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

Vulnerability : a view from different disciplines

Practitioners from different disciplines use different meanings and concepts of vulnerability, which, in turn, have led to diverse methods of measuring it. This paper presents a selective review of the literature from several disciplines to examine how they define and measure vulnerability. The disciplines include economics, sociology/anthropology, disaster management, environmental science, and health/nutrition. Differences between the disciplines can be explained by their tendency to focus on different components of risk, household responses to risk and welfare outcomes. In general, they focus either on the risks (at one extreme) or the underlying conditions (or outcomes) at the other. Trade-offs exist between simple measurement schemes and rich conceptual understanding.Environmental Economics&Policies,Health Economics&Finance,Insurance&Risk Mitigation,Economic Theory&Research,Rural Poverty Reduction

The LCO/Palomar 10,000 km/sec Cluster Survey. I. Properties of the Tully-Fisher Relation

The first results from a Tully-Fisher (TF) survey of cluster galaxies are presented. The galaxies are drawn from fifteen Abell clusters that lie in the redshift range 9000-12,000 km/sec and are distributed uniformly around the celestial sky. The data set consists of R-band CCD photometry and long- slit H-alpha spectroscopy. The rotation curves (RCs) are characterized by a turnover radius (r_t) and an asymptotic velocity v_a, while the surface brightness profiles are characterized in terms of an effective exponential surface brightness I_e and a scale length r_e. The TF scatter is minimized when the rotation velocity is measured at 2.0 +/- 0.2 r_e; a significantly larger scatter results when the rotation velocity is measured at > 3 or < 1.5 scale lengths. This effect demonstrates that RCs do not have a universal form, as has been suggested by Persic, Salucci, and Stel. In contrast to previous studies, a modest but statistically significant surface-brightness dependence of the TF relation is found, log v = const + 0.28*log L + 0.14*log I_e. This indicates a stronger parallel between the TF relation and the FP relations of elliptical galaxies than has previously been recognized. Future papers in this series will consider the implications of this cluster sample for deviations from Hubble flow on 100-200 Mpc scales.Comment: 35 pages, 8 figures, uses aaspp4.sty. Submitted to ApJ. Also available at http://astro.stanford.edu/jeff

Ages, metallicities and α\alpha-element enhancement for galaxies in Hickson compact groups

Central velocity dispersions and eight line-strength Lick indices have been determined from 1.3${\rm \AA}$ resolution long-slit spectra of 16 elliptical galaxies in Hickson compact groups. These data were used to determine galaxy properties (ages, metallicities and $\alpha$-element enhancements) and allowed a comparison with the parameters determined for a sample of galaxies in lower density environments, studied by Gonz\'alez (1993). The stellar population parameters were derived by comparison to single stellar population models of Thomas et al. (2003) and to a new set of SSP models for the indices Mg$_2$, Fe5270 and Fe5335 based on synthetic spetra. These models, based on an update version of the fitting functions presented in Barbuy et al. (2003), are fully described here. Our main results are: (1) the two samples have similar mean values for the metallicities and [$\alpha$/Fe] ratios, (2) the majority of the galaxies in compact groups seem to be old (median age of 14 Gyr for eight galaxies for which ages could be derived), in agreement with recent work by Proctor et al. (2004). These findings support two possible scenarios: compact groups are either young systems whose members have recently assembled and had not enough time to experience any merging yet or, instead, they are old systems that have avoided merging since their time of formation.Comment: Accepted for publication in A

The Star Formation Epoch of the Most Massive Early-Type Galaxies

We present new Keck spectroscopy of early-type galaxies in three galaxy clusters at z~0.5. We focus on the fundamental plane (FP) relation, and combine the kinematics with structural parameters determined from HST images. The galaxies obey clear FP relations, which are offset from the FP of the nearby Coma cluster due to passive evolution of the stellar populations. The z~0.5 data are combined with published data for 11 additional clusters at 0.18<z<1.28, to determine the evolution of the mean M/L(B) ratio of cluster galaxies with masses M>10^11 M_sun, as implied by the FP. We find dlog(M/L(B))/dz = -0.555+-0.042, stronger evolution than was previously inferred from smaller samples. The observed evolution depends on the luminosity-weighted mean age of the stars in the galaxies, the initial mass function (IMF), selection effects due to progenitor bias, and other parameters. Assuming a normal IMF but allowing for various other sources of uncertainty we find z* = 2.01+-0.20 for the luminosity-weighted mean star formation epoch. The main uncertainty is the slope of the IMF in the range 1-2 Solar masses: we find z* = 4.0 for a top-heavy IMF with slope x=0. The M/L(B) ratios of the cluster galaxies are compared to those of recently published samples of field early-type galaxies at 0.32<z<1.14. Assuming that progenitor bias and the IMF do not depend on environment we find that the present-day age of stars in massive field galaxies is 4.1 +- 2.0 % (~0.4 Gyr) less than that of stars in massive cluster galaxies, consistent with most, but not all, previous studies of local and distant early-type galaxies. This relatively small age difference is surprising in the context of expectations from ``standard'' hierarchical galaxy formation models. [ABRIDGED]Comment: Accepted for publication in ApJ. Minor corrections to match published versio

