2-adjoint equivalences in homotopy type theory

We introduce the notion of (half) 2-adjoint equivalences in Homotopy Type Theory and prove their expected properties. We formalized these results in the Lean Theorem Prover

Forecasting Chemical Abundance Precision for Extragalactic Stellar Archaeology

Increasingly powerful and multiplexed spectroscopic facilities promise detailed chemical abundance patterns for millions of resolved stars in galaxies beyond the Milky Way (MW). Here, we employ the Cram\'er-Rao Lower Bound (CRLB) to forecast the precision to which stellar abundances for metal-poor, low-mass stars outside the MW can be measured for 41 current (e.g., Keck, MMT, VLT, DESI) and planned (e.g., MSE, JWST, ELTs) spectrograph configurations. We show that moderate resolution ($R\lesssim5000$) spectroscopy at blue-optical wavelengths ($\lambda\lesssim4500$ \AA) (i) enables the recovery of 2-4 times as many elements as red-optical spectroscopy ($5000\lesssim\lambda\lesssim10000$ \AA) at similar or higher resolutions ($R\sim 10000$) and (ii) can constrain the abundances of several neutron capture elements to $\lesssim$0.3 dex. We further show that high-resolution ($R\gtrsim 20000$), low S/N ($\sim$10 pixel$^{-1}$) spectra contain rich abundance information when modeled with full spectral fitting techniques. We demonstrate that JWST/NIRSpec and ELTs can recover (i) $\sim$10 and 30 elements, respectively, for metal-poor red giants throughout the Local Group and (ii) [Fe/H] and [$\alpha$/Fe] for resolved stars in galaxies out to several Mpc with modest integration times. We show that select literature abundances are within a factor of $\sim$2 (or better) of our CRLBs. We suggest that, like ETCs, CRLBs should be used when planning stellar spectroscopic observations. We include an open source python package, \texttt{Chem-I-Calc}, that allows users to compute CRLBs for spectrographs of their choosing.Comment: 60 pages, 24 figures, accepted for publication in ApJ

Validating Stellar Abundance Measurements from Multi-Resolution Spectroscopy

Large-scale surveys will provide spectroscopy for $\sim$50 million resolved stars in the Milky Way and Local Group. However, these data will have a high degree of heterogeneity and most will be low-resolution ($R<10000$), posing challenges to measuring consistent and reliable stellar labels. Here, we introduce a framework for identifying and remedying these issues. By simultaneously fitting the full spectrum and Gaia photometry with the Payne, we measure $\sim$40 abundances for 8 red giants in M15. From degraded quality Keck/HIRES spectra, we evaluate trends with resolution and S/N and find that (i) $\sim$20 abundances are recovered consistently within $\lesssim$0.1 dex agreement and with $\lesssim$0.05-0.15~dex systematic uncertainties from $10000\lesssim R\lesssim80000$; (ii) for 9 elements (C, Mg, Ca, Sc, Ti, Fe, Ni, Y, Nd), this systematic precision and accuracy extends down to $R\sim2500$; and (iii) while most elements do not exhibit strong S/N-dependent systematics, there are non-negligible biases for 4 elements (C, Mg, Ca, and Dy) below $\text{S/N}\sim10$ pixel$^{-1}$. We compare statistical uncertainties from MCMC sampling to the easier-to-compute Cram\'er-Rao bounds and find that they agree for $\sim$75% of elements, indicating the latter to be a reliable and faster way to estimate uncertainties. Our analysis illustrates the great promise of low-resolution spectroscopy for stellar chemical abundance work, and ongoing improvements to stellar models (e.g., 3D-NLTE physics) will only further extend its viability to more elements and to higher precision and accuracy.Comment: 46 pages, 26 figures, submitted to ApJS. Comments welcome

Strong Outflows and Inefficient Star Formation in the Reionization-era Ultra-faint Dwarf Galaxy Eridanus II

We present novel constraints on the underlying galaxy formation physics (e.g., mass loading factor, star formation history, metal retention) at $z\gtrsim7$ for the low-mass ($M_*\sim10^5$ M$_\odot$) Local Group ultra-faint dwarf galaxy (UFD) Eridanus {\sc II} (Eri II). Using a hierarchical Bayesian framework, we apply a one-zone chemical evolution model to Eri II's CaHK-based photometric metallicity distribution function (MDF; [Fe/H]) and find that the evolution of Eri II is well-characterized by a short, exponentially declining star-formation history ($\tau_\text{SFH}=0.39\pm_{0.13}^{0.18}$ Gyr), a low star-formation efficiency ($\tau_\text{SFE}=27.56\pm_{12.92}^{25.14}$ Gyr), and a large mass-loading factor ($\eta=194.53\pm_{42.67}^{33.37}$). Our results are consistent with Eri II forming the majority of its stars before the end of reionization. The large mass-loading factor implies strong outflows in the early history of Eri II and is in good agreement with theoretical predictions for the mass-scaling of galactic winds. It also results in the ejection of $>$90\% of the metals produced in Eri II. We make predictions for the distribution of [Mg/Fe]-[Fe/H] in Eri II as well as the prevalence of ultra metal-poor stars, both of which can be tested by future chemical abundance measurements. Spectroscopic follow-up of the highest metallicity stars in Eri II ($\text{[Fe/H]} > -2$) will greatly improve model constraints. Our new framework can readily be applied to all UFDs throughout the Local Group, providing new insights into the underlying physics governing the evolution of the faintest galaxies in the reionization era.Comment: 20 pages; 12 figures, submitted to MNRA

