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

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

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

A More Fundamental Plane

We combine strong-lensing masses with SDSS stellar velocity dispersions and HST-ACS effective (half-light) radii for 36 lens galaxies from the Sloan Lens ACS (SLACS) Survey to study the mass dependence of mass-dynamical structure in early-type galaxies. We find that over a 180--390 km/s range in velocity dispersion, structure is independent of lensing mass to within 5%. This result suggests a systematic variation in the total (i.e., luminous plus dark matter) mass-to-light ratio as the origin of the tilt of the fundamental plane (FP) scaling relationship between galaxy size, velocity dispersion, and surface brightness. We construct the FP of the lens sample, which we find to be consistent with the FP of the parent SDSS early-type galaxy population, and present the first observational correlation between mass-to-light ratio and residuals about the FP. Finally, we re-formulate the FP in terms of surface mass density rather than surface brightness. By removing the complexities of stellar-population effects, this mass-plane formulation will facilitate comparison to numerical simulations and possible use as a cosmological distance indicator.Comment: 4+epsilon pages, 1 figure, emulateapj. Revised version accepted for publication in the ApJ Letter

Construction of Parseval wavelets from redundant filter systems

We consider wavelets in L^2(R^d) which have generalized multiresolutions. This means that the initial resolution subspace V_0 in L^2(R^d) is not singly generated. As a result, the representation of the integer lattice Z^d restricted to V_0 has a nontrivial multiplicity function. We show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary-matrix-valued functions on a torus acting on a certain vector bundle. Specifically, we show how the wavelet functions on R^d can be constructed directly from the generalized wavelet filters.Comment: 34 pages, AMS-LaTeX ("amsproc" document class) v2 changes minor typos in Sections 1 and 4, v3 adds a number of references on GMRA theory and wavelet multiplicity analysis; v4 adds material on pages 2, 3, 5 and 10, and two more reference

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

Near-Infrared Imaging of Early-Type Galaxies III. The Near-Infrared Fundamental Plane

Near-infrared imaging data on 251 early-type galaxies in clusters and groups are used to construct the near-infrared Fundamental Plane (FP) r_eff ~ sigma_0^1.53 _eff^-0.79. The slope of the FP therefore departs from the virial expectation of r_eff ~ sigma_0^2 _eff^-1 at all optical and near-infrared wavelengths, which could be a result of the variation of M/L along the elliptical galaxy sequence, or a systematic breakdown of homology among the family of elliptical galaxies. The slope of the near-infrared FP excludes metallicity variations as the sole cause of the slope of the FP. Age effects, dynamical deviations from a homology, or any combination of these (with or without metallicity), however, are not excluded. The scatter of both the near-infrared and optical FP are nearly identical and substantially larger than the observational uncertainties, demonstrating small but significant intrinsic cosmological scatter for the FP at all wavelengths. The lack of a correlation of the residuals of the near-infrared FP and the residuals from the Mg_2-sigma relation indicates that the thickness of these relations cannot be ascribed only to age or metallicity effects. Due to this metallicity independence, the small scatter of the near-infrared FP excludes a model in which age and metallicity effects ``conspire'' to keep the optical FP thin. All of these results suggest that the possible physical origins of the FP relations are complicated due to combined effects of variations of stellar populations and structural parameters among elliptical galaxies.Comment: to appear in The Astronomical Journal; 35 pages, including 13 Postscript figures and 1 table; uses AAS LaTeX style file