hp-dGFEM for Second-Order Elliptic Problems in Plyhedra I: Stability and Quasioptimality on Geometric Meshes

We introduce and analyze hp-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems in three-dimensional polyhedral domains. To resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined toward the corresponding neighborhoods. Similarly, the local polynomial degrees are increased linearly and possibly anisotropically away from singularities. We design interior penalty hp-dG methods and prove that they are well-defined for problems with singular solutions and stable under the proposed hp-refinements. We establish (abstract) error bounds that will allow us to prove exponential rates of convergence in the second part of this work.

hp-dGFEM for Second-Order Elliptic Problems in Plyhedra II: Exponential Convergence

The goal of this paper is to establish exponential convergence of $hp$-version interior penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and piecewise analytic data in three-dimensional polyhedral domains. More precisely, we shall analyze the convergence of the $hp$-IP dG methods considered in [D. Schötzau, C. Schwab, T. P. Wihler, SIAM J. Numer. Anal., 51 (2013), pp. 1610--1633] based on axiparallel $\sigma$-geometric anisotropic meshes and $\bm{s}$-linear anisotropic polynomial degree distributions.

Relative Predicativity and dependent recursion in second-order set theory and higher-orders theories

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universe V of sets is treated as the given totality (nor in the contexts of those of n+3-th order number or set theories, where the class of all n+2-th order objects is treated as the given totality).

hp-DGFEM for second-order mixed elliptic problems polyhedra

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. We consider piecewise analytic solutions which belong to a larger analytic class than those for the pure Dirichlet problem considered in [11, 12]. For such solutions, we establish the exponential convergence of a nonconforming dG interpolant given by local L 2 -projections on elements away from corners and edges, and by suitable local low-order quasi-interpolants on elements at corners and edges. Due to the appearance of non-homogeneous, weighted norms in the analytic regularity class, new arguments are introduced to bound the dG consistency errors in elements abutting on Neumann edges. The non-homogeneous norms also entail some crucial modifications of the stability and quasi-optimality proofs, as well as of the analysis for the anisotropic interpolation operators. The exponential convergence bounds for the dG interpolant constructed in this paper generalize the results of [11, 12] for the pure Dirichlet case.

A Research Roadmap for Complementary and Alternative Medicine – What We Need to Know by 2020

Background: The CAMbrella coordination action was funded within the Framework Programme 7. Its aim is to provide a research roadmap for clinical and epidemiological research for complementary and alternative medicine (CAM) that is appropriate for the health needs of European citizens and acceptable to their national research institutes and healthcare providers in both public and private sectors. One major issue in the European research agenda is the demographic change and its impact on health care. Our vision for 2020 is that there is an evidence base that enables European citizens to make informed decisions about CAM, both positive and negative. This roadmap proposes a strategic research agenda for the field of CAM designed to address future European health care challenges. This roadmap is based on the results of CAMbrella’s several work packages, literature reviews and expert discussions including a consensus meeting. Methods: We first conducted a systematic literature review on key issues in clinical and epidemiological research in CAM to identify the general concepts, methods and the strengths and weaknesses of current CAM research. These findings were discussed in a workshop (Castellaro, Italy, September 7–9th 2011) with international CAM experts and strategic and methodological recommendations were defined in order to improve the rigor and relevance of CAM research. These recommendations provide the basis for the research roadmap, which was subsequently discussed in a consensus conference (Järna, Sweden, May 9–11th 2012) with all CAMbrella members and the CAMbrella advisory board. The roadmap was revised after this discussion in CAMbrella Work Package (WP) 7 and finally approved by CAMbrella’s scientific steering committee on September 26th 2012. Results: Our main findings show that CAM is very heterogenous in terms of definitions and legal regulations between the European countries. In addition, citizens’ needs and attitudes towards CAM as well as the use and provision of CAM differ significantly between countries. In terms of research methodology, there was consensus that CAM researchers should make use of all the commonly accepted scientific research methods and employ those with utmost diligence combined in a mixed methods framework. Conclusions: We propose 6 core areas of research that should be investigated to achieve a robust knowledge base and to allow stakeholders to make informed decisions. These are: Research into the prevalence of CAM in Europe: Reviews show that we do not know enough about the circumstances in which CAM is used by Europeans. To enable a common European strategic approach, a clear picture of current use is of the utmost importance. Research into differences regarding citizens’ attitudes and needs towards CAM: Citizens are the driver for CAM utilization. Their needs and views on CAM are a key priority, and their interests must be investigated and addressed in future CAM research. Research into safety of CAM: Safety is a key issue for European citizens. CAM is considered safe, but reliable data is scarce although urgently needed in order to assess the risk and cost-benefit ratio of CAM. Research into the comparative effectiveness of CAM: Everybody needs to know in what situation CAM is a reasonable choice. Therefore, we recommend a clear emphasis on concurrent evaluation of the overall effectiveness of CAM as an additional or alternative treatment strategy in real-world settings. Research into effects of context and meaning: The impact of effects of context and meaning on the outcome of CAM treatments must be investigated; it is likely that they are significant. Research into different models of CAM health care integration: There are different models of CAM being integrated into conventional medicine throughout Europe, each with their respective strengths and limitations. These models should be described and concurrently evaluated; innovative models of CAM provision in health care systems should be one focus for CAM research. We also propose a methodological framework for CAM research. We consider that a framework of mixed methodological approaches is likely to yield the most useful information. In this model, all available research strategies including comparative effectiveness research utilising quantitative and qualitative methods should be considered to enable us to secure the greatest density of knowledge possible. Stakeholders, such as citizens, patients and providers, should be involved in every stage of developing the specific and relevant research questions, study design and the assurance of real-world relevance for the research. Furthermore, structural and sufficient financial support for research into CAM is needed to strengthen CAM research capacity if we wish to understand why it remains so popular within the EU. In order to consider employing CAM as part of the solution to the health care, health creation and self-care challenges we face by 2020, it is vital to obtain a robust picture of CAM use and reliable information about its cost, safety and effectiveness in real-world settings. We need to consider the availability, accessibility and affordability of CAM. We need to engage in research excellence and utilise comparative effectiveness approaches and mixed methods to obtain this data. Our recommendations are both strategic and methodological. They are presented for the consideration of researchers and funders while being designed to answer the important and implicit questions posed by EU citizens currently using CAM in apparently increasing numbers. We propose that the EU actively supports an EUwide strategic approach that facilitates the development of CAM research. This could be achieved in the first instance through funding a European CAM coordinating research office dedicated to foster systematic communication between EU governments, public, charitable and industry funders as well as researchers, citizens and other stakeholders. The aim of this office would be to coordinate research strategy developments and research funding opportunities, as well as to document and disseminate international research activities in this field. With the aim to develop sustainability as second step, a European Centre for CAM should be established that takes over the monitoring and further development of a coordinated research strategy for CAM, as well as it should have funds that can be awarded to foster high quality and robust independent research with a focus on citizens health needs and pan-European collaboration. We wish to establish a solid funding for CAM research to adequately inform health care and health creation decision-making throughout the EU. This centre would ensure that our vision of a common, strategic and scientifically rigorous approach to CAM research becomes our legacy and Europe’s reality. We are confident that our recommendations will serve these essential goals for EU citizens.

Nectar distribution and its relation to food quality in honeybee (Apis mellifera) colonies

In honeybees (Apis niellifera), the process of nectar collection is considered a straightforward example of task partitioning with two subtasks or two intersecting cycles of activity: (1) foraging and (2) storing of nectar, linked via its transfer between foragers and food processors. Many observations suggest, however, that nectar colleclion and processing in honeybees is a complex process, involving workers of other sub-castes and depending on variables such as resource profitability or the amount of stored honey. It has been observed that food processor bees often distribute food to other hive bees after receiving it from incoming foragers, instead of storing it immediately in honey cells. While there is little information about the sub-caste affiliation and the behaviour of these second-order receivers, this stage may be important for the rapid distribution of nutrients and related information. To investigate the identity of these second-order receivers, we quantified behaviours following nectar transfer and compared these behaviours with the behaviour of average worker hive-bees. Furthermore, we tested whether food quality (sugar concentration) affects the behaviour of the second-order receivers. Of all identified second-order receivers, 59.3% performed nurse duties, 18.5% performed food-processor duties and 22.2% performed forager duties. After food intake, these bees were more active, had more trophallaxes (especially offering contacts) compared to average workers and they were found mainly in the brood area, independent of food quality. Our results show that the liquid food can be distributed rapidly among many bees of the three main worker sub-castes, without being stored in honey cells first. Furthermore, the results suggest that the rapid distribution of food partly depends on the high activity of second-order receivers.

Hermitian curve interpolation for CT-data re-sampling

Purpose: Development of an interpolation algorithm for re‐sampling spatially distributed CT‐data with the following features: global and local integral conservation, avoidance of negative interpolation values for positively defined datasets and the ability to control re‐sampling artifacts. Method and Materials: The interpolation can be separated into two steps: first, the discrete CT‐data has to be continuously distributed by an analytic function considering the boundary conditions. Generally, this function is determined by piecewise interpolation. Instead of using linear or high order polynomialinterpolations, which do not fulfill all the above mentioned features, a special form of Hermitian curve interpolation is used to solve the interpolation problem with respect to the required boundary conditions. A single parameter is determined, by which the behavior of the interpolation function is controlled. Second, the interpolated data have to be re‐distributed with respect to the requested grid. Results: The new algorithm was compared with commonly used interpolation functions based on linear and second order polynomial. It is demonstrated that these interpolation functions may over‐ or underestimate the source data by about 10%–20% while the parameter of the new algorithm can be adjusted in order to significantly reduce these interpolation errors. Finally, the performance and accuracy of the algorithm was tested by re‐gridding a series of X‐ray CT‐images. Conclusion: Inaccurate sampling values may occur due to the lack of integral conservation. Re‐sampling algorithms using high order polynomialinterpolation functions may result in significant artifacts of the re‐sampled data. Such artifacts can be avoided by using the new algorithm based on Hermitian curve interpolation

Information processing in long delay memory-guided saccades: further insights from TMS

The performance of memory-guided saccades with two different delays (3 s and 30 s of memorisation) was studied in eight subjects. Single pulse transcranial magnetic stimulation (TMS) was applied simultaneously over the left and right dorsolateral prefrontal cortex (DLPFC) 1 s after target presentation. In both delays, stimulation significantly increased the percentage of error in amplitude of memory-guided saccades. Furthermore, the interfering effect of TMS was significantly higher in the short delay compared to that of the long delay paradigm. The results are discussed in the context of a mixed model of spatial working memory control including two components: First, serial information processing with a predominant role of the DLPFC during the early period of memorisation and, second, parallel information processing, which is independent from the DLPFC, operating during longer delays.