On the Early History of the Singular Value Decomposition

This paper surveys the contributions of five mathematicians\,---\,Eugenio Beltrami (1835--1899), Camille Jordan (1838--1921), James Joseph Sylvester (1814--1897), Erhard Schmidt (1876--1959), and Hermann Weyl (1885--1955)\,---\,who were responsible for establishing the existence of the singular value decomposition and developing its theory. (Also cross-referenced as UMIACS-TR-92-31

Objective Identification of Informative Wavelength Regions in Galaxy Spectra

Understanding the diversity in spectra is the key to determining the physical parameters of galaxies. The optical spectra of galaxies are highly convoluted with continuum and lines which are potentially sensitive to different physical parameters. Defining the wavelength regions of interest is therefore an important question. In this work, we identify informative wavelength regions in a single-burst stellar populations model by using the CUR Matrix Decomposition. Simulating the Lick/IDS spectrograph configuration, we recover the widely used Dn(4000), Hbeta, and HdeltaA to be most informative. Simulating the SDSS spectrograph configuration with a wavelength range 3450-8350 Angstrom and a model-limited spectral resolution of 3 Angstrom, the most informative regions are: first region-the 4000 Angstrom break and the Hdelta line; second region-the Fe-like indices; third region-the Hbeta line; fourth region-the G band and the Hgamma line. A Principal Component Analysis on the first region shows that the first eigenspectrum tells primarily the stellar age, the second eigenspectrum is related to the age-metallicity degeneracy, and the third eigenspectrum shows an anti-correlation between the strengths of the Balmer and the Ca K and H absorptions. The regions can be used to determine the stellar age and metallicity in early-type galaxies which have solar abundance ratios, no dust, and a single-burst star formation history. The region identification method can be applied to any set of spectra of the user's interest, so that we eliminate the need for a common, fixed-resolution index system. We discuss future directions in extending the current analysis to late-type galaxies.Comment: 36 Pages, 13 Figures, 4 Tables. AJ Accepte

Improvements In computed tomography perfusion output using complex singular value decomposition and the maximum slope algorithm

OBJECTIVE: Determine if complex singular value decomposition (cSVD) used as preprocessing in the maximum slope algorithm reduces image noise of resultant physiologic parametric images. Noise will be decreased in the parametric maps of cerebral blood flow (CBF), cerebral blood volume (CBV) as compared to the same algorithm and data set with no cSVD applied. MATERIALS AND METHODS: A set of 10 patients (n=15) underwent a total combined 15 CT perfusion studies upon presenting with stroke symptoms. It was determined these patients suffered from occlusions resulting in a prolonged arrival time of blood to the brain. DICOM data files of these patients scans were selected based on this increased arrival delay. We compared the output of estimation calculations for cerebral blood flow (CBF), and cerebral blood volume (CBV), using preprocessing cSVD against the same scan data with no preprocessing cSVD. Image noise was assessed through the calculation of the standard deviation within specific regions of interest copied to specific areas of grey and white matter as well as CSF space. A decrease in the standard deviation values will indicate improvement in the noise level of the resultant images.. Results for the mean value within the regions of interest are expected to be similar between the groups calculated using cSVD and those calculated under the standard method. This will indicate the presence of minimal bias. RESULTS: Between groups of the standard processing method and the cSVD method standard deviation (SD) reductions were seen in both CBF and CBV values across all three ROIs. In grey matter measures of CBV, SD was reduced an average of 0.0034 mL/100g while measures of CBF saw SD reduced by an average of 0.073 mL/100g/min. In samples of white matter, standard deviations of CBV values were reduced on average by 0.0041mL/100g while CBF SD's were reduced by 0.073 mL/100g/min. CSF ROIs in CBV calculations saw SD reductions averaging 0.0047 mL/100g and reductions of 0.074 mL/100g/min in measures of CBF. Bias within CBV calculations was at most minimal as determined by no significant changes in mean calculated values. Calculations of CBF saw large downward bias in the mean values. CONCLUSIONS: The application of the cSVD method to preprocessing of CT perfusion imaging studies produces an effective method of noise reduction. In calculations of CBV, cSVD noise reduction results in overall improvement. In calculations of CBF, cSVD, while effective in noise reduction, caused mean values to be statistically lower than the standard method. It should be noted that there is currently no evaluation of which values can be considered more accurate physiologically. Simulations of the effect of noise on CBF showed a positive correlation suggesting that the CBF algorithm itself is sensitive to the level of noise

Stellar Content from high resolution galactic spectra via Maximum A Posteriori

This paper describes STECMAP (STEllar Content via Maximum A Posteriori), a flexible, non-parametric inversion method for the interpretation of the integrated light spectra of galaxies, based on synthetic spectra of single stellar populations (SSPs). We focus on the recovery of a galaxy's star formation history and stellar age-metallicity relation. We use the high resolution SSPs produced by PEGASE-HR to quantify the informational content of the wavelength range 4000 - 6800 Angstroms. A detailed investigation of the properties of the corresponding simplified linear problem is performed using singular value decomposition. It turns out to be a powerful tool for explaining and predicting the behaviour of the inversion. We provide means of quantifying the fundamental limitations of the problem considering the intrinsic properties of the SSPs in the spectral range of interest, as well as the noise in these models and in the data. We performed a systematic simulation campaign and found that, when the time elapsed between two bursts of star formation is larger than 0.8 dex, the properties of each episode can be constrained with a precision of 0.04 dex in age and 0.02 dex in metallicity from high quality data (R=10 000, signal-to-noise ratio SNR=100 per pixel), not taking model errors into account. The described methods and error estimates will be useful in the design and in the analysis of extragalactic spectroscopic surveys.Comment: 31 pages, 23 figures, accepted for publication in MNRA

The Development of Intersection Homology Theory

This historical introduction is in two parts. The first is reprinted with permission from ``A century of mathematics in America, Part II,'' Hist. Math., 2, Amer. Math. Soc., 1989, pp.543-585. Virtually no change has been made to the original text. In particular, Section 8 is followed by the original list of references. However, the text has been supplemented by a series of endnotes, collected in the new Section 9 and followed by a second list of references. If a citation is made to the first list, then its reference number is simply enclosed in brackets -- for example, [36]. However, if a citation is made to the second list, then its number is followed by an `S' -- for example, [36S]. Further, if a subject in the reprint is elaborated on in an endnote, then the subject is flagged in the margin by the number of the corresponding endnote, and the endnote includes in its heading, between parentheses, the page number or numbers on which the subject appears in the reprint below. Finally, all cross-references appear as hypertext links in the dvi and pdf copies.Comment: 58 pages, hypertext links added; appeared in Part 3 of the special issue of Pure and Applied Mathematics Quarterly in honor of Robert MacPherson. However, the flags in the margin were unfortunately (and inexplicably) omitted from the published versio