show that this is a result of our selection procedure (approximately velocity dispersion sigma>240km/s). We conclude that SLACS lenses are a fair sample of high velocity dispersion early-type galaxies. By comparing the central stellar velocity dispersion that of the best fit lens model, we find == =1.01+-0.02 with 0.065 rms scatter. We conclude that within the Einstein radii the SLACS lenses are very well approximated by isothermal ellipsoids, requiring a fine tuning of the stellar and dark matter distribution (bulge-halo ``conspiracy''). Interpreting the offset from the local FP in terms of evolution of the stellar mass-to-light ratio, we find for the SLACS lenses d log M/L_B/dz=-0.69+-0.08 (rms 0.11) consistent with the rate found for field early-type galaxies and with a scenario where most of the stars were formed at high redshift (>2) with secondary episodes of star formation providing less than ~10% of the stellar mass below z=1. We discuss star formation history and structural homogeneity in the context of formation mechanisms such as collisionless (``dry'') mergers. [Abridged]Comment: 2006, ApJ, 604, 622; 13 pages, 7 figures, 2 tables. Replaced Table 2, since the previous version was incorrectly sorted. Updated references. No changes in plots or content. More info available at SLACS website www.slacs.or

Asymptotic Behavior of Ext functors for modules of finite complete intersection dimension

Let $R$ be a local ring, and let $M$ and $N$ be finitely generated $R$-modules such that $M$ has finite complete intersection dimension. In this paper we define and study, under certain conditions, a pairing using the modules \Ext_R^i(M,N) which generalizes Buchweitz's notion of the Herbrand diference. We exploit this pairing to examine the number of consecutive vanishing of \Ext_R^i(M,N) needed to ensure that \Ext_R^i(M,N)=0 for all $i\gg 0$. Our results recover and improve on most of the known bounds in the literature, especially when $R$ has dimension at most two