Monotone approximation in several variables

AbstractLet Ω denote the unit n-cube, [0, 1]n, and let M be the set of all real valued functions on Ω which are nondecreasing in each variable. If f is a bounded Lebesgue measurable function on Ω and 1 < p < ∞, let ƒp denote the best Lp-approximation to ƒ by elements of M. It is shown that ƒp converges almost everywhere as p decreases to one to a best L1-approximation to f by elements of M. If ƒ is continuous, then ƒp is continuous and converges uniformly as p ↓ 1 to a best L1-approximation to ƒ by elements of M and ƒp converges uniformly as p → ∞ to a best L∞-approximation to ƒ by elements of M

Internal Wave Scattering in Continental Slope Canyons, Part 1: Theory and Development of a Ray Tracing Algorithm

When internal waves interact with topography, such as continental slopes, they can transfer wave energy to local dissipation and diapycnal mixing. Submarine canyons comprise approximately ten percent of global continental slopes, and can enhance the local dissipation of internal wave energy, yet parameterizations of canyon mixing processes are currently missing from large-scale ocean models. As a first step in the development of such parameterizations, we conduct a parameter space study of M2 tidal-frequency, low-mode internal waves interacting with idealized V-shaped canyon topographies. Specifically, we examine the effects of varying the canyon mouth width, shape and slope of the thalweg (line of lowest elevation). This effort is divided into two parts. In the first part, presented here, we extend the theory of 3-dimensional internal wave reflection to a rotated coordinate system aligned with our idealized V-shaped canyons. Based on the updated linear internal wave reflection solution that we derive, we construct a ray tracing algorithm which traces a large number of rays (the discrete analog of a continuous wave) into the canyon region where they can scatter off topography. Although a ray tracing approach has been employed in other studies, we have, for the first time, used ray tracing to calculate changes in wavenumber and ray density which, in turn, can be used to calculate the Froude number (a measure of the likelihood of instability). We show that for canyons of intermediate aspect ratio, large spatial envelopes of instability can form in the presence of supercritical sidewalls. Additionally, the canyon height and length can modulate the Froude number. The second part of this study, a diagnosis of internal wave scattering in continental slope canyons using both numerical simulations and this ray tracing algorithm, as well as a test of robustness of the ray tracing, is presented in the companion article

Internal Wave Scattering in Continental Slope Canyons, Part 2: A Comparison of Ray Tracing and Numerical Simulations

When internal waves interact with topography, such as continental slopes, they can transfer wave energy to local dissipation and diapycnal mixing. Submarine canyons comprise approximately ten percent of global continental slopes, and can enhance the local dissipation of internal wave energy, yet parameterizations of canyon mixing processes are currently missing from large-scale ocean models. As a first step in the development of such parameterizations, we conduct a parameter space study of M2 tidal-frequency, low-mode internal waves interacting with idealized V-shaped canyon topographies. Specifically, we examine the effect of varying the canyon mouth width, shape and slope of the thalweg (line of lowest elevation) (i.e. flat bottom or near-critical slope). In Part 1 of this study (Nazarian and Legg, 2017a), we developed a ray tracing algorithm and used it to estimate how canyons can increase the wave Froude number, by increasing energy density and increasing vertical wavenumber. Here in Part 2 we examine the internal wave scattering in continental slope canyons using numerical simulations, and compare the results with the linear ray tracing predictions. We find that at intermediate canyon widths, a large fraction of incoming wave energy can be dissipated, which can be explained as a consequence of the increase in ray density and, for near-critical slope canyons, increase in vertical wave number, which leads to lower Richardson number followed by instability. Relative to a steep continental slope without a canyon, we find that V-shaped flat bottom canyons always dissipate more energy and are an effective geometry for wave trapping and subsequent energy loss. When both flat bottom canyons and near-critical slope canyons are made narrower, less wave energy enters the canyon, but a larger fraction of that energy is lost to dissipation due to subsequent reflections and wave trapping. There is agreement between the diagnostics calculated from the numerical model and the linear ray tracing, lending support for the use of linear theory to understand the fundamental dynamics of internal wave scattering in canyons

Sea Beam Survey of an Active Strike-Slip Fault: The San Clemente Fault in the California Continental Borderland

The San Clemente fault, located in the California Continental Borderland, is an active, northwest trending, right-lateral, wrench fault. Sea Beam data are used to map the major tectonic landforms associated with active submarine faulting in detail unavailable using conventional echo-sounding or seismic reflection data. In the area between North San Clemente Basin and Fortymile Bank, the major late Cenozoic faults are delineated by alignments of numerous tectonic landforms, including scarps, linear trenches, benches, and sags. Character and spatial patterns of these landforms are consistent with dextral wrench faulting, although vertical offsets may be substantial locally. The main trace of the San Clemente fault cuts a straight path directly across the rugged topography of the region, evidence of a steeply dipping fault surface. Basins or sags located at each right step in the en echelon pattern of faults are manifestations of pull-apart basin development in a right-slip fault zone. Seismic reflection profiles show offset reflectors and a graben in late Quaternary turbidites of the Navy Fan, where the fault zone follows a more northerly trend. Modern tectonic activity along the San Clemente fault zone is demonstrated by numerous earthquakes with epicenters located along the fault\u27s trend. The average strike of the San Clemente fault is parallel to the predicted Pacific-North American relative plate motion vector at this location. Therefore we conclude that the San Clemente fault zone is a part of the broad Pacific-North American transform plate boundary and that the southern California region may be considered as a broad shear zone

The Impact of Topographic Steepness on Tidal Dissipation at Bumpy Topography

Breaking internal waves are an important contributor to mixing in the stratified ocean interior. We use two-dimensional, nonhydrostatic numerical simulations to examine the breaking of internal waves generated by tidal flow over sinusoidal bottom topography. We explore the sensitivity of the internal wave breaking to the topographic steepness and Coriolis frequency, focusing on the vertical structure of kinetic energy dissipation and the ratio of local dissipation to the barotropic-to-baroclinic energy conversion. When the tidal frequency is twice the local Coriolis frequency, wave breaking above the topography is driven by wave–wave interactions which transfer wave energy from the tidal forcing frequency to the inertial frequency. The greater shear associated with the inertial frequency waves leads to enhanced dissipation in a thick layer above the topography. The topographic steepness strongly modulates this dependence of dissipation on Coriolis frequency; for some steep sinusoidal topographies, most wave energy propagates downward into the topographic troughs, eliminating the possibility for significant breaking above the topographic peaks. Current parameterizations of tidal dissipation in use in global ocean models need to be adapted to include the dependence of the local dissipation on both the Coriolis frequency and the topographic steepness

Sensitivity of the ocean state to the vertical distribution of internal-tide-driven mixing

Author Posting. © American Meteorological Society, 2013. This article is posted here by permission of American Meteorological Society for personal use, not for redistribution. The definitive version was published in Journal of Physical Oceanography 43 (2013): 602–615, doi:10.1175/JPO-D-12-055.1.The ocean interior stratification and meridional overturning circulation are largely sustained by diapycnal mixing. The breaking of internal tides is a major source of diapycnal mixing. Many recent climate models parameterize internal-tide breaking using the scheme of St. Laurent et al. While this parameterization dynamically accounts for internal-tide generation, the vertical distribution of the resultant mixing is ad hoc, prescribing energy dissipation to decay exponentially above the ocean bottom with a fixed-length scale. Recently, Polzin formulated a dynamically based parameterization, in which the vertical profile of dissipation decays algebraically with a varying decay scale, accounting for variable stratification using Wentzel–Kramers–Brillouin (WKB) stretching. This study compares two simulations using the St. Laurent and Polzin formulations in the Climate Model, version 2G (CM2G), ocean–ice–atmosphere coupled model, with the same formulation for internal-tide energy input. Focusing mainly on the Pacific Ocean, where the deep low-frequency variability is relatively small, the authors show that the ocean state shows modest but robust and significant sensitivity to the vertical profile of internal-tide-driven mixing. Therefore, not only the energy input to the internal tides matters, but also where in the vertical it is dissipated.This work is a component of the Internal- Wave Driven Mixing Climate Process Team funded by the National Science Foundation Grant OCE-0968721 and the National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Award NA08OAR4320752.2013-09-0

