Regularity for eigenfunctions of Schr\"odinger operators

We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator. More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0. Our result extends to the case when b_j and c_{ij} are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy

On the analogy between streamlined magnetic and solid obstacles

Analogies are elaborated in the qualitative description of two systems: the magnetohydrodynamic (MHD) flow moving through a region where an external local magnetic field (magnetic obstacle) is applied, and the ordinary hydrodynamic flow around a solid obstacle. The former problem is of interest both practically and theoretically, and the latter one is a classical problem being well understood in ordinary hydrodynamics. The first analogy is the formation in the MHD flow of an impenetrable region -- core of the magnetic obstacle -- as the interaction parameter $N$, i.e. strength of the applied magnetic field, increases significantly. The core of the magnetic obstacle is streamlined both by the upstream flow and by the induced cross stream electric currents, like a foreign insulated insertion placed inside the ordinary hydrodynamic flow. In the core, closed streamlines of the mass flow resemble contour lines of electric potential, while closed streamlines of the electric current resemble contour lines of pressure. The second analogy is the breaking away of attached vortices from the recirculation pattern produced by the magnetic obstacle when the Reynolds number $Re$, i.e. velocity of the upstream flow, is larger than a critical value. This breaking away of vortices from the magnetic obstacle is similar to that occurring past a real solid obstacle. Depending on the inlet and/or initial conditions, the observed vortex shedding can be either symmetric or asymmetric.Comment: minor changes, accepted for PoF, 26 pages, 7 figure

Tips for implementing multigrid methods on domains containing holes

As part of our development of a computer code to perform 3D `constrained evolution' of Einstein's equations in 3+1 form, we discuss issues regarding the efficient solution of elliptic equations on domains containing holes (i.e., excised regions), via the multigrid method. We consider as a test case the Poisson equation with a nonlinear term added, as a means of illustrating the principles involved, and move to a "real world" 3-dimensional problem which is the solution of the conformally flat Hamiltonian constraint with Dirichlet and Robin boundary conditions. Using our vertex-centered multigrid code, we demonstrate globally second-order-accurate solutions of elliptic equations over domains containing holes, in two and three spatial dimensions. Keys to the success of this method are the choice of the restriction operator near the holes and definition of the location of the inner boundary. In some cases (e.g. two holes in two dimensions), more and more smoothing may be required as the mesh spacing decreases to zero; however for the resolutions currently of interest to many numerical relativists, it is feasible to maintain second order convergence by concentrating smoothing (spatially) where it is needed most. This paper, and our publicly available source code, are intended to serve as semi-pedagogical guides for those who may wish to implement similar schemes.Comment: 18 pages, 11 figures, LaTeX. Added clarifications and references re. scope of paper, mathematical foundations, relevance of work. Accepted for publication in Classical & Quantum Gravit

Pharmacological Alterations of Anxious Behaviour in Mice Depending on Both Strain and the Behavioural Situation

A previous study comparing non-emotive mice from the strain C57BL/6/ByJ with ABP/Le mice showed ABP/Le to be more anxious in an open-field situation. In the present study, several compounds affecting anxiety were assayed on ABP/Le and C57BL/6/ByJ mice using three behavioural models of anxiety: the elevated plus-maze, the light-dark discrimination test and the free exploratory paradigm. The compounds used were the full benzodiazepine receptor agonist, chlordiazepoxide, and the antagonist, flumazenil, the GABAA antagonist, bicuculline, the full 5-HT1A agonist 8-OH-DPAT, and the mixed 5-HT1A/5-HT1B agonist, RU 24969. Results showed the effect of the compounds to be dependent on both the strain and the behavioural task. Several compounds found to be anxiolytic in ABP/Le mice had an anxiogenic effect on C57BL/6/ByJ mice. More behavioural changes were observed for ABP/Le in the elevated plus-maze, but the clearest findings for C57BL/6/ByJ mice were observed in the light-dark discrimination apparatus. These data demonstrate that anxious behaviour is a complex phenomenon which cannot be described by a single behavioural task nor by the action of a single compound

Transcultural body spaces: re-inventing and performing headwrap practice among young Congolese women in London

