Measure Theory in Noncommutative Spaces

The integral in noncommutative geometry (NCG) involves a non-standard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measure-theoretic view, however, the formulation contains several difficulties. We review results concerning the technical features of the integral in NCG and some outstanding problems in this area. The review is aimed for the general user of NCG

Noncommutative Residues and a Characterisation of the Noncommutative Integral

We continue the study of the relationship between Dixmier traces and noncommutative residues initiated by A. Connes. The utility of the residue approach to Dixmier traces is shown by a characterisation of the noncommutative integral in Connes' noncommutative geometry (for a wide class of Dixmier traces) as a generalised limit of vector states associated to the eigenvectors of a compact operator (or an unbounded operator with compact resolvent), i.e. as a generalised quantum limit. Using the characterisation, a criteria involving the eigenvectors of a compact operator and the projections of a von Neumann subalgebra of bounded operators is given so that the noncommutative integral associated to the compact operator is normal, i.e. satisfies a monotone convergence theorem, for the von Neumann subalgebra.Comment: 15 page

BASIS RISK FOR RICE

The objective of this paper is to develop a cross hedging model for rice that minimizes basis risk and accounts for the existence of the nonstationary nature of basis. Basis is treated as an endogenous variable and model for basis risk are developed.Financial Economics, Risk and Uncertainty,

The role of molecular clouds in the star formation process as observed in two grand design spiral galaxies.

We have compared the molecular, neutral and ionized hydrogen distributions in two nearby spiral galaxies. To estimate H\sb2 surface densities we acquired observations of the CO (J = 1 $\to$ 0) transition in 60 positions to a radius of 135 in the Sbc galaxy M51 (NGC 5194), and in 21 positions to a radius of 150 in the SAB galaxy M83 (NGC 5236) using the 13.7 m telescope of the Five College Radio Astronomy Observatory. The molecular component of the ISM was found to strongly dominate over the HI component in each galaxy. Extinction corrected H$\alpha$ intensities were used to compute the detailed massive star formation rates (MSFRs) in each galaxy. Estimates of the MSFR, gas density, and the ratio of these quantities, the massive star formation efficiency (MSFE), were then examined. In M51, the spiral arms exhibit an excess gas density of 1.4-1.6 times the interarm values. The MSFR contrast between the arms and interarms, measuring between 1.5 and 2.3 at the same resolution, exceeds the gas density contrast and implies a nonlinear relationship between star formation and gas surface density on the spiral arms. This follows the predictions of the cloud-cloud collision scenario fo star formation which relies on the occurrence of orbit crowding to bring clouds into close proximity. We note that the regions exhibiting the highest MSFEs are those in the spiral potential minimum inward of R = 124 , and those regions outward of R = 124 thought to be experiencing orbit crowding due to tidal distortion caused by the close passage of M51\u27s companion galaxy. The total (arm and interarm) gas content and massive star formation rates in concentric annuli in the disk of M51 were computed. The two quantities fall off together with radius, yielding a relatively constant MSFE with radius. This is consistent with the increased MSFE on the arms in that the majority of the gas shows a constant MSFE. The resulting time scale for gas depletion (total SFE\sp{-1}) in the disk is 2.5 $\pm$ 5 $\times$ 10\sp9 yr assuming a Salpeter-like initial mass function. In M83, the molecular gas component of the inner disk mimics the bar morphology. In this galaxy there is the suggestion of enhanced star formation at the ends of the central bar due to the compression of cloud orbits found there. The gas depletion time scale is 1.2 $\pm$ 0.3 $\times$ 10\sp9 yr

