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

Loudly sing cuckoo : More-than-human seasonalities in Britain

This research was funded by a grant from the Arts and Humanities Research Council, grant number AH/E009573/1.Peer reviewedPostprin

Experimental observations of tsunami induced scour at onshore structures

Tsunami inundation of the coastal environment can induce scour at structure foundations leading to failure. A series of experiments are made using a unique Pneumatic Long Wave Generator to generate tsunami wave periods of 25 - 147 s equating to 3 - 17.3 mins at 1:50 Froude scale. The waves propagate over a sloping bathymetry and impinge upon a square structure founded onshore in a flat sediment bed. Flow velocity, height and scour are recorded as a function of time during tsunami inundation. The rate of scour is observed to be time dependent. Equilibrium, which is not attained, is argued to be an inappropriate measure for time-dependent transient flows such as tsunami in which the flow velocity, depth and direction are variable. The maximum scour depth is recorded and critically is observed not to be equal to the final depth due to significant sediment slumping when flow velocities reduce in the latter stages of inundation. Current and wave scour predictor equations over predict the scour, while the ASCE 7-16 method under predicts. Comparisons with available data in the literature show longer inundation durations increase the amount of scour

Dwarf carbon stars are likely metal-poor binaries and unlikely hosts to carbon planets

Dwarf carbon stars make up the largest fraction of carbon stars in the Galaxy with around 1200 candidates known to date primarily from the Sloan Digital Sky Survey. They either possess primordial carbon-enhancements, or are polluted by mass transfer from an evolved companion such that C/O is enhanced beyond unity. To directly test the binary hypothesis, a radial velocity monitoring survey has been carried out on 28 dwarf carbon stars, resulting in the detection of variations in 21 targets. Using Monte Carlo simulations, this detection fraction is found to be consistent with a 100% binary population and orbital periods on the order of hundreds of days. This result supports the post-mass transfer nature of dwarf carbon stars, and implies they are not likely hosts to carbon planets.Comment: 5 pages, 1 figure, accepted to MNRA

Heavy Isotope Composition of Oxygen in Zircon from Soil Sample 14163: Lunar Perspective of an Early Ocean on the Earth

Thirty oxygen analyses of a large (sub-millimetre) zircon grain from the lunar soil sample 14163 have been determined using CAMECA 1270 ion microprobe. The sample 14163 was returned form the Fra Mauro region by Apollo 14 mission. Zircon grain of 0.6-0.8 mm in size extracted from the sample was imaged using CL detector fitted to the Philips Electron Microscope in order to reveal internal structure. Oxygen isotopes have been analysed during two sessions. The first set of data was collected using the original mount where the grain was set in the resin attached to the glass slide. This resulted in the two complications: (i) standard zircon has to be analysed from the separate mount and (ii) the lunar zircon grain was rased in the holder compared to the standard. In order to investigate, if the elevated oxygen compositions observed during this session could have resulted from this difference in geometric configuration during the standard and sample analyses, the lunar zircon was extracted from the original mount, remounted with the standard chip in the new resin disk and reanalysed during the second session. All analyses made during the first session show delta O-18 values heavier than 6.0%. The second set of data has a wider spread of delta O-18 values with some values as low as 5.6%. Nevertheless, a half of observed delta O-18 values in this set is also higher than 6.0%. Slightly lighter oxygen compositions observed during the second session indicate possible dependence of measured delta O-18 values on the geometry of analysed samples. Presence of zircons with similar heavy oxygen isotope compositions on the Moon, which neither had liquid water or felic crust similar to that on the Earth nor ever developed regime similar to plate tectonics, suggests that other mechanisms can be responsible for elevated delta O-18 values in zircons. This implies that there is no support for the presence of an ocean on the surface of the early Earth and as the ocean appears to be an essential ingredient for the plate tectonics, there is no basis for belief that this mechanism was operating in the early history of the planet

Are autistic traits in the general population stable across development?

There is accumulating evidence that autistic traits (AT) are on a continuum in the general population, with clinical autism representing the extreme end of a quantitative distribution. While the nature and severity of symptoms in clinical autism are known to persist over time, no study has examined the long-term stability of AT among typically developing toddlers. The current investigation measured AT in 360 males and 400 males from the general population close to two decades apart, using the Pervasive Developmental Disorder subscale of the Child Behavior Checklist in early childhood (M = 2.14 years; SD = 0.15), and the Autism-Spectrum Quotient in early adulthood (M = 19.50 years; SD = 0.70). Items from each scale were further divided into social (difficulties with social interaction and communication) and non-social (restricted and repetitive behaviours and interests) AT. The association between child and adult measurements of AT as well the influence of potentially confounding sociodemographic, antenatal and obstetric variables were assessed using Pearson's correlations and linear regression. For males, Total AT in early childhood were positively correlated with total AT (r = .16, p = .002) and social AT (r = .16, p = .002) in adulthood. There was also a positive correlation for males between social AT measured in early childhood and Total (r = .17, p = .001) and social AT (r = .16, p = .002) measured in adulthood. Correlations for non-social AT did not achieve significance in males. Furthermore, there was no significant longitudinal association in AT observed for males or females. Despite the constraints of using different measures and different raters at the two ages, this study found modest developmental stability of social AT from early childhood to adulthood in boys

Zircon U-Pb, Hf and O isotope constraints on growth versus reworking of continental crust in the subsurface Grenville orogen, Ohio, USA

Combined U-Pb, O and Hf isotope data in zircon allows discrimination between juvenile and reworked crust, and is therefore a useful tool for understanding formation and evolution of the continental crust. The crustal evolution of basement rocks in central North America (Laurentia) is poorly constrained, as it is almost entirely overlain by Palaeozoic cover. In order to improve our understanding of the evolution of this region we present U-Pb, O and Hf isotope data from zircon in drill-core samples from the subsurface basement of Ohio. The Hf isotope data suggests juvenile crust formation at similar to 1650 Ma followed by continued reworking of a single reservoir. This similar to 1650 Ma reservoir was tapped at similar to 1450 Ma during the formation of the Granite-Rhyolite Province and subsequently reworked again during the Grenvillian orogeny. The similar to 1650 Ma crust formation model age for the suite of samples along with the presence of similar to 1650 Ma magmatic rocks suggests an eastward extension of the Mazatzal Province (or Mazatzal-like crust) and makes it a possible protolith to the subsurface basement of Ohio and surrounding Mesoproterozoic (i.e. Grenville-age) rocks. The eastward extension of this similar to 1650 Ma crustal reservoir into Ohio requires a revision of the crustal boundary defined by Nd isotopic data to be located further east, now overlapping with the Grenville front magnetic lineament in Ohio. In fact, the easternmost sample in this study is derived from a more depleted reservoir. This limits the extent of >1.5 Ga basement in subsurface Ohio and constrains the location of the crustal boundary. Further, syn-orogenic magmatism at similar to 1050 Ma suggests a potential extrapolation of the Interior Magmatic Belt into Ohio. Oxygen isotopic data in zircon suggests that during Grenvillian metamorphism, zircon recrystallisation occurred in the presence of heavy delta O-18 fluids resulting in zircon with elevated delta O-18 values. (C) 2015 Elsevier B.V. All rights reserved