On tt-core and self-conjugate (2t−1)(2t-1)-core partitions in arithmetic progressions

We extend recent results of Ono and Raji, relating the number of self-conjugate $7$-core partitions to Hurwitz class numbers. Furthermore, we give a combinatorial explanation for the curious equality $2\operatorname{sc}_7(8n+1) = \operatorname{c}_4(7n+2)$. We also conjecture that an equality of this shape holds if and only if $t=4$, proving the cases $t\in\{2,3,5\}$ and giving partial results for $t>5$

What do teachers attend to in curriculum materials?

In this paper, we describe an emerging methodology using eye tracking to explore teachers’ curricular attending as they interact with curriculum materials to design a lesson in order to learn what teachers pay attention to and how this attention shifts during planning. We propose affordances of this new method, remark on some of its limitations, and propose future directions

Curricular noticing: A comprehensive framework to describe teachers’ interactions with curriculum materials

Building on the work of Professional Noticing of Children’s Mathematical Thinking, we introduce the Curricular Noticing Framework to describe how teachers recognize opportunities within curriculum materials, understand their affordances and limitations, and use strategies to act on them. This framework builds on Remillard’s (2005) notion of participation with curriculum materials, connects with and broadens existing research on the relationship between teachers and written curriculum, and highlights new are as for research. We argue that once mathematics educators better understand the strategic curricular practices that support ambitious teaching, which we refer to as professional curricular noticing, then this knowledge can lead to recommendations for how to support the curricular work of teachers, particularly novice teachers

Ground-based adaptive optics coronagraphic performance under closed-loop predictive control

The discovery of the exoplanet Proxima b highlights the potential for the coming generation of giant segmented mirror telescopes (GSMTs) to characterize terrestrial --- potentially habitable --- planets orbiting nearby stars with direct imaging. This will require continued development and implementation of optimized adaptive optics systems feeding coronagraphs on the GSMTs. Such development should proceed with an understanding of the fundamental limits imposed by atmospheric turbulence. Here we seek to address this question with a semi-analytic framework for calculating the post-coronagraph contrast in a closed-loop AO system. We do this starting with the temporal power spectra of the Fourier basis calculated assuming frozen flow turbulence, and then apply closed-loop transfer functions. We include the benefits of a simple predictive controller, which we show could provide over a factor of 1400 gain in raw PSF contrast at 1 $\lambda/D$ on bright stars, and more than a factor of 30 gain on an I = 7.5 mag star such as Proxima. More sophisticated predictive control can be expected to improve this even further. Assuming a photon noise limited observing technique such as High Dispersion Coronagraphy, these gains in raw contrast will decrease integration times by the same large factors. Predictive control of atmospheric turbulence should therefore be seen as one of the key technologies which will enable ground-based telescopes to characterize terrrestrial planets.Comment: Accepted to JATI

The Magellan Adaptive Secondary VisAO Camera: Diffraction- Limited Broadband Visible Imaging and 20mas Fiber Array IFS

The Magellan Adaptive Secondary AO system, scheduled for first light in the fall of 2011, will be able to simultaneously perform diffraction limited AO science in both the mid-IR, using the BLINC/MIRAC4 10\{mu}m camera, and in the visible using our novel VisAO camera. The VisAO camera will be able to operate as either an imager, using a CCD47 with 8.5 mas pixels, or as an IFS, using a custom fiber array at the focal plane with 20 mas elements in its highest resolution mode. In imaging mode, the VisAO camera will have a full suite of filters, coronagraphic focal plane occulting spots, and SDI prism/filters. The imaging mode should provide ~20% mean Strehl diffraction-limited images over the band 0.5-1.0 \{mu}m. In IFS mode, the VisAO instrument will provide R~1,800 spectra over the band 0.6-1.05 \{mu}m. Our unprecedented 20 mas spatially resolved visible spectra would be the highest spatial resolution achieved to date, either from the ground or in space. We also present lab results from our recently fabricated advanced triplet Atmospheric Dispersion Corrector (ADC) and the design of our novel wide-field acquisition and active optics lens. The advanced ADC is designed to perform 58% better than conventional doublet ADCs and is one of the enabling technologies that will allow us to achieve broadband (0.5-1.0\{mu}m) diffraction limited imaging and wavefront sensing in the visible.Comment: Proceedings of the SPIE, 2010, Vol. 7736, 77362

Orbital Differential Imaging: A New High-Contrast Post-Processing Technique For Direct Imaging of Exoplanets

Current post-processing techniques in high contrast imaging depend on some source of diversity between the exoplanet signal and the residual star light at that location. The two main techniques are angular differential imaging (ADI), which makes use of parallactic sky rotation to separate planet from star light, and spectral differential imaging (SDI), which makes use of differences in the spectrum of planet and star light and the wavelength dependence of the point spread function (PSF). Here we introduce our technique for exploiting another source of diversity: orbital motion. Given repeated observations of an exoplanetary system with sufficiently short orbital periods, the motion of the planets allows us to discriminate them from the PSF. In addition to using powerful PSF subtraction algorithms, such an observing strategy enables temporal filtering. Once an orbit is determined, the planet can be ``de-orbited'' to further increase the signal-to-noise ratio. We call this collection of techniques Orbital Differential Imaging (ODI). Here we present the motivation for this technique, present a noise model, and present results from simulations. We believe ODI will be an enabling technique for imaging Earth-like planets in the habitable zones of Sun-like stars with dedicated space missions.Comment: 9 page, 5 figures. Presented at SPIE 2015 (9605-42). See other ACESat papers by Belikov, Bendek, and Thoma

MagAO: Status and on-sky performance of the Magellan adaptive optics system

MagAO is the new adaptive optics system with visible-light and infrared science cameras, located on the 6.5-m Magellan "Clay" telescope at Las Campanas Observatory, Chile. The instrument locks on natural guide stars (NGS) from 0$^\mathrm{th}$ to 16$^\mathrm{th}$ $R$-band magnitude, measures turbulence with a modulating pyramid wavefront sensor binnable from 28x28 to 7x7 subapertures, and uses a 585-actuator adaptive secondary mirror (ASM) to provide flat wavefronts to the two science cameras. MagAO is a mutated clone of the similar AO systems at the Large Binocular Telescope (LBT) at Mt. Graham, Arizona. The high-level AO loop controls up to 378 modes and operates at frame rates up to 1000 Hz. The instrument has two science cameras: VisAO operating from 0.5-1 $\mu$m and Clio2 operating from 1-5 $\mu$m. MagAO was installed in 2012 and successfully completed two commissioning runs in 2012-2013. In April 2014 we had our first science run that was open to the general Magellan community. Observers from Arizona, Carnegie, Australia, Harvard, MIT, Michigan, and Chile took observations in collaboration with the MagAO instrument team. Here we describe the MagAO instrument, describe our on-sky performance, and report our status as of summer 2014.Comment: 13 pages, 11 figures, to appear in Proc. SPIE 9148-