Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This curse is well known. It also occurs for finite-dimensional linear operators. We circumvent it by developing a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. It extends the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics are relevant, including memoryful stochastic processes, open non unitary quantum systems, and far-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrary powers of an operator. In particular, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a general method to construct it. We provide new formulae for constructing projection operators and delineate the relations between projection operators, eigenvectors, and generalized eigenvectors. By way of illustrating its application, we explore several, rather distinct examples.Comment: 29 pages, 4 figures, expanded historical citations; http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht

Diffraction Patterns of Layered Close-packed Structures from Hidden Markov Models

We recently derived analytical expressions for the pairwise (auto)correlation functions (CFs) between modular layers (MLs) in close-packed structures (CPSs) for the wide class of stacking processes describable as hidden Markov models (HMMs) [Riechers \etal, (2014), Acta Crystallogr.~A, XX 000-000]. We now use these results to calculate diffraction patterns (DPs) directly from HMMs, discovering that the relationship between the HMMs and DPs is both simple and fundamental in nature. We show that in the limit of large crystals, the DP is a function of parameters that specify the HMM. We give three elementary but important examples that demonstrate this result, deriving expressions for the DP of CPSs stacked (i) independently, (ii) as infinite-Markov-order randomly faulted 2H and 3C stacking structures over the entire range of growth and deformation faulting probabilities, and (iii) as a HMM that models Shockley-Frank stacking faults in 6H-SiC. While applied here to planar faulting in CPSs, extending the methods and results to planar disorder in other layered materials is straightforward. In this way, we effectively solve the broad problem of calculating a DP---either analytically or numerically---for any stacking structure---ordered or disordered---where the stacking process can be expressed as a HMM.Comment: 18 pages, 6 figures, 3 tables; http://csc.ucdavis.edu/~cmg/compmech/pubs/dplcps.ht

Photometric Redshifts and Photometry Errors

We examine the impact of non-Gaussian photometry errors on photometric redshift performance. We find that they greatly increase the scatter, but this can be mitigated to some extent by incorporating the correct noise model into the photometric redshift estimation process. However, the remaining scatter is still equivalent to that of a much shallower survey with Gaussian photometry errors. We also estimate the impact of non-Gaussian errors on the spectroscopic sample size required to verify the photometric redshift rms scatter to a given precision. Even with Gaussian {\it photometry} errors, photometric redshift errors are sufficiently non-Gaussian to require an order of magnitude larger sample than simple Gaussian statistics would indicate. The requirements increase from this baseline if non-Gaussian photometry errors are included. Again the impact can be mitigated by incorporating the correct noise model, but only to the equivalent of a survey with much larger Gaussian photometry errors. However, these requirements may well be overestimates because they are based on a need to know the rms, which is particularly sensitive to tails. Other parametrizations of the distribution may require smaller samples.Comment: submitted to ApJ

A Closed-Form Shave from Occam's Quantum Razor: Exact Results for Quantum Compression

The causal structure of a stochastic process can be more efficiently transmitted via a quantum channel than a classical one, an advantage that increases with codeword length. While previously difficult to compute, we express the quantum advantage in closed form using spectral decomposition, leading to direct computation of the quantum communication cost at all encoding lengths, including infinite. This makes clear how finite-codeword compression is controlled by the classical process' cryptic order and allows us to analyze structure within the length-asymptotic regime of infinite-cryptic order (and infinite Markov order) processes.Comment: 21 pages, 13 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/eqc.ht

[CII] line emission in BRI1335-0417 at z=4.4

Using the 12m APEX telescope, we have detected redshifted emission from the 157.74micron [CII] line in the z=4.4074 quasar BRI1335-0417. The linewidth and redshift are in good agreement with previous observations of high-J CO line emission. We measure a [CII] line luminosity, L_[CII] = (16.4 +/- 2.6)x10^9 Lsun, making BRI~1335-0417 the most luminous, unlensed [CII] line emitter known at high-redshift. The [CII]-to-FIR luminosity ratio of (5.3+/-0.8)x10^-4 is ~3x higher than expected for an average object with a FIR luminosity L_FIR = 3.1x10^13 Lsun, if this ratio were to follow the trend observed in other FIR-bright galaxies that have been detected in [CII] line emission. These new data suggest that the scatter in the [CII]-to-FIR luminosity ratio could be larger than previously expected for high luminosity objects. BR1335-0417 has a similar FIR luminosity and [CII]/CO luminosity compared to local ULIRGS and appears to be a gas-rich merger forming stars at a rate of a few thousand solar masses per year.Comment: A&A accepte