This article examines embodied representation of race, ethnicity, and gender, questioning ideas of cultural appropriation. Using the London-based Congolese transnational fashion brand Kiyana Wraps as a case study, the article addresses how young Congolese designers re-invent their cultural heritage to conceive the label stylisation and construct meanings of Blackness/Africanness. The article also explores the brand’s social spaces, where the headwrap ritual is used by different actors to perform hybrid identities. In addition, wearing the headwrap reveals symbolic metaphors of empowerment, through which intertwined ‘feminist’ and ‘feminine’ identities are evoked. The paper examines how Congolese women are creatively taking inspiration from the environment of London to produce innovative fashion trajectories as lived socio-cultural experiences. It argues how the headwrap ritual signifies an aesthetic and material process through which specific racial and ethnic boundaries are transcended, fabricating transcultural body spaces which encompass individuals with diverse cultural backgrounds

An introduction to the Australian and New Zealand flux tower network - OzFlux

Published: 31 October 2016OzFlux is the regional Australian and New Zealand flux tower network that aims to provide a continental-scale national research facility to monitor and assess trends, and improve predictions, of Australia's terrestrial biosphere and climate. This paper describes the evolution, design, and current status of OzFlux as well as provides an overview of data processing. We analyse measurements from all sites within the Australian portion of the OzFlux network and two sites from New Zealand. The response of the Australian biomes to climate was largely consistent with global studies except that Australian systems had a lower ecosystem water-use efficiency. Australian semi-arid/arid ecosystems are important because of their huge extent (70 %) and they have evolved with common moisture limitations. We also found that Australian ecosystems had a similar radiation-use efficiency per unit leaf area compared to global values that indicates a convergence toward a similar biochemical efficiency. The two New Zealand sites represented extremes in productivity for a moist temperate climate zone, with the grazed dairy farm site having the highest GPP of any OzFlux site (2620 gC m⁻² yr⁻¹) and the natural raised peat bog site having a very low GPP (820 gC m⁻² yr⁻¹). The paper discusses the utility of the flux data and the synergies between flux, remote sensing, and modelling. Lastly, the paper looks ahead at the future direction of the network and concludes that there has been a substantial contribution by OzFlux, and considerable opportunities remain to further advance our understanding of ecosystem response to disturbances, including drought, fire, land-use and land-cover change, land management, and climate change, which are relevant both nationally and internationally. It is suggested that a synergistic approach is required to address all of the spatial, ecological, human, and cultural challenges of managing the delicately balanced ecosystems in Australasia.Jason Beringer ... Wayne Meyer ... et al

Adaptive fast multipole methods on the GPU

We present a highly general implementation of fast multipole methods on graphics processing units (GPUs). Our two-dimensional double precision code features an asymmetric type of adaptive space discretization leading to a particularly elegant and flexible implementation. All steps of the multipole algorithm are efficiently performed on the GPU, including the initial phase which assembles the topological information of the input data. Through careful timing experiments we investigate the effects of the various peculiarities of the GPU architecture.Comment: Software available at http://user.it.uu.se/~stefane/freeware.htm

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Scalability of Incompressible Flow Computations on Multi-GPU Clusters Using Dual-Level and Tri-Level Parallelism

High performance computing using graphics processing units (GPUs) is gaining popularity in the scientific computing field, with many large compute clusters being augmented with multiple GPUs in each node. We investigate hybrid tri-level (MPI-OpenMP-CUDA) parallel implementations to explore the efficiency and scalability of incompressible flow computations on GPU clusters up to 128 GPUS. This work details some of the unique issues faced when merging fine-grain parallelism on the GPU using CUDA with coarse-grain parallelism using OpenMP for intra-node and MPI for inter-node communication. Comparisons between the tri-level MPI-OpenMP-CUDA and dual-level MPI-CUDA implementations are shown using computationally large computational fluid dynamics (CFD) simulations. Our results demonstrate that a tri-level parallel implementation does not provide a significant advantage in performance over the dual-level implementation, however further research is needed to justify our conclusion for a cluster with a high GPU per node density or when using software that can utilize OpenMP’s fine-grain parallelism more effectively