Remarks on computing the Grothendieck rings of C*-algebras

In this paper, we present a captivating construction by Grothendieck, originally formulated for algebraic varieties, and adapt it to the realm of C*-algebras. Our main objective is to investigate the conditions under which this particular class of C*-algebras possesses a nontrivial Grothendieck ring. To achieve this, we explore the existence of nontrivial characters, which significantly enriches our understanding of these algebras. In particular, we conduct a detailed study of rings of C*-algebras over $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{H}$

Noncommutative resolutions of discriminants

We give an introduction to the McKay correspondence and its connection to quotients of $\mathbb{C}^n$ by finite reflection groups. This yields a natural construction of noncommutative resolutions of the discriminants of these reflection groups. This paper is an extended version of E.F.'s talk with the same title delivered at the ICRA.Comment: 15 pages, 4 figures. Final version to appear in Contemporary Mathematics 705, "Representations of Algebras

Spitzer Infrared Spectrograph Detection of Molecular Hydrogen Rotational Emission towards Translucent Clouds

Using the Infrared Spectrograph on board the Spitzer Space Telescope, we have detected emission in the S(0), S(1), and S(2) pure-rotational (v = 0-0) transitions of molecular hydrogen (H_2) toward six positions in two translucent high Galactic latitude clouds, DCld 300.2–16.9 and LDN 1780. The detection of these lines raises important questions regarding the physical conditions inside low-extinction clouds that are far from ultraviolet radiation sources. The ratio between the S(2) flux and the flux from polycyclic aromatic hydrocarbons (PAHs) at 7.9 μm averages 0.007 for these six positions. This is a factor of about four higher than the same ratio measured toward the central regions of non-active Galaxies in the Spitzer Infrared Nearby Galaxies Survey. Thus, the environment of these translucent clouds is more efficient at producing rotationally excited H_2 per PAH-exciting photon than the disks of entire galaxies. Excitation analysis finds that the S(1) and S(2) emitting regions are warm (T ≳ 300 K), but comprise no more than 2% of the gas mass. We find that UV photons cannot be the sole source of excitation in these regions and suggest mechanical heating via shocks or turbulent dissipation as the dominant cause of the emission. The clouds are located on the outskirts of the Scorpius-Centaurus OB association and may be dissipating recent bursts of mechanical energy input from supernova explosions. We suggest that pockets of warm gas in diffuse or translucent clouds, integrated over the disks of galaxies, may represent a major source of all non-active galaxy H_2 emission

Radar studies of the planets

The radar measurements phase of the lunar studies involving reflectivity and topographic mapping of the visible lunar surface was ended in December 1972, but studies of the data and production of maps have continued. This work was supported by Manned Spacecraft Center, Houston. Topographic mapping of the equatorial regions of Mars has been carried out during the period of each opposition since that of 1967. The method comprised extended precise traveling time measurements to a small area centered on the subradar point. As measurements continued, planetary motions caused this point to sweep out extensive areas in both latitude and longitude permitting the development of a fairly extensive topographical map in the equatorial region. Radar observations of Mercury and Venus have also been made over the past few years. Refinements of planetary motions, reflectivity maps and determinations of rotation rates have resulted

A McKay correspondence for reflection groups

We construct a noncommutative desingularization of the discriminant of a finite reflection group G as a quotient of the skew group ring A=S∗G. If G is generated by order 2 reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement A(G) viewed as a module over the coordinate ring SG/(Δ) of the discriminant of G. This yields, in particular, a correspondence between the nontrivial irreducible representations of G to certain maximal Cohen–Macaulay modules over the coordinate ring SG/(Δ). These maximal Cohen–Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement A(G) viewed as a module over SG/(Δ). We identify some of the corresponding matrix factorizations, namely, the so-called logarithmic (co-)residues of the discriminant

Random Forests Applied to High-precision Photometry Analysis with Spitzer IRAC

We present a new method employing machine-learning techniques for measuring astrophysical features by correcting systematics in IRAC high-precision photometry using random forests. The main systematic in IRAC light-curve data is position changes due to unavoidable telescope motions coupled with an intrapixel response function. We aim to use the large amount of publicly available calibration data for the single pixel used for this type of work (the sweet-spot pixel) to make a fast, easy-to-use, accurate correction to science data. This correction on calibration data has the advantage of using an independent data set instead of the science data themselves, which has the disadvantage of including astrophysical variations. After focusing on feature engineering and hyperparameter optimization, we show that a boosted random forest model can reduce the data such that we measure the median of 10 archival eclipse observations of XO-3b to be 1459 ± 200 ppm. This is a comparable depth to the average of those in the literature done by seven different methods; however, the spread in measurements is 30%–100% larger than those literature values, depending on the reduction method. We also caution others attempting similar methods to check their results with the fiducial data set of XO-3b, as we were also able to find models providing initially great scores on their internal test data sets but whose results significantly underestimated the eclipse depth of that planet

Random Forests Applied to High-precision Photometry Analysis with Spitzer IRAC

We present a new method employing machine-learning techniques for measuring astrophysical features by correcting systematics in IRAC high-precision photometry using random forests. The main systematic in IRAC light-curve data is position changes due to unavoidable telescope motions coupled with an intrapixel response function. We aim to use the large amount of publicly available calibration data for the single pixel used for this type of work (the sweet-spot pixel) to make a fast, easy-to-use, accurate correction to science data. This correction on calibration data has the advantage of using an independent data set instead of the science data themselves, which has the disadvantage of including astrophysical variations. After focusing on feature engineering and hyperparameter optimization, we show that a boosted random forest model can reduce the data such that we measure the median of 10 archival eclipse observations of XO-3b to be 1459 ± 200 ppm. This is a comparable depth to the average of those in the literature done by seven different methods; however, the spread in measurements is 30%–100% larger than those literature values, depending on the reduction method. We also caution others attempting similar methods to check their results with the fiducial data set of XO-3b, as we were also able to find models providing initially great scores on their internal test data sets but whose results significantly underestimated the eclipse depth of that planet

Protein Supplements for Beef Calves on Winter Range

These trials were to compare the effect of different sources and levels of supplemental protein on the winter and subsequent summer gains of beef calves grazing native range at the Fort Robinson Beef Cattle Research Station, Crawford, Nebraska

The influence of deposit-feeding on chlorophyll-a degradation in coastal marine sediments

To determine how macrofaunal activity affects rates and mechanisms of Chlorophyll-a (Chl-a) decomposition, we measured Chl-a concentrations during laboratory incubations of surface sediment with varying abundances of a subsurface deposit-feeder, Yoldia limatula. Decomposition patterns of Chl-a in sediment cores with and without a layer of algal-enriched sediment added to the surface were compared. Decomposition rate constants, kd, were calculated from the loss of reactive Chl-a and further quantified using a nonsteady state, depth-dependent, reaction-diffusion model. Values of kd decreased approximately exponentially with depth and were directly proportional to the number of Yoldia present. Yoldia increased the kd of both natural sedimentary Chl-a and algal enriched Chl-a in the upper 2 cm by up to 5.7×. Surface sediment porosity, penetration depths of a conservative tracer of diffusion (Br-), and oxidized metabolic substrates (e.g. Fe(III)) all increased significantly in the presence of Yoldia. Macrofaunal bioturbation increased the importance of suboxic degradation pathways. These experiments demonstrated that organic compounds from a single source can have a continuum of degradation rate constants as a function of biogenically determined environmental conditions (Chl-a kd ˜ 0.0043-0.20 d-1). In particular, Chl-a can have a continuum of kd values related to redox conditions, transport, and macrofauna abundance as a function of depth