Least squares approximations of measures via geometric condition numbers

For a probability measure on a real separable Hilbert space, we are interested in "volume-based" approximations of the d-dimensional least squares error of it, i.e., least squares error with respect to a best fit d-dimensional affine subspace. Such approximations are given by averaging real-valued multivariate functions which are typically scalings of squared (d+1)-volumes of (d+1)-simplices. Specifically, we show that such averages are comparable to the square of the d-dimensional least squares error of that measure, where the comparison depends on a simple quantitative geometric property of it. This result is a higher dimensional generalization of the elementary fact that the double integral of the squared distances between points is proportional to the variance of measure. We relate our work to two recent algorithms, one for clustering affine subspaces and the other for Monte-Carlo SVD based on volume sampling

High-Dimensional Menger-Type Curvatures-Part II: d-Separation and a Menagerie of Curvatures

This is the second of two papers wherein we estimate multiscale least squares approximations of certain measures by Menger-type curvatures. More specifically, we study an arbitrary d-regular measure on a real separable Hilbert space. The main result of the paper bounds the least squares error of approximation at any ball by an average of the discrete Menger-type curvature over certain simplices in in the ball. A consequent result bounds the Jones-type flatness by an integral of the discrete curvature over all simplices. The preceding paper provided the opposite inequalities. Furthermore, we demonstrate some other discrete curvatures for characterizing uniform rectifiability and additional continuous curvatures for characterizing special instances of the (p, q)-geometric property. We also show that a curvature suggested by Leger (Annals of Math, 149(3), p. 831-869, 1999) does not fit within our framework.Comment: 32 pages, no figure

Morphological and Pathological Evolution of the Brain Microcirculation in Aging and Alzheimer’s Disease

Key pathological hallmarks of Alzheimer’s disease (AD), including amyloid plaques, cerebral amyloid angiopathy (CAA) and neurofibrillary tangles do not completely account for cognitive impairment, therefore other factors such as cardiovascular and cerebrovascular pathologies, may contribute to AD. In order to elucidate the microvascular changes that contribute to aging and disease, direct neuropathological staining and immunohistochemistry, were used to quantify the structural integrity of the microvasculature and its innervation in three oldest-old cohorts: 1) nonagenarians with AD and a high amyloid plaque load; 2) nonagenarians with no dementia and a high amyloid plaque load; 3) nonagenarians without dementia or amyloid plaques. In addition, a non-demented (ND) group (average age 71 years) with no amyloid plaques was included for comparison. While gray matter thickness and overall brain mass were reduced in AD compared to ND control groups, overall capillary density was not different. However, degenerated string capillaries were elevated in AD, potentially suggesting greater microvascular “dysfunction” compared to ND groups. Intriguingly, apolipoprotein ε4 carriers had significantly higher string vessel counts relative to non-ε4 carriers. Taken together, these data suggest a concomitant loss of functional capillaries and brain volume in AD subjects. We also demonstrated a trend of decreasing vesicular acetylcholine transporter staining, a marker of cortical cholinergic afferents that contribute to arteriolar vasoregulation, in AD compared to ND control groups, suggesting impaired control of vasodilation in AD subjects. In addition, tyrosine hydroxylase, a marker of noradrenergic vascular innervation, was reduced which may also contribute to a loss of control of vasoconstriction. The data highlight the importance of the brain microcirculation in the pathogenesis and evolution of AD

The James Webb Space Telescope Mission

Twenty-six years ago a small committee report, building on earlier studies, expounded a compelling and poetic vision for the future of astronomy, calling for an infrared-optimized space telescope with an aperture of at least $4m$. With the support of their governments in the US, Europe, and Canada, 20,000 people realized that vision as the $6.5m$ James Webb Space Telescope. A generation of astronomers will celebrate their accomplishments for the life of the mission, potentially as long as 20 years, and beyond. This report and the scientific discoveries that follow are extended thank-you notes to the 20,000 team members. The telescope is working perfectly, with much better image quality than expected. In this and accompanying papers, we give a brief history, describe the observatory, outline its objectives and current observing program, and discuss the inventions and people who made it possible. We cite detailed reports on the design and the measured performance on orbit.Comment: Accepted by PASP for the special issue on The James Webb Space Telescope Overview, 29 pages, 4 figure

