598 research outputs found
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 (adopting a
CO-to- conversion factor ). We find that the molecular gas properties of IR QSOs,
such as the molecular gas mass, star formation efficiency () 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 and 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
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
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