Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^s(\Omega)$ and $\widetilde{H}^s(\Omega)$, for $s\in \mathbb{R}$ and an open $\Omega\subset \mathbb{R}^n$. We exhibit examples in one and two dimensions of sets $\Omega$ for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if $\Omega$ is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large

Concentrating Membrane Proteins Using Asymmetric Traps and AC Electric Fields

Membrane proteins are key components of the plasma membrane and are responsible for control of chemical ionic gradients, metabolite and nutrient transfer, and signal transduction between the interior of cells and the external environment. Of the genes in the human genome, 30% code for membrane proteins (Krogh et al. J. Mol. Biol.2001, 305, 567). Furthermore, many FDA-approved drugs target such proteins (Overington et al. Nat. Rev. Drug Discovery2006, 5, 993). However, the structure-function relationships of these are notably sparse because of difficulties in their purification and handling outside of their membranous environment. Methods that permit the manipulation of membrane components while they are still in the membrane would find widespread application in separation, purification, and eventual structure-function determination of these species (Poo et al. Nature1977, 265, 602). Here we show that asymmetrically patterned supported lipid bilayers in combination with AC electric fields can lead to efficient manipulation of charged components. We demonstrate the concentration and trapping of such components through the use of a “nested trap” and show that this method is capable of yielding an approximately 30-fold increase in the average protein concentration. Upon removal of the field, the material remains trapped for several hours as a result of topographically restricted diffusion. Our results indicate that this method can be used for concentrating and trapping charged membrane components while they are still within their membranous environment. We anticipate that our approach could find widespread application in the manipulation and study of membrane proteins

Independent control of polar and azimuthal anchoring

Monte Carlo simulation, experiment and continuum theory are used to examine the anchoring exhibited by a nematic liquid crystal at a patterned substrate comprising a periodic array of rectangles that, respectively, promote vertical and planar alignment. It is shown that the easy axis and effective anchoring energy promoted by such surfaces can be readily controlled by adjusting the design of the pattern. The calculations reveal rich behavior: for strong anchoring, as exhibited by the simulated system, for rectangle ratios $\geq 2$ the nematic aligns in the direction of the long edge of the rectangles, the azimuthal anchoring coefficient changing with pattern shape. In weak anchoring scenarios, however, including our experimental systems, preferential anchoring is degenerate between the two rectangle diagonals. Bistability between diagonally-aligned and edge-aligned arrangement is predicted for intermediate combinations of anchoring coefficient and system length-scale.Comment: 12 pages, 12 figure

Nematic liquid crystal alignment on chemical patterns

Patterned Self-Assembled Monolayers (SAMs) promoting both homeotropic and planar degenerate alignment of 6CB and 9CB in their nematic phase, were created using microcontact printing of functionalised organothiols on gold films. The effects of a range of different pattern geometries and sizes were investigated, including stripes, circles and checkerboards. EvanescentWave Ellipsometry was used to study the orientation of the liquid crystal (LC) on these patterned surfaces during the isotropic-nematic phase transition. Pretransitional growth of a homeotropic layer was observed on 1 ¹m homeotropic aligning stripes, followed by a homeotropic mono-domain state prior to the bulk phase transition. Accompanying Monte-Carlo simulations of LCs aligned on nano-patterned surfaces were also performed. These simulations also showed the presence of the homeotropic mono-domain state prior to the transition.</p

Some Comments on Multigrid Methods for Computing Propagators

I make three conceptual points regarding multigrid methods for computing propagators in lattice gauge theory: 1) The class of operators handled by the algorithm must be stable under coarsening. 2) Problems related by symmetry should have solution methods related by symmetry. 3) It is crucial to distinguish the vector space $V$ from its dual space $V^*$. All the existing algorithms violate one or more of these principles.Comment: 16 pages, LaTeX plus subeqnarray.sty (included at end), NYU-TH-93/07/0

