Polynominals related to powers of the Dedekind eta function

The vanishing properties of Fourier coefficients of integral powers of the Dedekind eta function correspond to the existence of integral roots of integer-valued polynomials Pn(x) introduced by M. Newman. In this paper we study the derivatives of these polynomials. We obtain non-vanishing results at integral points. As an application we prove that integral roots are simple if the index n of the polynomial is equal to a prime power pm or to pm + 1. We obtain a formula for the derivative of Pn(x) involving the polynomials of lower degree

Hudson's Theorem for finite-dimensional quantum systems

We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson's Theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counter-example. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.Comment: 17 pages, 3 figures; References updated. Title changed to match published version. See also quant-ph/070200

Rotational periods of very young brown dwarfs and very low-mass stars in ChaI

We have studied the photometric variability of very young brown dwarfs and very low-mass stars (masses well below 0.2 M_sun) in the ChaI star forming region. We have determined photometric periods in the Gunn i and R band for the three M6.5-M7 type brown dwarf candidates ChaHa2, ChaHa3 and ChaHa6 of 2.2 to 3.4 days. These are the longest photometric periods found for any brown dwarf so far. If interpreted as rotationally induced they correspond to moderately fast rotational velocities, which is fully consistent with their v sini values and their relatively large radii. We have also determined periods for the two M5-M5.5 type very low-mass stars B34 and CHXR78C. In addition to the Gunn i and R band data, we have analysed JHK_s monitoring data of the targets, which have been taken a few weeks earlier and confirm the periods found in the optical data. Upper limits for the errors in the period determination are between 2 and 9 hours. The observed periodic variations of the brown dwarf candidates as well as of the T Tauri stars are interpreted as modulation of the flux at the rotation period by magnetically driven surface features, on the basis of a consistency with v sini values as well as (R-i) color variations typical for spots. Furthermore, the temperatures even for the brown dwarfs in the sample are relatively high (>2800K) because the objects are very young. Therefore, the atmospheric gas should be sufficiently ionized for the formation of spots on one hand and the temperatures are too high for significant dust condensation and hence variabilities due to clouds on the other hand.Comment: 18 pages, 6 figures, accepted for publication in Ap

High-order time-splitting Hermite and Fourier spectral methods

In this paper, we are concerned with the numerical solution of the time-dependent Gross-Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider discretisations that are based on spectral methods in space and higher-order exponential operator splitting methods in time. The resulting methods are favourable in view of accuracy and efficiency; moreover, geometric properties of the equation such as particle number and energy conservation are well captured. Regarding the spatial discretisation of the GPE, we consider two approaches. In the unbounded domain, we employ a spectral decomposition of the solution into Hermite basis functions: on the other hand. restricting the equation to a sufficiently large bounded domain, Fourier techniques are applicable. For the time integration of the GPE, we study various exponential operator splitting methods of convergence orders two, four, and six. Our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting Fourier and Hermite pseudospectral methods for the time evolution of the GPE. Furthermore, we illustrate the effectiveness of higher-order time-splitting methods compared to standard integrators in a long-term integration

Discovery of Two T Dwarf Companions with the Spitzer Space Telescope

We report the discovery of T dwarf companions to the nearby stars HN Peg (G0V, 18.4 pc, ~0.3 Gyr) and HD 3651 (K0V, 11.1 pc, ~7 Gyr). During an ongoing survey of 5'x5' fields surrounding stars in the solar neighborhood with IRAC aboard the Spitzer Space Telescope, we identified these companions as candidate T dwarfs based on their mid-IR colors. Using near-IR spectra obtained with SpeX at the NASA IRTF, we confirm the presence of methane absorption that characterizes T dwarfs and measure spectral types of T2.5+/-0.5 and T7.5+/-0.5 for HN Peg B and HD 3651 B, respectively. By comparing our Spitzer data to images from 2MASS obtained several years earlier, we find that the proper motions of HN Peg B and HD 3651 B are consistent with those of the primaries, confirming their companionship. HN Peg B and HD 3651 B have angular separations of 43.2" and 42.9" from their primaries, which correspond to projected physical separations of 795 and 476 AU, respectively. A comparison of their luminosities to the values predicted by theoretical evolutionary models implies masses of 0.021+/-0.009 and 0.051+/-0.014 Msun for HN Peg B and HD 3651 B. In addition, the models imply an effective temperature for HN Peg B that is significantly lower than the values derived for other T dwarfs at similar spectral types, which is the same behavior reported by Metchev & Hillenbrand for the young late-L dwarf HD 203030 B. Thus, the temperature of the L/T transition appears to depend on surface gravity. Meanwhile, HD 3651 B is the first substellar companion directly imaged around a star that is known to harbor a close-in planet from RV surveys. The discovery of this companion supports the notion that the high eccentricities of close-in planets like the one near HD 3651 may be the result of perturbations by low-mass companions at wide separations.Comment: Astrophysical Journal, in pres

First optical images of circumstellar dust surrounding the debris disk candidate HD 32297

Near-infrared imaging with the Hubble Space Telescope recently revealed a circumstellar dust disk around the A star HD 32297. Dust scattered light is detected as far as 400 AU radius and the linear morphology is consistent with a disk ~10 degrees away from an edge-on orientation. Here we present the first optical images that show the dust scattered light morphology from 560 to 1680 AU radius. The position angle of the putative disk midplane diverges by 31 degrees and the color of dust scattering is most likely blue. We associate HD 32297 with a wall of interstellar gas and the enigmatic region south of the Taurus molecular cloud. We propose that the extreme asymmetries and blue disk color originate from a collision with a clump of interstellar material as HD 32297 moves southward, and discuss evidence consistent with an age of 30 Myr or younger.Comment: 5 pages; Accepted for publication in ApJ Letter