Analyzing Vertical Mergers: Accounting for the Unilateral Effects Tradeoff and Thinking Holistically About Efficiencies

With the adoption of the 2020 Vertical Merger Guidelines, the U.S. antitrust agencies have updated their guidance on vertical mergers for the Twenty-First Century. Although economists have long recognized the procompetitive benefits most vertical mergers generate, the law has not always followed suit, and has sometimes condemned vertical mergers for making the merged firm more efficient. In this article, we attempt to catalogue the extensive list of efficiencies that vertical mergers can generate, trace the often halting efforts to incorporate these insights into the law, and propose a framework that courts and agencies can use to assess the likely competitive effects of vertical transactions. We draw heavily upon leading cases, particularly Baker Hughes and AT&T, with two refinements. First, consistent with the final Guidelines (but not the earlier draft) and the economic literature noting a symmetry between unilateral anticompetitive effects (raising rivals’ costs) and procompetitive effects (the elimination of double marginalization), which we call the “unilateral effects tradeoff,” we argue a plaintiff alleging a raising rivals cost (RRC) theory of harm must also address EDM as part of its prima facie case. Second, if the plaintiff carries its prima facie burden, then the defendant should be able to argue, and courts and Agencies should seriously consider, the full range of procompetitive efficiencies, which we call a “holistic efficiency analysis.

Detection of Cometary Amines in Samples Returned by the Stardust Spacecraft

The delivery of amino acids to the early Earth by comets and their fragments could have been a significant source of the early Earth's prebiotic organic inventory that led to the emergence of life (Chyba and Sagan, 1992). Over 20 organic molecules including methane, ethane, ammonia, cyanic acid, formaldehyde, formamide, acetaldehyde, acetonitrile, and methanol have been identified by radio spectroscopic observations of the comae of comets Hale-Bopp and Hyakutake (Crovisier et al. 2004). These simple molecules could have provided the organic reservoir to allow the formation of more complex prebiotic organic compounds such as amino acids. After a 7-year mission, the Stardust spacecraft returned to Earth samples from comet Wild 2 on January 15, 2006 providing the opportunity to analyze the organic composition and isotopic distribution of cometary material with state-of-the-art laboratory instrumentation. The Preliminary Examination Team analyses of organics in samples returned by Stardust were largely focused on particles that impacted the collector aerogel and aluminum foil (Sandford et al. 2006). However, it is also possible that Stardust returned a "diffuse" sample of gas-phase organic molecules that struck the aerogel directly or diffused away from the grains after impact. To test this possibility, samples of Stardust flight aerogel and foil were carried through a hot water extraction and acid hydrolysis procedure to see if primary amine compounds were present in excess of those seen in controls. Here we report highly sensitive liquid chromatography time-of-flight mass spectrometry measurements of amino acids and amines in samples returned from a comet (Glavin et al. 2008). A suite of amino acids and amines including glycine, L-alanine, methylamine (MA), and ethylamine (EA) were identified in the Stardust bulk aerogel. With the exception of MA and EA, all other primary amines detected in comet-exposed aerogels were also present in the aerogel witness tile that was not exposed to Wild 2, suggesting that most amines are terrestrial in origin. However, the enhanced abundances of MA, EA, and possibly glycine in comet-exposed aerogel compared to controls, coupled with MA to EA ratios (1 to 2) that are distinct from preflight aerogels (7 to 10), suggest that these amines were captured from Wild 2. It is possible that MA and EA were formed on energetically processed icy grains containing methane, ethane, and ammonia. The presence of cometary amines in Stardust material supports the hypothesis that comets were an important source of prebiotic organics on the early Earth. To better understand their origin, a systematic compound specific carbon isotopic analysis (C-CSIA) via gas chromatography quadrupole mass spectrometry in with parallel with combustion isotope ratio mass spectrometry (GCQMS/ IRMS) is being conducted. We will discuss our latest C-CSIA measurements and what they indicate about the origin of amino acids extracted from Stardust samples

2-adjoint equivalences in homotopy type theory

We introduce the notion of (half) 2-adjoint equivalences in Homotopy Type Theory and prove their expected properties. We formalized these results in the Lean Theorem Prover

No Conclusive Evidence for Transits of Proxima b in MOST photometry

The analysis of Proxima Centauri's radial velocities recently led Anglada-Escud\'e et al. (2016) to claim the presence of a low mass planet orbiting the Sun's nearest star once every 11.2 days. Although the a-priori probability that Proxima b transits its parent star is just 1.5%, the potential impact of such a discovery would be considerable. Independent of recent radial velocity efforts, we observed Proxima Centauri for 12.5 days in 2014 and 31 days in 2015 with the MOST space telescope. We report here that we cannot make a compelling case that Proxima b transits in our precise photometric time series. Imposing an informative prior on the period and phase, we do detect a candidate signal with the expected depth. However, perturbing the phase prior across 100 evenly spaced intervals reveals one strong false-positive and one weaker instance. We estimate a false-positive rate of at least a few percent and a much higher false-negative rate of 20-40%, likely caused by the very high flare rate of Proxima Centauri. Comparing our candidate signal to HATSouth ground-based photometry reveals that the signal is somewhat, but not conclusively, disfavored (1-2 sigmas) leading us to argue that the signal is most likely spurious. We expect that infrared photometric follow-up could more conclusively test the existence of this candidate signal, owing to the suppression of flare activity and the impressive infrared brightness of the parent star.Comment: Accepted to ApJ. Posterior samples, MOST photometry and HATSouth photometry are all available at https://github.com/CoolWorlds/Proxim