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

Predicting Lotto Numbers

We investigate the "law of small numbers" using a unique panel data set on lotto gambling. Because we can track individual players over time, we can measure how they react to outcomes of recent lotto drawings. We can therefore test whether they behave as if they believe they can predict lotto numbers based on recent drawings. While most players pick the same set of number week after week without regards of numbers drawn or anything else, we find that those who do change, act on average in the way predicted by the law of small numbers as formalized in recent behavioral theory. In particular, on average they move away from numbers that are on streak, i.e. have been drawn several weeks in a row, consistent with the "hot hand fallacy".gambler's fallacy;hot hand fallacy;representativeness;law of small numbers

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

Observations of nitrogen isotope fractionation in deeply embedded protostars

(Abridged) The terrestrial planets, comets, and meteorites are significantly enriched in 15N compared to the Sun and Jupiter. While the solar and jovian nitrogen isotope ratio is believed to represent the composition of the protosolar nebula, a still unidentified process has caused 15N-enrichment in the solids. Several mechanisms have been proposed to explain the variations, including chemical fractionation. However, observational results that constrain the fractionation models are scarce. While there is evidence of 15N-enrichment in prestellar cores, it is unclear how the signature evolves into the protostellar phases. Our aim is to measure the 14N/15N ratio around three nearby, embedded low-to-intermediate-mass protostars. Isotopologues of HCN and HNC were used to probe the 14N/15N ratio. A selection of H13CN, HC15N, HN13C, and H15NC transitions was observed with the APEX telescope. The 14N/15N ratios were derived from the integrated intensities assuming a standard 12C/13C ratio. The assumption of optically thin emission was verified using radiative transfer modeling and hyperfine structure fitting. Two sources, IRAS 16293A and R CrA IRS7B, show 15N-enrichment by a factor of around 1.5-2.5 in both HCN and HNC with respect to the solar composition. Solar composition cannot be excluded for the third source, OMC-3 MMS6. Furthermore, there are indications of a trend toward increasing 14N/15N ratios with increasing outer envelope temperature. The enhanced 15N abundances in HCN and HNC found in two Class~0 sources (14N/15N of 160-290) and the tentative trend toward a temperature-dependent 14N/15N ratio are consistent with the chemical fractionation scenario, but 14N/15N ratios from additional tracers are indispensable for testing the models. Spatially resolved observations are needed to distinguish between chemical fractionation and isotope-selective photochemistry.Comment: Accepted for publication in Astronomy and Astrophysics. 16 pages, 13 figure

Dependence of the Superconducting Transition Temperature of MgB2 on Pressure to 20 GPa

The dependence of Tc on nearly hydrostatic pressure has been measured for an isotopically pure (11B) MgB2 sample in a helium-loaded diamond-anvil-cell to nearly 20 GPa. Tc decreases monotonically with pressure from 39.1 K at ambient pressure to 20.9 K at 19.2 GPa. The initial dependence is the same as that obtained earlier (dTc/dP = -1.11(2) K/GPa) on the same sample in a He-gas apparatus to 0.7 GPa. The observed pressure dependence Tc(P) to 20 GPa can be readily described in terms of simple lattice stiffening within standard phonon-mediated BCS superconductivity.Comment: 9 pages, 3 figure