The ISM in the M82 starburst

We have observed (O 1) (63 microns) and (Si 2) (35 microns) in the central 700 pc of the starburst galaxy M82. The luminosities in these transitions are 7.1 x 10(exp 7) solar luminosity and 6.2 x 10(exp 7) solar luminosity, respectively, which are each approx. 0.15% of the bolometric luminosity from this region. The ratios of (O 1) line luminosity to (O 3), (Si 2) (35 microns), and to bolometric luminosities in M82 are similar to those in M42, M17, and Sgr A. These similarities, and the association of the bulk of the (O 1) and (Si 2) emission with the ionized emission, suggest that the dominant emission mechanism for (O 1) and (Si 2) in M82 is the same as in these Galactic regions, namely warm gas photodissociated by UV flux from the OB stars responsible for the nearby H 2 regions. We argue that shock or x ray heated gas or H 2 plasma is a minor contributor to the intensities of these fine structure lines. Both the (O 1) (53 microns) and the (Si 2) (35 microns) spectrum show an asymmetric line profile indistinguishable in shape from those of the (O 3) (52 and 88 microns) and (N 3) (57 microns) lines and similar to that of the more extended (C 2) 158 micron line measured previously in M82. We detect two distinct velocity components, which we attribute to emission from two regions at either end of the central bar, where the bar connects to an orbiting torus of neutral gas seen in H 1 and CO J = 1-0. We model separately the two velocity components and derive the physical conditions in these two regions. The clouds in these regions are small, R approx. 1-2 pc, have warm neutral gas surfaces, T approx. 200 K, and are concentrated with volume filling factors of approx. 0.02 and area filling factors of 1-5. The entire central region (R approx. 700 pc) is characterized by a large number, approx. 5 x 10(exp 4), of 2 x 10(exp 3) solar mass clouds with surface densities of approx. 3 x 10(exp 4) cm(exp -3), illuminated by FUV fluxes 10(exp 4) times the average local interstellar value for the Milky Way. These clouds reside in the harsh conditions of a starburst nucleus, with photoevaporation times of 10(exp 6) yr, and collision timescales only about an order of magnitude longer

A new software tool for computing Earth's atmospheric transmission of near- and far-infrared radiation

This report describes a new software tool, ATRAN, which computes the transmittance of Earth's atmosphere at near- and far-infrared wavelengths. We compare the capabilities of this program with others currently available and demonstrate its utility for observational data calibration and reduction. The program employs current water-vapor and ozone models to produce fast and accurate transmittance spectra for wavelengths ranging from 0.8 microns to 10 mm

Dixmier Traces as Singular Symmetric Functionals and Applications to Measurable Operators

This paper introduces a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. By unifying various constructions, and translating the situation of Dixmier traces into the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces, we obtain the results (i) and (ii) below. The results are stated here, for the reader, in terms of the ideal $L^{(1,\infty)}$ of compact operators whose partial sums of singular values are of logarithmic divergence. (i) a positive compact operator $x$ in $L^{(1,\infty)}$ yields the same value for an arbitrary Connes-Dixmier trace (ie. $x$ is measurable in the sense of Connes) if and only if $\lim_{N\to\infty} \frac{1}{Log N}\sum_{n=1}^N s_n(x)$ exists, where $s_n(x)$ are the singular values of the compact operator $x$; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space $L^{(1,\infty)}/L^{(1,\infty)}_0$, where the space $L^{(1,\infty)}_0$ is the closure of all finite rank operators in the norm $||.||_{(1,\infty)}$.Comment: 31 pages, LaTex source, to appear in J. Funct. Ana

Life Cycle Costing and Food Systems: Concepts, Trends, and Challenges of Impact Valuation

Our global food systems create pervasive environmental, social, and health impacts. Impact valuation is an emerging concept that aims to quantify all environmental, social, and health costs of food systems in an attempt to make the true cost of food more transparent. It also is designed to facilitate the transformation of global food systems. The concept of impact valuation is emerging at the same time as, and partly as a response to, calls for the development of legal mechanisms to address environmental, social, and health concerns. Information has long been understood both as a necessary precursor for regulation and as a regulatory tool in and of itself. With global supply chains and widespread impacts, data necessary to produce robust and complete impact valuation requires participation and cooperation from a variety of food system actors. New costing methods, beyond basic accounting, are necessary to incorporate the scope of impacts and stakeholders. Furthermore, there are a range of unanswered questions surrounding realizations of impact valuation methods, e.g. data sharing, international privacy, corporate transparency, limitations on valuation itself, and data collection standardization. Because of the proliferation of calls for costing tools, this article steps back and assesses the current development of impact valuation methods. In this article, we review current methods and initiatives for the implementation of food system impact valuation. We conclude that in some instances, calls for the implementation of costing have outpaced available and reliable data collection and current costing techniques. Many existing initiatives are being developed without adequate consideration of the legal challenges that hinder implementation. Finally, we conclude with a reminder that although impact valuation tools are most often sought and implemented in service of market-based tools for reform, they can also serve as a basis for robust public policies

Riemannian manifolds in noncommutative geometry

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spinc manifolds; and conversely, in the presence of a spinc structure. We also show how to obtain an analogue of Kasparov\u27s fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples