Irreducible character degrees and normal subgroups

Let N be a normal subgroup of a finite group G and consider the set cd(G|N) of degrees of irreducible characters of G whose kernels do not contain N. A number of theorems are proved relating the set cd(G|N) to the structure of N. For example, if N is solvable, its derived length is bounded above by a function of |cd(G|N)|. Also, if |cd(G|N)| is at most 2, then N is solvable and its derived length is at most |cd(G|N)|. If G is solvable and |cd(G|N)| = 3, then the derived length of N is at most 3

Theoretical Spectral Models of the Planet HD 209458b with a Thermal Inversion and Water Emission Bands

We find that a theoretical fit to all the HD 209458b data at secondary eclipse requires that the dayside atmosphere of HD 209458b have a thermal inversion and a stratosphere. This inversion is caused by the capture of optical stellar flux by an absorber of uncertain origin that resides at altitude. One consequence of stratospheric heating and temperature inversion is the flipping of water absorption features into emission features from the near- to the mid-infrared and we see evidence of such a water emission feature in the recent HD 209458b IRAC data of Knutson et al. In addition, an upper-atmosphere optical absorber may help explain both the weaker-than-expected Na D feature seen in transit and the fact that the transit radius at 24 $\mu$m is smaller than the corresponding radius in the optical. Moreover, it may be a factor in why HD 209458b's optical transit radius is as large as it is. We speculate on the nature of this absorber and the planets whose atmospheres may, or may not, be affected by its presence.Comment: Accepted to the Astrophysical Journal Letters on August 28, 2007, six pages in emulateapj forma

XMM-Newton observations of HD189733 during planetary transits

We report on two XMM-Newton observations of the planetary host star HD189733. The system has a close in planet and it can potentially affect the coronal structure via interactions with the magnetosphere. We have obtained X-ray spectra and light curves from EPIC and RGS on board XMM-Newton which we have analyzed and interpreted. We reduced X-ray data from primary transit and secondary eclipse occurred in April 17th 2007 and May 18th 2009, respectively. In the April 2007 observation only variability due to weak flares is recognized. In 2009 HD189733 exhibited a X-ray flux always larger than in the 2007 observation. The average flux in 2009 was higher than in 2007 observation by a factor of 45%. During the 2009 secondary eclipse we observed a softening of the X-ray spectrum significant at level of ~3 sigma. Further, we observed the most intense flare recorded at either epochs. This flare occurred 3 ks after the end of the eclipse.The flare decay shows several minor ignitions perhaps linked to the main event and hinting for secondary loops that emit triggered by the main loop. Magneto-Hydro-Dynamical (MHD) simulations show that the magnetic interaction between planet and star enhances the density and the magnetic field in a region comprised between the planet and the star because of their relative orbital/rotation motion. X-ray observations and model predictions are globally found in agreement, despite the quite simple MHD model and the lack of precise estimate of parameters including the alignment and the intensity of stellar and planetary magnetic fields. Future observations should confirm or disprove this hypothesis, by determining whether flares are systematically recurring in the light curve at the same planetary phase.Comment: Accepted for publication on The Astrophysical Journa

Of `Cocktail Parties' and Exoplanets

The characterisation of ever smaller and fainter extrasolar planets requires an intricate understanding of one's data and the analysis techniques used. Correcting the raw data at the 10^-4 level of accuracy in flux is one of the central challenges. This can be difficult for instruments that do not feature a calibration plan for such high precision measurements. Here, it is not always obvious how to de-correlate the data using auxiliary information of the instrument and it becomes paramount to know how well one can disentangle instrument systematics from one's data, given nothing but the data itself. We propose a non-parametric machine learning algorithm, based on the concept of independent component analysis, to de-convolve the systematic noise and all non-Gaussian signals from the desired astrophysical signal. Such a `blind' signal de-mixing is commonly known as the `Cocktail Party problem' in signal-processing. Given multiple simultaneous observations of the same exoplanetary eclipse, as in the case of spectrophotometry, we show that we can often disentangle systematic noise from the original light curve signal without the use of any complementary information of the instrument. In this paper, we explore these signal extraction techniques using simulated data and two data sets observed with the Hubble-NICMOS instrument. Another important application is the de-correlation of the exoplanetary signal from time-correlated stellar variability. Using data obtained by the Kepler mission we show that the desired signal can be de-convolved from the stellar noise using a single time series spanning several eclipse events. Such non-parametric techniques can provide important confirmations of the existent parametric corrections reported in the literature, and their associated results. Additionally they can substantially improve the precision exoplanetary light curve analysis in the future.Comment: ApJ accepte

The GROUSE project II: Detection of the Ks-band secondary eclipse of exoplanet HAT-P-1b

Context: Only recently it has become possible to measure the thermal emission from hot-Jupiters at near-Infrared wavelengths using ground-based telescopes, by secondary eclipse observations. This allows the planet flux to be probed around the peak of its spectral energy distribution, which is vital for the understanding of its energy budget. Aims: The aim of the reported work is to measure the eclipse depth of the planet HAT-P-1b at 2.2micron. This planet is an interesting case, since the amount of stellar irradiation it receives falls in between that of the two best studied systems (HD209458 and HD189733), and it has been suggested to have a weak thermal inversion layer. Methods: We have used the LIRIS instrument on the William Herschel Telescope (WHT) to observe the secondary eclipse of HATP-1b in the Ks-band, as part of our Ground-based secondary eclipse (GROUSE) project. The observations were done in staring mode, while significantly defocusing the telescope to avoid saturation on the K=8.4 star. With an average cadence of 2.5 seconds, we collected 6520 frames during one night. Results: The eclipse is detected at the 4sigma level, the measured depth being 0.109+/-0.025%. The uncertainties are dominated by residual systematic effects, as estimated from different reduction/analysis procedures. The measured depth corresponds to a brightness temperature of 2136+150-170K. This brightness temperature is significantly higher than those derived from longer wavelengths, making it difficult to fit all available data points with a plausible atmospheric model. However, it may be that we underestimate the true uncertainties of our measurements, since it is notoriously difficult to assign precise statistical significance to a result when systematic effects are important.Comment: 7 pages, 10 figures, Accepted for publication in A&

Ground-based detection of sodium in the transmission spectrum of exoplanet HD209458b

[Context] The first detection of an atmosphere around an extrasolar planet was presented by Charbonneau and collaborators in 2002. In the optical transmission spectrum of the transiting exoplanet HD209458b, an absorption signal from sodium was measured at a level of 0.023+-0.006%, using the STIS spectrograph on the Hubble Space Telescope. Despite several attempts, so far only upper limits to the Na D absorption have been obtained using telescopes from the ground, and the HST result has yet to be confirmed. [Aims] The aims of this paper are to re-analyse data taken with the High Dispersion Spectrograph on the Subaru telescope, to correct for systematic effects dominating the data quality, and to improve on previous results presented in the literature. [Methods] The data reduction process was altered in several places, most importantly allowing for small shifts in the wavelength solution. The relative depth of all lines in the spectra, including the two sodium D lines, are found to correlate strongly with the continuum count level in the spectra. These variations are attributed to non-linearity effects in the CCDs. After removal of this empirical relation the uncertainties in the line depths are only a fraction above that expected from photon statistics. [Results] The sodium absorption due to the planet's atmosphere is detected at >5 sigma, at a level of 0.056+-0.007% (2x3.0 Ang band), 0.070+-0.011% (2x1.5 Ang band), and 0.135+-0.017% (2x0.75 Ang band). There is no evidence that the planetary absorption signal is shifted with respect to the stellar absorption, as recently claimed for HD189733b. The measurements in the two most narrow bands indicate that some signal is being resolved.[abridged]Comment: Latex, 7 pages: accepted for publication in Astronomy & Astrophysic

Schubert calculus of Richardson varieties stable under spherical Levi subgroups

We observe that the expansion in the basis of Schubert cycles for $H^*(G/B)$ of the class of a Richardson variety stable under a spherical Levi subgroup is described by a theorem of Brion. Using this observation, along with a combinatorial model of the poset of certain symmetric subgroup orbit closures, we give positive combinatorial descriptions of certain Schubert structure constants on the full flag variety in type $A$. Namely, we describe $c_{u,v}^w$ when $u$ and $v$ are inverse to Grassmannian permutations with unique descents at $p$ and $q$, respectively. We offer some conjectures for similar rules in types $B$ and $D$, associated to Richardson varieties stable under spherical Levi subgroups of SO(2n+1,\C) and SO(2n,\C), respectively.Comment: Section 4 significantly shortened, and other minor changes made as suggested by referees. Final version, to appear in Journal of Algebraic Combinatoric

Classification of Reductive Monoid Spaces Over an Arbitrary Field

In this semi-expository paper we review the notion of a spherical space. In particular we present some recent results of Wedhorn on the classification of spherical spaces over arbitrary fields. As an application, we introduce and classify reductive monoid spaces over an arbitrary field.Comment: This is the final versio

The solution of the quantum A1A_1 T-system for arbitrary boundary

We solve the quantum version of the $A_1$ $T$-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum $A_1$ $Q$-system and generalize it to the fully non-commutative case. We give the relation between the quantum $T$-system and the quantum lattice Liouville equation, which is the quantized $Y$-system.Comment: 24 pages, 18 figure

An Operational Meaning of Discord in terms of Teleportation Fidelity

Quantum discord is a prominent measure of quantum correlations, playing an important role in expanding its horizon beyond entanglement. Here we provide an operational meaning of (geometric) discord, which quantifies the amount of non-classical correlation of an arbitrary quantum system in terms of its minimal distance from the set of classical states, in terms of teleportation fidelity for general two qubit and $d \otimes d$ dimensional isotropic and Werner states. A critical value of the discord is found beyond which the two qubit state must violate the Bell inequality. This is illustrated by an open system model of a dissipative two qubit. For the $d \otimes d$ dimensional states the lower bound of discord is shown to be obtainable from an experimentally measurable witness operator.Comment: 5 pages and 1 figur