Search

Astronomers have been measuring the separations and position angles between the two components of binary stars since William Herschel began his observations in 1781. In 1970, Anton Labeyrie pioneered a method, speckle interferometry, that overcomes the usual resolution limits induced by atmospheric turbulence by taking hundreds or thousands of short exposures and reducing them in Fourier space. Our 2022 automation of speckle interferometry allowed us to use a fully robotic 1.0-meter PlaneWave Instruments telescope, located at the El Sauce Observatory in the Atacama Desert of Chile, to obtain observations of many known binaries with established orbits. The long-term objective of these observations is to establish the precision, accuracy, and limitations of this telescope's automated speckle interferometry measurements. This paper provides an early overview of the Known Binaries Project and provide example results on a small-separation (0.27") binary, WDS 12274-2843 B 228.

Marginal likelihood, also known as model evidence, is a fundamental quantity in Bayesian statistics. It is used for model selection using Bayes factors or for empirical Bayes tuning of prior hyper-parameters. Yet, the calculation of evidence has remained a longstanding open problem in Gaussian graphical models. Currently, the only feasible solutions that exist are for special cases such as the Wishart or G-Wishart, in moderate dimensions. We develop an approach based on a novel telescoping block decomposition of the precision matrix that allows the estimation of evidence by application of Chib's technique under a very broad class of priors under mild requirements. Specifically, the requirements are: (a) the priors on the diagonal terms on the precision matrix can be written as gamma or scale mixtures of gamma random variables and (b) those on the off-diagonal terms can be represented as normal or scale mixtures of normal. This includes structured priors such as the Wishart or G-Wishart, and more recently introduced element-wise priors, such as the Bayesian graphical lasso and the graphical horseshoe. Among these, the true marginal is known in an analytically closed form for Wishart, providing a useful validation of our approach. For the general setting of the other three, and several more priors satisfying conditions (a) and (b) above, the calculation of evidence has remained an open question that this article resolves under a unifying framework.

Located on the campus of the Thacher School in Southern California, the Thacher Observatory has a legacy of astronomy research and education that dates back to the late 1950's. In 2016, the observatory was fully renovated with upgrades including a new 0.7-m telescope, a research grade camera, and a slit dome with full automation capabilities. The low-elevation site is bordered by the Los Padres National Forest and therefore affords dark to very dark skies allowing for accurate and precise photometric observations. We present a characterization of the site including sky brightness, weather, and seeing, and we demonstrate the on-sky performance of the facility. Our primary research programs are based around our multi-band photometric capabilities and include photometric monitoring of variable sources, a nearby supernova search and followup program, a quick response transient followup effort, and exoplanet and eclipsing binary light curves. Select results from these programs are included in this work which highlight the broad range of science available to an automated observatory with a moderately sized telescope., Comment: 20 pages, 18 figures, PASP accepted

The main focus of this paper is on models of quartic surfaces, especially so-called complex surfaces. These are special fourth-degree surfaces that Julius Pl\"ucker introduced in the 1860s for visualizing the local structure of a quadratic line complex. Pl\"ucker's complex surfaces turned out to be closely related to Kummer surfaces and both of these types of quartics are examples of caustic surfaces, which arise in geometrical optics. Indeed, Kummer surfaces represent a natural generalization of the wave surface, first introduced by Augustin Fresnel to explain double refraction in biaxial crystals.

Special types of quartic surfaces were much studied objects during the 1860s. Quartics were thus very much in the air when Sophus Lie and Felix Klein first met in Berlin in 1869. As this study shows, such surfaces played a major role in their subsequent work, much of which centered on linear and quadratic line complexes. This mutual interest led them to a number of new results on the quartic surfaces of Steiner, Pl\"ucker, and Kummer, as well as various types of ruled quartics studied earlier by Cremona. This paper, which draws on unpublished archival sources as well as published work from the period 1869-1872, underscores the importance of this aspect of the early geometrical work of these two famous figures. A highlight was Lie's line-to-sphere transformation, which led to surprising new findings on properties of asymptotic curves on Kummer surfaces.

This paper discusses a central theorem in birational geometry first proved by Eugenio Bertini in 1891. J.L. Coolidge described the main ideas behind Bertini's proof, but he attributed the theorem to Clebsch. He did so owing to a short note that Felix Klein appended to the republication of Bertini's article in 1894. The precise circumstances that led to Klein's intervention can be easily reconstructed from letters Klein exchanged with Max Noether, who was then completing work on the lengthy report he and Alexander Brill published on the history of algebraic functions [Brill/Noether 1894]. This correspondence sheds new light on Noether's deep concerns about the importance of this report in substantiating his own priority rights and larger intellectual legacy.

During the First World War, the status of energy conservation in general relativity was one of the most hotly debated questions surrounding Einstein's new theory of gravitation. His approach to this aspect of general relativity differed sharply from another set forth by Hilbert, even though the latter conjectured in 1916 that both theories were probably equivalent. Rather than pursue this question himself, Hilbert chose to charge Emmy Noether with the task of probing the mathematical foundations of these two theories. Indirect references to her results came out two years later when Klein began to examine this question again with Noether's assistance. Over several months, Klein and Einstein pursued these matters in a lengthy correspondence, which culminated with several publications, including Noether's now famous paper "Invariante Variationsprobleme". The present account focuses on the earlier discussions from 1916 involving Einstein, Hilbert, and Noether. In these years, a Swiss student named R.J. Humm was studying relativity in G\"ottingen, during which time he transcribed part of Noether's lost manuscript on Hilbert's invariant energy vector. By making use of this 9-page manuscript, it is possible to reconstruct the arguments Noether set forth in response to Hilbert's conjecture. Her results turn out to be closely related to the findings Klein published two years later, thereby highlighting, once again, how her work significantly deepened contemporary understanding of the mathematical underpinnings of general relativity.

A fully quantal algebraic version of the Bohr-Mottelson unified model is presented with the important property that its quantisation is defined by its irreducible unitary representations which span the many-nucleon Hilbert space of every nucleus. The model is uniquely defined by the requirement that its Lie algebra of observables includes the nuclear quadrupole moments and kinetic energy. It then follows that there can be no non-zero isoscalar E2 transitions between any states belonging to its different irreducible representations and, as a result, the states of the model are uniquely defined with the property that observed transitions between rotational states of nuclei are to be expressed in terms of mixtures of the model irreps. The algebraic version of the unified model parallels the Bohr-Mottelson model in most respects, including the possibility of including the effects of Coriolis and centrifugal forces as subsequent perturbations. However, it corrects its treatment of angular momentum quantisation and no longer uses an over-complete set of coordinates. These changes have significant implications for the dynamics of nuclear rotations which are hidden when its moments of inertia are considered as inertial masses in the standard expression of rotational kinetic energies. Thus, the developments put a new perspective on the phenomenon of shape coexistence., Comment: 10 pages, 1 figure, 1 table

It is determined that a many-nucleon version of the Bohr-Mottelson unified model that contains the essential observables of that model and has irreducible representations that span the Hilbert space of fully anti-symmetric states of nuclei, is given uniquely by the symplectic model. This model is shown to provide a framework for an examination of the dynamics of nuclear rotations. A first discovery is that rotational energies are mixtures of potential and kinetic energies even in an adiabatic limit., Comment: 3 pages, 1 figure, 28 references