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

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

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

Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings

In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but generalizes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring R, that is, the set of all possible finite global dimensions of endomorphism rings of MCM R-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings

Intra-pixel gain variations and high-precision photometry with the Infrared Array Camera (IRAC)

The Infrared Array Camera (IRAC) on the Spitzer Space Telescope has been used to measure < 10^(-4) temporal variations in point sources (such as transiting extrasolar planets) at 3.6 and 4.5 μm. Due to the under-sampled nature of the PSF, the warm IRAC arrays show variations of as much as 8% in sensitivity as the center of the PSF moves across a pixel due to normal spacecraft pointing wobble and drift. These intra-pixel gain variations are the largest source of correlated noise in IRAC photometry. Usually this effect is removed by fitting a model to the science data themselves (self-calibration), which could result in the removal of astrophysically interesting signals. We describe a new technique for significantly reducing the gain variations and improving photometric precision in a given observation, without using the data to be corrected. This comprises: (1) an adaptive centroiding and repositioning method ("Peak-Up") that uses the Spitzer Pointing Control Reference Sensor (PCRS) to repeatedly position a target to within 0.1 IRAC pixels of an area of minimal gain variation; and (2) the high-precision, high-resolution measurement of the pixel gain structure using non-variable stars. We show that the technique currently allows the reduction of correlated noise by almost an order of magnitude over raw data, which is comparable to the improvement due to self-calibration. We discuss other possible sources of correlated noise, and proposals for reducing their impact on photometric precision

Sensitivity analysis of circadian entrainment in the space of phase response curves

Sensitivity analysis is a classical and fundamental tool to evaluate the role of a given parameter in a given system characteristic. Because the phase response curve is a fundamental input--output characteristic of oscillators, we developed a sensitivity analysis for oscillator models in the space of phase response curves. The proposed tool can be applied to high-dimensional oscillator models without facing the curse of dimensionality obstacle associated with numerical exploration of the parameter space. Application of this tool to a state-of-the-art model of circadian rhythms suggests that it can be useful and instrumental to biological investigations.Comment: 22 pages, 8 figures. Correction of a mistake in Definition 2.1. arXiv admin note: text overlap with arXiv:1206.414

The Infrared Spectrograph on the Spitzer Space Telescope

The Infrared Spectrograph (IRS) is one of three science instruments on the Spitzer Space Telescope. The IRS comprises four separate spectrograph modules covering the wavelength range from 5.3 to 38micron with spectral resolutions, R \~90 and 600, and it was optimized to take full advantage of the very low background in the space environment. The IRS is performing at or better than the pre-launch predictions. An autonomous target acquisition capability enables the IRS to locate the mid-infrared centroid of a source, providing the information so that the spacecraft can accurately offset that centroid to a selected slit. This feature is particularly useful when taking spectra of sources with poorly known coordinates. An automated data reduction pipeline has been developed at the Spitzer Science Center.Comment: Accepted in ApJ Sup. Spitzer Special Issue, 6 pages, 4 figure

Absolute photometric calibration of IRAC: lessons learned using nine years of flight data

Significant improvements in our understanding of various photometric effects have occurred in the more than nine years of flight operations of the Infrared Array Camera aboard the Spitzer Space Telescope. With the accumulation of calibration data, photometric variations that are intrinsic to the instrument can now be mapped with high fidelity. Using all existing data on calibration stars, the array location-dependent photometric correction (the variation of flux with position on the array) and the correction for intra-pixel sensitivity variation (pixel-phase) have been modeled simultaneously. Examination of the warm mission data enabled the characterization of the underlying form of the pixelphase variation in cryogenic data. In addition to the accumulation of calibration data, significant improvements in the calibration of the truth spectra of the calibrators has taken place. Using the work of Engelke et al. (2006), the KIII calibrators have no offset as compared to the AV calibrators, providing a second pillar of the calibration scheme. The current cryogenic calibration is better than 3% in an absolute sense, with most of the uncertainty still in the knowledge of the true flux densities of the primary calibrators. We present the final state of the cryogenic IRAC calibration and a comparison of the IRAC calibration to an independent calibration methodology using the HST primary calibrators