Randomised, double-blind, placebo-controlled trials of non-individualised homeopathic treatment: systematic review and meta-analysis

Background: A rigorous systematic review and meta-analysis focused on randomised controlled trials (RCTs) of non-individualised homeopathic treatment has not previously been reported. We tested the null hypothesis that the main outcome of treatment using a non-individualised (standardised) homeopathic medicine is indistinguishable from that of placebo. An additional aim was to quantify any condition-specific effects of non-individualised homeopathic treatment. Methods: Literature search strategy, data extraction and statistical analysis all followed the methods described in a pre-published protocol. A trial comprised ‘reliable evidence’ if its risk of bias was low or it was unclear in one specified domain of assessment. ‘Effect size’ was reported as standardised mean difference (SMD), with arithmetic transformation for dichotomous data carried out as required; a negative SMD indicated an effect favouring homeopathy. Results: Forty-eight different clinical conditions were represented in 75 eligible RCTs. Forty-nine trials were classed as ‘high risk of bias’ and 23 as ‘uncertain risk of bias’; the remaining three, clinically heterogeneous, trials displayed sufficiently low risk of bias to be designated reliable evidence. Fifty-four trials had extractable data: pooled SMD was –0.33 (95% confidence interval (CI) –0.44, –0.21), which was attenuated to –0.16 (95% CI –0.31, –0.02) after adjustment for publication bias. The three trials with reliable evidence yielded a non-significant pooled SMD: –0.18 (95% CI –0.46, 0.09). There was no single clinical condition for which meta-analysis included reliable evidence. Conclusions: The quality of the body of evidence is low. A meta-analysis of all extractable data leads to rejection of our null hypothesis, but analysis of a small sub-group of reliable evidence does not support that rejection. Reliable evidence is lacking in condition-specific meta-analyses, precluding relevant conclusions. Better designed and more rigorous RCTs are needed in order to develop an evidence base that can decisively provide reliable effect estimates of non-individualised homeopathic treatment

Geoarchaeological Modeling of Late Paleoindian Site Location in the Northwestern Great Lakes Region

Full-time occupation of recently deglaciated landscapes in the northwestern Great Lakes by late Paleoindian groups marks a key milestone in the colonization of the region, yet settlement-subsistence systems of these colonizing populations remains poorly understood. Here we apply geoarchaeological modeling and early Holocene environmental reconstruction to analyze environmental settings of known late Paleoindian sites in Michigan’s Upper Peninsula. Our results reveal significant settlement patterning associated with this early Holocene record, highlighting the spatial correlation between site locations and high ground adjacent to hilly terrain and inland lakes – prime locations for monitoring the movement of large game. The analysis highlights a core area with a high likelihood for undiscovered late Paleoindian sites in the northwest corner of Marquette County and suggests the possibility of a north-south travel corridor into the region from upper Wisconsin along the Michigamme River

Error rate on the director's task is influenced by the need to take another's perspective but not the type of perspective.

Adults are prone to responding erroneously to another's instructions based on what they themselves see and not what the other person sees. Previous studies have indicated that in instruction-following tasks participants make more errors when required to infer another's perspective than when following a rule. These inference-induced errors may occur because the inference process itself is error-prone or because they are a side effect of the inference process. Crucially, if the inference process is error-prone, then higher error rates should be found when the perspective to be inferred is more complex. Here, we found that participants were no more error-prone when they had to judge how an item appeared (Level 2 perspective-taking) than when they had to judge whether an item could or could not be seen (Level 1 perspective-taking). However, participants were more error-prone in the perspective-taking variants of the task than in a version that only required them to follow a rule. These results suggest that having to represent another's perspective induces errors when following their instructions but that error rates are not directly linked to errors in inferring another's perspective