High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities

We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness''. The latter quantity adds up scaled errors of approximations of a measure by d-planes at different scales and locations, and is commonly used to characterize uniform rectifiability. We thus obtain a characterization of uniform rectifiability by using the Menger-type curvature. In the current paper (part I) we control the continuous Menger-type curvature of an Ahlfors d-regular measure by its Jones-type flatness.Comment: 47 pages, 13 figures. Minor revisions and the inclusion of figure

Quasi-uniformity of Minimal Weighted Energy Points on Compact Metric Spaces

For a closed subset $K$ of a compact metric space $A$ possessing an $\alpha$-regular measure $\mu$ with $\mu(K)>0$, we prove that whenever $s>\alpha$, any sequence of weighted minimal Riesz $s$-energy configurations $\omega_N=\{x_{i,N}^{(s)}\}_{i=1}^N$ on $K$ (for `nice' weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as $N$ grows large. Furthermore, if $K$ is an $\alpha$-rectifiable compact subset of Euclidean space ($\alpha$ an integer) with positive and finite $\alpha$-dimensional Hausdorff measure, it is possible to generate such a quasi-uniform sequence of configurations that also has (as $N\to \infty$) a prescribed positive continuous limit distribution with respect to $\alpha$-dimensional Hausdorff measure. As a consequence of our energy related results for the unweighted case, we deduce that if $A$ is a compact $C^1$ manifold without boundary, then there exists a sequence of $N$-point best-packing configurations on $A$ whose mesh-separation ratios have limit superior (as $N\to \infty$) at most 2

The summertime plankton community at South Georgia (Southern Ocean): comparing the historical (1926/27) and modern (post 1995) records.

The earliest comprehensive plankton sampling programme in the Southern Ocean was 32 undertaken during the early part of last century by Discovery Investigations to gain a 33 greater scientific understanding of whale stocks and their summer feeding grounds. An 34 initial survey was carried out around South Georgia during December 1926 and January 35 1927 to describe the distribution of plankton during the summer, and to serve as a 36 baseline against which to compare future surveys. We have reanalysed phytoplankton and 37 zooplankton data from this survey and elucidated patterns of community distribution and 38 compared them with our recent understanding of the ecosystem based on contemporary 39 data. Analysis of Discovery data identified five groups of stations with characteristic 40 phytoplankton communities which were almost entirely consistent with the original 41 analysis conducted by Hardy and Gunther (1935). Major groupings were located at the 42 western end of the island and over the northern shelf where Corethron spp. were 43 dominant, and to the south and east where a more diverse flora included high abundances 44 of Nitzschia seriata. Major zooplankton-station groupings were located over the inner 45 shelf which was characterised by a high abundance of Drepanopus forcipatus and in 46 oceanic water >500 m deep that were dominated by Foraminifera, Oithona spp., 47 Ctenocalanus vanus, and Calanoides acutus. Stations along the middle and outer shelf 48 regions to the north and west, were characterised by low overall abundance. There was 49 some evidence that groupings of stations to the north of the island originated in different 50 water masses on either side of the Southern Antarctic Circumpolar Current Front, the 51 major frontal system in the deep ocean close to South Georgia. However, transect lines 52 during 1926/27 did not extend far enough offshore to sample this frontal region 53 3 adequately. Interannual variability of zooplankton abundance was assessed from stations 54 which were sampled repeatedly during 7 recent British Antarctic Survey cruises (1995-55 2005) to the region and following taxonomic harmonization and numerical 56 standardization (ind. m-3), a subset of 45 taxonomic categories of zooplankton (species 57 and higher taxa) from 1926/27, were compared with similar data obtained during the 58 BAS cruises using a linear model. Initially comparisons were restricted to BAS stations 59 that lay within 40 km of Discovery stations although a comparison was also made using 60 all available data. Despite low abundance values in 1926/27, in neither comparison did 61 Discovery data differ significantly from BAS data. Calculation of the percentage 62 similarity index across cruises did not reveal any systematic differences in species 63 composition between 1926/27 and the present. In the light of ocean warming trends, the 64 existence of more subtle changes in species composition is not ruled out, but an absence 65 of finely resolved time-series data make this impossible to determine

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

Describing the Dying Days of Alzheimer\u27s disease

There is no doubt that individuals, families, communities, and governments face an enormous challenge in the growing number of persons with dementia in our societies. Accompanying this demographic boomis an opportunity to examine the way we think about the vast, miscellaneous category we currently call “Alzheimer’s disease”, (AD), which has just eclipsed its 100th year in the medical lexicon

Marketplace of Memory: What the Brain Fitness Technology Industry Says About Us and How We Can Do Better

In the therapeutic void created by over 20 failed Alzheimer\u27s disease drugs during the past decade, a new marketplace of brain fitness technology products has emerged. Ranging from video games and computer software to mobile phone apps and hand-held devices, these commercial products promise to maintain and enhance the memory, concentration, visual and spatial skills, verbal recall, and executive functions of individual users. It is instructive to view these products as sociocultural objects deeply imbued with the values and ideologies of our age; consequently, this article offers a critique of the brain fitness technology marketplace while identifying limitations in the capacity of commercial products to realistically improve cognitive health. A broader conception of brain health is presented, going beyond the reductionism of the commercial brain fitness marketplace and asking how our most proximate relationships and local communities can play a role in supporting cognitive and psychosocial well-being. This vision is grounded in recent experiences at The Intergenerational School in Cleveland, OH, a multigenerational community-oriented learning environment that is implementing brain fitness technology in novel ways

A Tale of Two Reports: What Recent Publications from the Alzheimer’s Association and Institute of Medicine Say About the State of the Field

Scientific reports hold a mirror up to a culture, reflecting the beliefs, values, assumptions, practices, technologies, and political-economic interests that shape discourse around particular diseases in particular cultural moments [1]. Indeed, despite their authoritative patina, such reports are “living documents” that invite critical analysis about where discourse in a particular field has been and where it is going. In this spirit, two recent reports from the Alzheimer’s Association (Changing the Trajectory of Alzheimer’s Disease: How a Treatment by 2025 Saves Lives and Dollars) and the Institute of Medicine in the United States (Cognitive Aging: Progress in Understanding and Opportunities for Action) are worthy of attention from those endeavoring to understand and contextualize the science and politics of brain health, aging, and dementia [2]