The Possible z=0.83 Precursors of z=0 M* Early-type Cluster Galaxies

We examine the distribution of stellar masses of galaxies in MS 1054-03 and RX J0152.7-1357, two X-ray selected clusters of galaxies at z=0.83. Our stellar mass estimates, from spectral energy distribution fitting, reproduce the dynamical masses as measured from velocity dispersions and half-light radii with a scatter of 0.2 dex in the mass for early-type galaxies. When we restrict our sample of members to high stellar masses, > 1e11.1 Msun (M* in the Schechter mass function for cluster galaxies), we find that the fraction of early-type galaxies is 79 +/- 6% at z=0.83 and 87 +/- 6% at z=0.023 for the Coma cluster, consistent with no evolution. Previous work with luminosity-selected samples finds that the early-type fraction in rich clusters declines from =~80% at z=0 to =~60% at z=0.8. The observed evolution in the early-type fraction from luminosity-selected samples must predominately occur among sub-M* galaxies. As M* for field and group galaxies, especially late-types, is below M* for clusters galaxies, infall could explain most of the recent early-type fraction growth. Future surveys could determine the morphological distributions of lower mass systems which will confirm or refute this explanation.Comment: 5 pages in emulate ApJ format with three color figures. Accepted for publication in ApJ Letters, v642n2. Updated to correct grammatical and typographic errors found by the journa

The Fundamental Plane in RX J0142.0+2131: a galaxy cluster merger at z=0.28

We present the Fundamental Plane (FP) in the z = 0.28 cluster of galaxies RX J0142.0+2131. There is no evidence for a difference in the slope of the FP when compared with the Coma cluster, although the internal scatter is larger. On average, stellar populations in RX J0142.0+2131 have rest-frame V-band mass-to-light ratios (M/L_V) 0.29+-0.03 dex lower than in Coma. This is significantly lower than expected for a passively-evolving cluster formed at z_f=2. Lenticular galaxies have lower average M/L_V and a distribution of M/L_V with larger scatter than ellipticals. Lower mass-to-light ratios are not due to recent star formation: our previous spectroscopic observations of RX J0142.0+2131 E/S0 galaxies showed no evidence for significant star-formation within the past ~4 Gyr. However, cluster members have enhanced alpha-element abundance ratios, which may act to decrease M/L_V. The increased scatter in the RX J0142.0+2131 FP reflects a large scatter in M/L_V implying that galaxies have undergone bursts of star formation over a range of epochs. The seven easternmost cluster galaxies, including the second brightest member, have M/L_V consistent with passive evolution and z_f = 2. We speculate that RX J0142.0+2131 is a cluster-cluster merger where the galaxies to the east are yet to fall into the main cluster body or have not experienced star formation as a result of the merger.Comment: 4 pages, 2 figures, accepted for publication in ApJ Letter

RXJ0142.0+2131: I. The galaxy content of an X-ray-luminous galaxy cluster at z=0.28

We present a photometric and spectroscopic study of stellar populations in the X-ray-luminous cluster of galaxies RXJ0142.0+2131 at z=0.280. This paper analyses the results of high signal-to-noise spectroscopy, as well as g'-, r'-, and i'-band imaging, using the Gemini Multi-Object Spectrograph on Gemini North. Of 43 spectroscopic targets, we find 30 cluster members over a range in color. Central velocity dispersions and absorption-line strengths for lines in the range 3700A < lambda_rest < 5800A are derived for cluster members, and are compared with a low-redshift sample of cluster galaxies, and single stellar population (SSP) models. We use a combination of these indicators to estimate luminosity-weighted mean ages, metallicities ([M/H]), and alpha-element abundance ratios ([alpha/Fe]). RXJ0142.0+2131 is a relatively poor cluster and lacks galaxies with high central velocity dispersions. Although the red sequence and the Faber-Jackson relation are consistent with pure passive evolution of the early-type population with a formation redshift of z_form = 2, the strengths of the 4000A break and scaling relations between metal line indices and velocity dispersion reject this model with high significance. By inverting SSP models for the Hbeta_G, Mgb, and line indices, we calculate that, at a given velocity dispersion and metallicity, galaxies in RXJ0142.0+2131 have luminosity-weighted mean ages 0.14 +- 0.07 dex older than the low-redshift sample. We also find that [alpha/Fe] in stellar populations in RXJ0142.0+2131 is 0.14 +- 0.03 greater than at low redshift. All scaling relations are consistent with these estimated offsets. (abridged)Comment: AJ, accepted. 31 pages, 13 figures, uses emulateapj.cls. High-resolution figures available on request from first autho

The Universality of the Fundamental Plane of E and S0 Galaxies. Spectroscopic data

We present here central velocity dispersion measurements for 325 early-type galaxies in eight clusters and groups of galaxies, including new observations for 212 galaxies. The clusters and groups are the A262, A1367, Coma (A1656), A2634, Cancer and Pegasus clusters, and the NGC 383 and NGC 507 groups. The new measurements were derived from medium dispersion spectra, that cover 600 A centered on the Mg Ib triplet at lambda ~ 5175. Velocity dispersions were measured using the Tonry & Davis cross-correlation method, with a typical accuracy of 6%. A detailed comparison with other data sources is made.Comment: 12 pages, 5 tables, 3 figures, to appear in AJ. Note that tables 2 and 3 are in separate files, as they should be printed in landscape forma