Relaxation of monotone coupling conditions: Poisson approximation and beyond

It is well-known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also give explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincar\'e inequality and in a normal approximation result.Comment: 19 page

Infrared Period-Luminosity Relations of Evolved Variable Stars in the Large Magellanic Cloud

We combine variability information from the MAssive Compact Halo Objects (MACHO) survey of the Large Magellanic Cloud (LMC) with infrared photometry from the Spitzer Space Telescope Surveying the Agents of a Galaxy's Evolution (SAGE) survey to create a dataset of ~30 000 variable red sources. We photometrically classify these sources as being on the first ascent of the Red Giant Branch (RGB), or as being in one of three stages along the Asymptotic Giant Branch (AGB): oxygen-rich, carbon-rich, or highly reddened with indeterminate chemistry ("extreme" AGB candidates). We present linear period-luminosity relationships for these sources using 8 separate infrared bands (J, H, K, 3.6, 4.5, 5.8, 8.0, and 24 micron) as proxies for the luminosity. We find that the wavelength dependence of the slope of the period-luminosity relationship is different for different photometrically determined classes of AGB stars. Stars photometrically classified as O-rich show the least variation of slope with wavelength, while dust enshrouded extreme AGB stars show a pronounced trend toward steeper slopes with increasing wavelength. We find that O-rich AGB stars pulsating in the fundamental mode obey a period-magnitude relation with a slope of -3.41 +/- 0.04 when magnitude is measured in the 3.6 micron band, in contrast to C-rich AGB stars, which obey a relation of slope -3.77 +/- 0.05

Web invariants for flamingo Specht modules

Webs yield an especially important realization of certain Specht modules, irreducible representations of symmetric groups, as they provide a pictorial basis with a convenient diagrammatic calculus. In recent work, the last three authors associated polynomials to noncrossing partitions without singleton blocks, so that the corresponding polynomials form a web basis of the pennant Specht module $S^{(d,d,1^{n-2d})}$. These polynomials were interpreted as global sections of a line bundle on a 2-step partial flag variety. Here, we both simplify and extend this construction. On the one hand, we show that these polynomials can alternatively be situated in the homogeneous coordinate ring of a Grassmannian, instead of a 2-step partial flag variety, and can be realized as tensor invariants of classical (but highly nonplanar) tensor diagrams. On the other hand, we extend these ideas from the pennant Specht module $S^{(d,d,1^{n-2d})}$ to more general flamingo Specht modules $S^{(d^r,1^{n-rd})}$. In the hook case $r=1$, we obtain a spanning set that can be restricted to a basis in various ways. In the case $r>2$, we obtain a basis of a well-behaved subspace of $S^{(d^r,1^{n-rd})}$, but not of the entire module.Comment: 32 pages, 7 figure

SDSS J210014.12+004446.0: A New Dwarf Nova with Quiescent Superhumps?

We report follow-up observations of the Sloan Digital Sky Survey Cataclysmic Variable SDSS J210014.12+004446.0 (hereafter SDSS J2100). We obtained photometry and spectroscopy in both outburst and quiescent states, providing the first quiescent spectrum of this source. In both states, non-sinusoidal photometric modulations are apparent, suggestive of superhumps, placing SDSS J2100 in the SU UMa subclass of dwarf novae. However, the periods during outburst and quiescence differ significantly, being 2.099 plus or minus 0.002 hr and 1.96 plus or minus 0.02 hr respectively. Our phase-resolved spectroscopy during outburst yielded an estimate of about 2 hr for the orbital period, consistent with the photometry. The presence of the shorter period modulation at quiescence is unusual, but not unique. Another atypical feature is the relative weakness of the Balmer emission lines in quiescence. Overall, we find a close similarity between SDSS J2100 and the well-studied superhump cataclysmic Variable V503 Cygni. By analogy, we suggest that the quiescent modulation is due to a tilted accretion disk -- producing negative superhumps -- and the modulation in outburst is due to positive superhumps from the precession of an elliptical disk.Comment: 6 pages, 5 eps figures, accepted by PASP Dec. 16th, 200