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 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 Fundamental Plane at z=1.27: First Calibration of the Mass Scale of Red Galaxies at Redshifts z>1

We present results on the Fundamental Plane (FP) of early-type galaxies in the cluster RDCS J0848+4453 at z=1.27. Internal velocity dispersions of three K-selected early-type galaxies are determined from deep Keck spectra. Structural parameters are determined from HST NICMOS images. The galaxies show substantial offsets from the FP of the nearby Coma cluster, as expected from passive evolution of their stellar populations. The offsets from the FP can be expressed as offsets in M/L ratio. The M/L ratios of the two most massive galaxies are consistent with an extrapolation of results obtained at z=0.02-0.83. The evolution of early-type galaxies with masses >10^11 M_sun is well described by ln M/L(B) = (-1.06 +- 0.09) z, corresponding to passive evolution of -1.50 +- 0.13 mag at z=1.3. Ignoring selection effects, the best fitting stellar formation redshift is z*=2.6, corresponding to a luminosity weighted age at the epoch of observation of ~2 Gyr. The M/L ratios of these two galaxies are also in excellent agreement with predictions from models that include progenitor bias. The third galaxy is a factor ~10 less massive than the other two, shows strong Balmer absorption lines in its spectrum, and is offset from the Coma Fundamental Plane by 2.9 mag in rest-frame B. Despite their large range in M/L ratios, all three galaxies fall in the ``Extremely Red Object'' (ERO) class with I-H>3 and R-K>5, and our results show that it is hazardous to use simple models for converting luminosity to mass for these objects. Measurements of M/L ratios at high redshift can be considered first steps to empirically disentangle luminosity and mass evolution at the high mass end of the galaxy population, lifting an important degeneracy in the interpretation of evolution of the luminosity function. [SHORTENED]Comment: Accepted for publication in the Astrophysical Journa

Tuning of magnetic and electronic states by control of oxygen content in lanthanum strontium cobaltites

We report on the magnetic, resistive, and structural studies of perovskite La$_{1/3}$Sr$_{2/3}$CoO$_{3-\delta}$. By using the relation of synthesis temperature and oxygen partial pressure to oxygen stoichiometry obtained from thermogravimetric analysis, we have synthesized a series of samples with precisely controlled $\delta=0.00-0.49$. These samples show three structural phases at $\delta=0.00-0.15$, $\approx0.25$, $\approx0.5$, and two-phase behavior for other oxygen contents. The stoichiometric material with $\delta=0.00$ is a cubic ferromagnetic metal with the Curie temperature $T_{\rm C}=274$ K. The increase of $\delta$ to 0.15 is followed by a linear decrease of $T_{\rm C}$ to $\approx$ 160 K and a metal-insulator transition near the boundary of the cubic structure range. Further increase of $\delta$ results in formation of a tetragonal $2a_p\times 2a_p \times 4a_p$ phase for $\delta\approx 0.25$ and a brownmillerite phase for $\delta\approx0.5$. At low temperatures, these are weak ferromagnetic insulators (canted antiferromagnets) with magnetic transitions at $T_{\rm m}\approx230$ and 120 K, respectively. At higher temperatures, the $2a_p\times 2a_p \times 4a_p$ phase is $G$-type antiferromagnetic between 230 K and $\approx$360 K. Low temperature magnetic properties of this system for $\delta<1/3$ can be described in terms of a mixture of Co$^{3+}$ ions in the low-spin state and Co$^{4+}$ ions in the intermediate-spin state and a possible spin transition of Co$^{3+}$ to the intermediate-spin state above $T_{\rm C}$. For $\delta>1/3$, there appears to be a combination of Co$^{2+}$ and Co$^{3+}$ ions, both in the high-spin state with dominating antiferromagnetic interactions.Comment: RevTeX, 9 pages, 7 figures, to be published in Physical Review

The Fundamental Plane of Gravitational Lens Galaxies and The Evolution of Early-Type Galaxies in Low Density Environments

Most gravitational lenses are early-type galaxies in relatively low density environments -- a ``field'' rather than a ``cluster'' population. We show that field early-type galaxies with 0 < z < 1, as represented by the lens galaxies, lie on the same fundamental plane as those in rich clusters at similar redshifts. We then use the fundamental plane to measure the combined evolutionary and K-corrections for early-type galaxies in the V, I and H bands. Only for passively evolving stellar populations formed at z > 2 (H_0=65 km/s Mpc, Omega_0=0.3, Lambda_0=0.7) can the lens galaxies be matched to the local fundamental plane. The high formation epoch and the lack of significant differences between the field and cluster populations contradict many current models of the formation history of early-type galaxies. Lens galaxy colors and the fundamental plane provide good photometric redshift estimates with an empirical accuracy of -0.03 +/- 0.11 for the 17 lenses with known redshifts. A mass model dominated by dark matter is more consistent with the data than either an isotropic or radially anisotropic constant M/L mass model, and a radially anisotropic model is better than an isotropic model.Comment: 36 pages, 9 figures, 6 tables. ApJ in press. Final version contains more observational dat