Molecular Gas in Infrared Ultraluminous QSO Hosts

We report CO detections in 17 out of 19 infrared ultraluminous QSO (IR QSO) hosts observed with the IRAM 30m telescope. The cold molecular gas reservoir in these objects is in a range of 0.2--2.1$\times 10^{10}M_\odot$ (adopting a CO-to-${\rm H_2}$ conversion factor $\alpha_{\rm CO}=0.8 M_\odot {\rm (K km s^{-1} pc^2)^{-1}}$). We find that the molecular gas properties of IR QSOs, such as the molecular gas mass, star formation efficiency ($L_{\rm FIR}/L^\prime_{\rm CO}$) and the CO (1-0) line widths, are indistinguishable from those of local ultraluminous infrared galaxies (ULIRGs). A comparison of low- and high-redshift CO detected QSOs reveals a tight correlation between L$_{\rm FIR}$ and $L^\prime_{\rm CO(1-0)}$ for all QSOs. This suggests that, similar to ULIRGs, the far-infrared emissions of all QSOs are mainly from dust heated by star formation rather than by active galactic nuclei (AGNs), confirming similar findings from mid-infrared spectroscopic observations by {\it Spitzer}. A correlation between the AGN-associated bolometric luminosities and the CO line luminosities suggests that star formation and AGNs draw from the same reservoir of gas and there is a link between star formation on $\sim$ kpc scale and the central black hole accretion process on much smaller scales.Comment: 30 pages, 9 figures, accepted for publication in The Astrophysical Journa

Discovery of an extremely bright submillimeter galaxy at z=3.93

Serendipitously we have discovered a rare, bright submillimeter galaxy (SMG) with a flux density of 30 +/- 2 mJy at lambda=1.2mm, using MAMBO2 at the IRAM 30-meter millimeter telescope. Although no optical counterpart is known for MM18423+5938, we were able to measure the redshift z=3.92960 +/- 0.00013 from the detection of CO lines using the IRAM Eight MIxer Receiver (EMIR). In addition, by collecting all available photometric data in the far-infrared and radio to constrain its spectral energy distribution, we derive the FIR luminosity 4.8 10^14/m Lsol and mass 6.0 10^9/m Msol for its dust, allowing for a magnification factor m caused by a probable gravitational lens. The corresponding star-formation rate is 8.3 10^4/m Msol/yr. The detection of three lines of the CO rotational ladder, and a significant upper limit for a fourth CO line, allow us to estimate an H2 mass of between 1.9 10^11/m Msol and 1.1 10^12/m Msol. The two lines CI(3p1-3p0) and CI(3p2-3p1) were clearly detected and yield a [CI]/[H2] number abundance between 1.4 10^-5 and 8.0 10^-5. Upper limits are presented for emission lines of HCN, HCO^+, HNC, H_2O and other molecules observed. The moderate excitation of the CO lines is indicative of an extended starburst, and excludes the dominance of an AGN in heating this high-redshift SMG.Comment: Model revised. Accepted as an Astronomy and Astrophysics Letter, 4 pages, 3 figure

A Kiloparsec-Scale Hyper-Starburst in a Quasar Host Less than 1 Gigayear after the Big Bang

The host galaxy of the quasar SDSS J114816.64+525150.3 (at redshift z=6.42, when the Universe was <1 billion years old) has an infrared luminosity of 2.2x10^13 L_sun, presumably significantly powered by a massive burst of star formation. In local examples of extremely luminous galaxies such as Arp220, the burst of star formation is concentrated in the relatively small central region of <100pc radius. It is unknown on which scales stars are forming in active galaxies in the early Universe, which are likely undergoing their initial burst of star formation. We do know that at some early point structures comparable to the spheroidal bulge of the Milky Way must have formed. Here we report a spatially resolved image of [CII] emission of the host galaxy of J114816.64+525150.3 that demonstrates that its star forming gas is distributed over a radius of ~750pc around the centre. The surface density of the star formation rate averaged over this region is ~1000 M_sun/yr/kpc^2. This surface density is comparable to the peak in Arp220, though ~2 orders of magnitudes larger in area. This vigorous star forming event will likely give rise to a massive spheroidal component in this system.Comment: Nature, in press, Feb 5 issue, p. 699-70

Radio observations of the cool gas, dust, and star formation in the first galaxies

We summarize cm through submm observations of the host galaxies of z ~ 6 quasars. These observations reveal the cool molecular gas (the fuel for star formation), the warm dust (heated by star formation), the fine structure line emission (tracing the CNM and PDRs), and the synchrotron emission. Our results imply active star formation in ~ 30% of the host galaxies, with star formation rates ~ 10^3 M_sun/year, and molecular gas masses ~ 10^10 M_sun. Imaging of the [CII] emission from the most distant quasar reveals a 'maximal starburst disk' on a scale ~ 1.5 kpc. Gas dynamical studies suggest a departure of these galaxies from the low-z M_{BH} -- M_{bulge} relation, with the black holes being, on average, 15 times more massive than expected. Overall, we are witnessing the co-eval formation of massive galaxies and supermassive black holes within 1 Gyr of the Big Bang.Comment: First Stars and Galaxies: Challenges in the Next Decade, AIP, 2010; Austin TX (eds Whelan, Bromm, Yoshida); 7 page