The Multiband Imaging Photometer for Spitzer (MIPS) provides long-wavelength capability for the mission in imaging bands at 24, 70, and 160 m and measurements of spectral energy distributions between 52 and 100 m at a spectral resolution of about 7%. By using true detector arrays in each band, it provides both critical sampling of the Spitzer point-spread function and relatively large imaging fields of view, allowing for substantial advances in sensitivity, angular resolution, and efficiency of areal coverage compared with previous space far-infrared capabilities. The 24 m array has excellent photometric properties, and measurements with rms relative errors of about 1% can be obtained. The two longer-wavelength arrays use detectors with poor photometric stability, but a system of onboard stimulators used for relative calibration, combined with a unique data pipeline, produce good photometry with rms relative errors of less than 10%.

We present 70 and 160 m observations from the Spitzer extragalactic First Look Survey (xFLS). The data reduction techniques and the methods for producing co-added mosaics and source catalogs are discussed. Currently, 26% of the 70 m sample and 49% of the 160 m-selected sources have redshifts. The majority of sources with redshifts are star-forming galaxies at z < 0:5, while about 5% have infrared colors consistent with active galactic nuclei. The observed infrared colors agree with the spectral energy distributions (SEDs) of local galaxies previously determined from IRAS and Infrared Space Observatory data. The average 160 m/70 m color temperature for the dust is T d ' 30 AE 5 K, and the average 70 m/24 m spectral index is ' 2:4 AE 0:4. The observed infrared-to-radio correlation varies with redshift as expected out to z $ 1 based on the SEDs of local galaxies. The xFLS number counts at 70 and 160 m are consistent within uncertainties with the models of galaxy evolution, but there are indications that the current models may require slight modifications. Deeper 70 m observations are needed to constrain the models, and redshifts for the faint sources are required to measure the evolution of the infrared luminosity function.

In this paper we address sampling and approximation of functions on combinatorial graphs. We develop filtering on graphs by using Schrödinger's group of operators generated by combinatorial Laplace operator. Then we construct a sampling theory by proving Poincare and Plancherel-Polya-type inequalities for functions on graphs. These results lead to a theory of sparse approximations on graphs and have potential applications to filtering, denoising, data dimension reduction, image processing, image compression, computer graphics, visualization and learning theory.

Compositional data are ubiquitous in chemistry and materials science: analysis of elements in multicomponent systems, combinatorial problems, etc., lead to data that are non-negative and sum to a constant (for example, atomic concentrations). The constant sum constraint restricts the sampling space to a simplex instead of the usual Euclidean space. Since statistical measures such as mean and standard deviation are defined for the Euclidean space, traditional correlation studies, multivariate analysis, and hypothesis testing may lead to erroneous dependencies and incorrect inferences when applied to compositional data. Furthermore, composition measurements that are used for data analytics may not include all of the elements contained in the material; that is, the measurements may be subcompositions of a higher-dimensional parent composition. Physically meaningful statistical analysis must yield results that are invariant under the number of composition elements, requiring the application of specialized statistical tools. We present specifics and subtleties of compositional data processing through discussion of illustrative examples. We introduce basic concepts, terminology, and methods required for the analysis of compositional data and utilize them for the spatial interpolation of composition in a sputtered thin film. The results demonstrate the importance of this mathematical framework for compositional data analysis (CDA) in the fields of materials science and chemistry.