Might children rust? What are the risks of supplemental oxygen in acute illness

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On p-adic lattices and Grassmannians

It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive group G over a field k, carry the geometric structure of an inductive limit of projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for G. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of p-typical Witt vectors, k is a perfect field of characteristic p and G is a reductive group scheme over W(k). The present paper is an attempt to describe which constructions carry over from the function field case to the p-adic case, more precisely to the situation of the p-adic affine Grassmannian for the special linear group G=SL_n. We start with a description of the R-valued points of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R), where R is a perfect k-algebra. In order to obtain a link with geometry we further construct projective k-subvarieties of the multigraded Hilbert scheme which map equivariantly to the p-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the p-adic setting. Further, for any reduced k-algebra R these morphisms induce bijective maps between the sets of R-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the p-adic affine Grassmannian for SL_n.Comment: 36 pages. This is a thorough revision, in the form accepted by Math. Zeitschrift, of the previously published preprint "On p-adic loop groups and Grassmannians

Infinite product expansion of the Fokker-Planck equation with steady-state solution

We present an analytical technique for solving Fokker–Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples

Inference of riverine nitrogen processing from longitudinal and diel variation in dual nitrate isotopes

Longitudinal and diel measurements of dual isotope composition (δ¹⁵N and δ¹⁸O) in nitrate (NO<inf>3</inf>-N) were made in the Ichetucknee River, a large (∼8m³ s ^-1), entirely spring-fed river in North Florida, to determine whether isotopic variation can deconvolve assimilatory and dissimilatory removal. Comparing nitrate concentrations and isotope composition during the day and night we predicted (1) daytime declines in total fractionation due to low assimilatory fractionation and (2) diurnal variation in dual isotope coupling between 1:1 (assimilation) and 2:1 (denitrification). Five daytime longitudinal transects comprising 10 sampling stations showed consistent NO<inf>3</inf>-N removal (25-35% of inputs) and modest fractionation (¹⁵ε <inf>total</inf> between -2 and -6‰, enriching the residual nitrate pool). Lower fractionation (by ∼1‰) during two nighttime transects, suggests higher fractionation due to assimilation than denitrification. Total fractionation was significantly negatively associated with discharge, input [NO<inf>3</inf>-N], N mass removal, and fractional water loss. Despite well-constrained mass balance estimates that denitrification dominated total N removal, isotope coupling was consistently 1:1, both for longitudinal and diel sampling. Hourly samples on two dates at the downstream location showed significant diel variation in concentration ([NO<inf>3</inf>-N] amplitude = 60 to 90 μg N L^-1) and isotope composition (δ¹⁵N amplitude = -0.7‰ to -1.6‰). Total fractionation differed between day and night only on one date but estimated assimilatory fractionation assuming constant denitrification was highly variable and implausibly large (for N, ¹⁵ε = -2 to -25‰), suggesting that fractionation and removal due to denitrification is not diurnally constant. Pronounced counterclockwise hysteresis in the relationship between [NO<inf>3</inf>-N] and δ¹⁵N suggests diel variation in N isotope dynamics. Together, low fractionation, isotope versus concentration hysteresis, and consistent 1:1 isotope coupling suggests that denitrification is controlled by NO <inf>3</inf>^- diffusion into the benthic sediments, the length of which is mediated by riverine oxygen dynamics. While using dual isotope behavior to deconvolve removal pathways was not possible, isotope measurements did yield valuable information about riverine N cycling and transformations. Copyright © 2012 by the American Geophysical Union

Analytical approximation to the multidimensional Fokker--Planck equation with steady state

The Fokker--Planck equation is a key ingredient of many models in physics, and related subjects, and arises in a diverse array of settings. Analytical solutions are limited to special cases, and resorting to numerical simulation is often the only route available; in high dimensions, or for parametric studies, this can become unwieldy. Using asymptotic techniques, that draw upon the known Ornstein--Uhlenbeck (OU) case, we consider a mean-reverting system and obtain its representation as a product of terms, representing short-term, long-term, and medium-term behaviour. A further reduction yields a simple explicit formula, both intuitive in terms of its physical origin and fast to evaluate. We illustrate a breadth of cases, some of which are `far' from the OU model, such as double-well potentials, and even then, perhaps surprisingly, the approximation still gives very good results when compared with numerical simulations. Both one- and two-dimensional examples are considered

The effect of intervertebral cartilage on neutral posture and range of motion in the necks of sauropod dinosaurs

The necks of sauropod dinosaurs were a key factor in their evolution. The habitual posture and range of motion of these necks has been controversial, and computer-aided studies have argued for an obligatory sub-horizontal pose. However, such studies are compromised by their failure to take into account the important role of intervertebral cartilage. This cartilage takes very different forms in different animals. Mammals and crocodilians have intervertebral discs, while birds have synovial joints in their necks. The form and thickness of cartilage varies significantly even among closely related taxa. We cannot yet tell whether the neck joints of sauropods more closely resembled those of birds or mammals. Inspection of CT scans showed cartilage:bone ratios of 4.5% for Sauroposeidon and about 20% and 15% for two juvenile Apatosaurus individuals. In extant animals, this ratio varied from 2.59% for the rhea to 24% for a juvenile giraffe. It is not yet possible to disentangle ontogenetic and taxonomic signals, but mammal cartilage is generally three times as thick as that of birds. Our most detailed work, on a turkey, yielded a cartilage:bone ratio of 4.56%. Articular cartilage also added 11% to the length of the turkey's zygapophyseal facets. Simple image manipulation suggests that incorporating 4.56% of neck cartilage into an intervertebral joint of a turkey raises neutral posture by 15°. If this were also true of sauropods, the true neutral pose of the neck would be much higher than has been depicted. An additional 11% of zygapophyseal facet length translates to 11% more range of motion at each joint. More precise quantitative results must await detailed modelling. In summary, including cartilage in our models of sauropod necks shows that they were longer, more elevated and more flexible than previously recognised