Galerkin FEM for fractional order parabolic equations with initial data in H−s, 0<s≤1H^{-s},~0 < s \le 1

We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that $\Omega\subset \mathbb{R}^d$, $d=1,2,3$ is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in $L_2$- and $H^1$-norms for initial data in $H^{-s}(\Omega),~0\le s \le 1$. We confirm our theoretical findings with a number of numerical tests that include initial data $v$ being a Dirac $\delta$-function supported on a $(d-1)$-dimensional manifold.Comment: 13 pages, 3 figure

High-temperature environments of human evolution in East Africa based on bond ordering in paleosol carbonates

Many important hominid-bearing fossil localities in East Africa are in regions that are extremely hot and dry. Although humans are well adapted to such conditions, it has been inferred that East African environments were cooler or more wooded during the Pliocene and Pleistocene when this region was a central stage of human evolution. Here we show that the Turkana Basin, Kenya—today one of the hottest places on Earth—has been continually hot during the past 4 million years. The distribution of ^(13)C-^(18)O bonds in paleosol carbonates indicates that soil temperatures during periods of carbonate formation were typically above 30 °C and often in excess of 35 °C. Similar soil temperatures are observed today in the Turkana Basin and reflect high air temperatures combined with solar heating of the soil surface. These results are specific to periods of soil carbonate formation, and we suggest that such periods composed a large fraction of integrated time in the Turkana Basin. If correct, this interpretation has implications for human thermophysiology and implies a long-standing human association with marginal environments

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

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

A standardized comparison of commercially available prion decontamination reagents using the Standard Steel-Binding Assay

Prions are comprised principally of aggregates of a misfolded host protein and cause fatal transmissible neurodegenerative disorders of mammals, such as variant Creutzfeldt–Jakob disease in humans and bovine spongiform encephalopathy in cattle. Prions pose significant public health concerns through contamination of blood products and surgical instruments, and can resist conventional hospital sterilization methods. Prion infectivity binds avidly to surgical steel and can efficiently transfer infectivity to a suitable host, and much research has been performed to achieve effective prion decontamination of metal surfaces. Here, we exploit the highly sensitive Standard Steel-Binding Assay (SSBA) to perform a direct comparison of a variety of commercially available decontamination reagents marketed for the removal of prions, alongside conventional sterilization methods. We demonstrate that the efficacy of marketed prion decontamination reagents is highly variable and that the SSBA is able to rapidly evaluate current and future decontamination reagents

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144

Smoothness-Increasing Accuracy-Conserving (SIAC) filtering and quasi interpolation: A unified view

Filtering plays a crucial role in postprocessing and analyzing data in scientific and engineering applications. Various application-specific filtering schemes have been proposed based on particular design criteria. In this paper, we focus on establishing the theoretical connection between quasi-interpolation and a class of kernels (based on B-splines) that are specifically designed for the postprocessing of the discontinuous Galerkin (DG) method called Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. SIAC filtering, as the name suggests, aims to increase the smoothness of the DG approximation while conserving the inherent accuracy of the DG solution (superconvergence). Superconvergence properties of SIAC filtering has been studied in the literature. In this paper, we present the theoretical results that establish the connection between SIAC filtering to long-standing concepts in approximation theory such as quasi-interpolation and polynomial reproduction. This connection bridges the gap between the two related disciplines and provides a decisive advancement in designing new filters and mathematical analysis of their properties. In particular, we derive a closed formulation for convolution of SIAC kernels with polynomials. We also compare and contrast cardinal spline functions as an example of filters designed for image processing applications with SIAC filters of the same order, and study their properties

Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

Discontinuous Galerkin (DG) methods are a popular class of numerical techniques to solve partial differential equations due to their higher order of accuracy. However, the inter-element discontinuity of a DG solution hinders its utility in various applications, including visualization and feature extraction. This shortcoming can be alleviated by postprocessing of DG solutions to increase the inter-element smoothness. A class of postprocessing techniques proposed to increase the inter-element smoothness is SIAC filtering. In addition to increasing the inter-element continuity, SIAC filtering also raises the convergence rate from order k+1k+1 to order 2k+12k+1 . Since the introduction of SIAC filtering for univariate hyperbolic equations by Cockburn et al. (Math Comput 72(242):577–606, 2003), many generalizations of SIAC filtering have been proposed. Recently, the idea of dimensionality reduction through rotation has been the focus of studies in which a univariate SIAC kernel has been used to postprocess a two-dimensional DG solution (Docampo-Sánchez et al. in Multi-dimensional filtering: reducing the dimension through rotation, 2016. arXiv preprint arXiv:1610.02317). However, the scope of theoretical development of multidimensional SIAC filters has never gone beyond the usage of tensor product multidimensional B-splines or the reduction of the filter dimension. In this paper, we define a new SIAC filter called hexagonal SIAC (HSIAC) that uses a nonseparable class of two-dimensional spline functions called hex splines. In addition to relaxing the separability assumption, the proposed HSIAC filter provides more symmetry to its tensor-product counterpart. We prove that the superconvergence property holds for a specific class of structured triangular meshes using HSIAC filtering and provide numerical results to demonstrate and validate our theoretical results