Effect of cadence on locomotor–respiratory coupling during upper-body exercise

Introduction: Asynchronous arm-cranking performed at high cadences elicits greater cardiorespiratory responses compared to low cadences. This has been attributed to increased postural demand and locomotor–respiratory coupling (LRC), and yet, this has not been empirically tested. This study aimed to assess the effects of cadence on cardiorespiratory responses and LRC during upper-body exercise. Methods: Eight recreationally-active men performed arm-cranking exercise at moderate and severe intensities that were separated by 10 min of rest. At each intensity, participants exercised for 4 min at each of three cadences (50, 70, and 90 rev min−1) in a random order, with 4 min rest-periods applied in-between cadences. Exercise measures included LRC via whole- and half-integer ratios, cardiorespiratory function, perceptions of effort (RPE and dyspnoea), and diaphragm EMG using an oesophageal catheter. Results: The prevalence of LRC during moderate exercise was highest at 70 vs. 50 rev min−1 (27 ± 10 vs. 13 ± 9%, p = 0.000) and during severe exercise at 90 vs. 50 rev min−1 (24 ± 7 vs. 18 ± 5%, p = 0.034), with a shorter inspiratory time and higher mean inspiratory flow (p < 0.05) at higher cadences. During moderate exercise, (Formula presented.) and fC were higher at 90 rev min−1 (p < 0.05) relative to 70 and 50 rev min−1 ((Formula presented.) 1.19 ± 0.25 vs. 1.05 ± 0.21 vs. 0.97 ± 0.24 L min−1; fC 116 ± 11 vs. 101 ± 13 vs. 101 ± 12 b min−1), with concomitantly elevated dyspnoea. There were no discernible cadence-mediated effects on diaphragm EMG. Conclusion: Participants engage in LRC to a greater extent at moderate-high cadences which, in turn, increase respiratory airflow. Cadence rate should be carefully considered when designing aerobic training programmes involving the upper-limbs

Corporatism as a process of working class containment and roll-back: The recent experiences of South Africa and South Korea

In this article we argue that recent debates in the corporatist literature about whether corporatism is best understood as a process of structured interest representation or political dialogue miss the point as to corporatism's central task - the shift of material resources and power away from the working class to the capitalist class, in which two processes are evident - containment and roll-back. We discuss these processes in the context of successive waves of corporatism in Western Europe from the 1940s to the 1990s before moving on to an analysis of the contrasting fortunes of corporatism in South Africa and South Korea during democratic transition. We conclude that the ability of corporatism to carry out the processes of containment and roll back in these two cases have been dependent on the existence (or absence) of supportive prior political relationships between organised labour and the state

Market and Organizational Factors Associated with Teaching Hospital Participation in Strategic Hospital Alliances

This research investigated market and organizational factors that influence the strategic decisions of teaching hospitals to participate in strategic hospital alliances (SHAs). It described the characteristics of both teaching hospitals and the health care environment in which they operate. This research also examines the association of these factors with the strategic position of teaching hospitals in terms of their dominance in the market or within their organizations. The theoretical model used two concepts from institutional theory--coercive and normative pressures. It was argued that coercive pressures in the market facilitated the decision to participate in SHAs and gain market and Organizational dominant positions. Alternatively, normative organizational pressures were argued to hinder the process of participating in SRAs and gaining market and organizational dominance. An important finding of this research was that high levels of SRA penetration had a negative influence on all three dependent variables , SRA participation, market dominance, and organizational dominance. This finding suggests that as market consolidation advances, teaching hospitals may find it difficult to participate in SRAs or gain positions of dominance. In addition to the SRA penetration measure there were a number of other relationships of interest. SRA participation was related to the percent of large employers in the market and the teaching hospital\u27s net revenue . Market dominance was related to the percent of large group practices and the percent of primary care physicians in the market as well as the profit status of the teaching hospital. Organizational dominance was related to the profit status and the administrative structure of the teaching hospital

Protein content prediction in single wheat kernels using hyperspectral imaging

Hyperspectral imaging (HSI) combines Near-infrared (NIR) spectroscopy and digital imaging to give information about the chemical properties of objects and their spatial distribution. Protein content is one of the most important quality factors in wheat. It is known to vary widely depending on the cultivar, agronomic and climatic conditions. However, little information is known about single kernel protein variation within batches. The aim of the present work was to measure the distribution of protein content in whole wheat kernels on a single kernel basis, and to apply HSI to predict this distribution. Wheat samples from 2013 and 2014 harvests were sourced from UK millers and wheat breeders, and individual kernels were analysed by HSI and by the Dumas combustion method for total protein content. HSI was applied in the spectral region 980-2500 nm in reflectance mode using the push-broom approach. Single kernel spectra were used to develop partial least squares (PLS) regression models for protein prediction of intact single grains. The protein content ranged from 6.2 to 19.8% (“as-is” basis), with significantly higher values for hard wheats. The performance of the calibration model was evaluated using the coefficient of determination (R2) and the root mean square error (RMSE) from 3250 samples used for calibration and 868 used for external validation. The calibration performance for single kernel protein content was R2 of 0.82 and 0.79, and RMSE of 0.86 and 0.94% for the calibration and validation dataset, enabling quantification of the protein distribution between kernels and even visualisation within the same kernel. The performance of the single kernel measurement was poorer than that typically obtained for bulk samples, but is acceptable for some specific applications. The use of separate calibrations built by separating hard and soft wheat, or on kernels placed on similar orientation did not greatly improve the prediction ability. We simulated the use of the lower cost InGaAs detector (1000-1700 nm), and reported that the use of proposed HgCdTe detectors over a restricted spectral range gave a lower prediction error (RMSEC=0.86% vs 1.06%, for HgCdTe and InGaAs, respectively), and 26 increased R2 value (Rc2=0.82 vs 0.73)