Harmonic analysis of iterated function systems with overlap

In this paper we extend previous work on IFSs without overlap. Our method involves systems of operators generalizing the more familiar Cuntz relations from operator algebra theory, and from subband filter operators in signal processing.Comment: 37 page

Modeling cluster-level constructs measured by individual responses:Configuring a shared approach

When multiple items are used to measure cluster-level constructs with individual-level responses, multilevel confirmatory factor models are useful. How to model constructs across levels is still an active area of research in which competing methods are available to capture what can be interpreted as a valid representation of cluster-level phenomena. Moreover, the terminology used for the cluster-level constructs in such models varies across researchers. We therefore provide an overview of used terminology and modeling approaches for cluster-level constructs measured through individual responses. We classify the constructs based on whether (a) the target of measurement is at the cluster level or at the individual level and (b) the construct requires a measurement model. Next, we discuss various two-level factor models that have been proposed for multilevel constructs that require a measurement model, and we show that the so-called doubly latent model with cross-level invariance of factor loadings is appropriate for all types of constructs that require a measurement model. We provide two illustrations using empirical data from students and organizational teams on stimulating teaching and on conflict in organizational teams, respectively.</p

Chemical ionization of phenyl n-propyl ether and methyl substituted analogs: propene loss initiated by competing proton transfer to the oxygen atom and the aromatic ring.

The mechanism of propene loss from protonated phenyl n-propyl ether and a series of mono-, di-, and trimethylphenyl n-propyl ethers has been examined by chemical ionization (CI) mass spectrometry in combination with tandem mass spectrometry experiments. The role of initial proton transfer to the oxygen atom and the aromatic ring, respectively, has been probed with the use of deuterated CI reagents, D2O, CD3OD, and CD3CN (given in order of increasing proton affinity), in combination with deuterium labeling of the β position of the n-propyl group or the phenyl ring. The metastable [M + D]+ ions of phenyl n-propyl ether—formed with D2O as the CI reagent—eliminate C3H5D and C3H6 in a ratio of 10:90, which indicates that the added deuteron is incorporated to a minor extent in the expelled neutral species. In the experiments with CD3OD as the CI reagent, the ratio between the losses of C3H5D and C3H6 from the metastable [M + D]+ ions of phenyl n-propyl ether is 18:82, whereas the ratio becomes 27:73 with CD3CN as the reagent. A similar trend in the tendency to expel a propene molecule that contains the added deuteron is observed for the metastable [M + D]+ ions of phenyl n-propyl ether labeled at the β position of the alkyl group. Incorporation of a hydrogen atom that originates from the aromatic ring in the expelled propene molecule is of negligible importance as revealed by the minor loss of C3H5D from the metastable [M + H]+ ions of C6D5OCH2CH2CH3 irrespective of whether H2O, CH3OH, or CH3CN is the CI reagent. The combined results for the [M + D]+ ions of phenyl n-propyl ether and deuterium-labeled analogs are suggested to be in line with a model that assumes that propene loss occurs not only from species formed by deuteron transfer to the oxygen atom, but also from ions generated by deuteron transfer to the ring. This is substantiated by the results for the methyl-substituted ethers, which reveal that the position as well as the number of methyl groups bonded to the ring exert a marked effect on the relative importances of the losses of C3H5D and C3H6 from the metastable [M + D]+ ions of the unlabeled methyl-substituted species

Reproductive ecology of interior least tern and piping plover in relation to Platte River hydrology and sandbar dynamics

Historical and contemporary use of large, economically important rivers by threatened and/or endangered species in the United States is a subject of great interest to a wide range of stakeholders. In a recent study of the Platte River in Nebraska, Farnsworth et al. (2017) (hereinafter referred to as “the authors” or “Farnsworth et al.”) used distributions of nest initiation dates taken mostly from human-created, off-channel habitats and a model of emergent sandbar habitat to evaluate the hypothesis that least terns (Sternula antillarum) and piping plovers (Charadrius melodus) are physiologically adapted to initiate nests concurrent with the cessation of spring river flow rises. The authors conclude that (1) these species are not now, nor were they in the past, physiologically adapted to the hydrology of the Platte River, (2) habitats in the Platte River did not, and cannot support reproductive levels sufficient to maintain species subpopulations, (3) the gap in local elevation between peak river stage and typical sandbar height, in combination with the timing of the average spring flood, creates a physical environment which limits opportunities for successful nesting and precludes persistence by either species, and (4) the presence of off-channel habitats, including human-created sand and gravel mines, natural lakes, and a playa wetland, allowed the species to expand into the Platte River basin