The Science Performance of JWST as Characterized in Commissioning

This paper characterizes the actual science performance of the James Webb Space Telescope (JWST), as determined from the six month commissioning period. We summarize the performance of the spacecraft, telescope, science instruments, and ground system, with an emphasis on differences from pre-launch expectations. Commissioning has made clear that JWST is fully capable of achieving the discoveries for which it was built. Moreover, almost across the board, the science performance of JWST is better than expected; in most cases, JWST will go deeper faster than expected. The telescope and instrument suite have demonstrated the sensitivity, stability, image quality, and spectral range that are necessary to transform our understanding of the cosmos through observations spanning from near-earth asteroids to the most distant galaxies.Comment: 5th version as accepted to PASP; 31 pages, 18 figures; https://iopscience.iop.org/article/10.1088/1538-3873/acb29

Mass balance of the Greenland and Antarctic ice sheets from 1992 to 2020

Ice losses from the Greenland and Antarctic ice sheets have accelerated since the 1990s, accounting for a significant increase in the global mean sea level. Here, we present a new 29-year record of ice sheet mass balance from 1992 to 2020 from the Ice Sheet Mass Balance Inter-comparison Exercise (IMBIE). We compare and combine 50 independent estimates of ice sheet mass balance derived from satellite observations of temporal changes in ice sheet flow, in ice sheet volume, and in Earth's gravity field. Between 1992 and 2020, the ice sheets contributed 21.0±1.9g€¯mm to global mean sea level, with the rate of mass loss rising from 105g€¯Gtg€¯yr-1 between 1992 and 1996 to 372g€¯Gtg€¯yr-1 between 2016 and 2020. In Greenland, the rate of mass loss is 169±9g€¯Gtg€¯yr-1 between 1992 and 2020, but there are large inter-annual variations in mass balance, with mass loss ranging from 86g€¯Gtg€¯yr-1 in 2017 to 444g€¯Gtg€¯yr-1 in 2019 due to large variability in surface mass balance. In Antarctica, ice losses continue to be dominated by mass loss from West Antarctica (82±9g€¯Gtg€¯yr-1) and, to a lesser extent, from the Antarctic Peninsula (13±5g€¯Gtg€¯yr-1). East Antarctica remains close to a state of balance, with a small gain of 3±15g€¯Gtg€¯yr-1, but is the most uncertain component of Antarctica's mass balance. The dataset is publicly available at 10.5285/77B64C55-7166-4A06-9DEF-2E400398E452 (IMBIE Team, 2021)

Evaluation of appendicitis risk prediction models in adults with suspected appendicitis

Background Appendicitis is the most common general surgical emergency worldwide, but its diagnosis remains challenging. The aim of this study was to determine whether existing risk prediction models can reliably identify patients presenting to hospital in the UK with acute right iliac fossa (RIF) pain who are at low risk of appendicitis. Methods A systematic search was completed to identify all existing appendicitis risk prediction models. Models were validated using UK data from an international prospective cohort study that captured consecutive patients aged 16–45 years presenting to hospital with acute RIF in March to June 2017. The main outcome was best achievable model specificity (proportion of patients who did not have appendicitis correctly classified as low risk) whilst maintaining a failure rate below 5 per cent (proportion of patients identified as low risk who actually had appendicitis). Results Some 5345 patients across 154 UK hospitals were identified, of which two‐thirds (3613 of 5345, 67·6 per cent) were women. Women were more than twice as likely to undergo surgery with removal of a histologically normal appendix (272 of 964, 28·2 per cent) than men (120 of 993, 12·1 per cent) (relative risk 2·33, 95 per cent c.i. 1·92 to 2·84; P < 0·001). Of 15 validated risk prediction models, the Adult Appendicitis Score performed best (cut‐off score 8 or less, specificity 63·1 per cent, failure rate 3·7 per cent). The Appendicitis Inflammatory Response Score performed best for men (cut‐off score 2 or less, specificity 24·7 per cent, failure rate 2·4 per cent). Conclusion Women in the UK had a disproportionate risk of admission without surgical intervention and had high rates of normal appendicectomy. Risk prediction models to support shared decision‐making by identifying adults in the UK at low risk of appendicitis were identified