Computational science and re-discovery: open-source implementations of ellipsoidal harmonics for problems in potential theory

We present two open-source (BSD) implementations of ellipsoidal harmonic expansions for solving problems of potential theory using separation of variables. Ellipsoidal harmonics are used surprisingly infrequently, considering their substantial value for problems ranging in scale from molecules to the entire solar system. In this article, we suggest two possible reasons for the paucity relative to spherical harmonics. The first is essentially historical---ellipsoidal harmonics developed during the late 19th century and early 20th, when it was found that only the lowest-order harmonics are expressible in closed form. Each higher-order term requires the solution of an eigenvalue problem, and tedious manual computation seems to have discouraged applications and theoretical studies. The second explanation is practical: even with modern computers and accurate eigenvalue algorithms, expansions in ellipsoidal harmonics are significantly more challenging to compute than those in Cartesian or spherical coordinates. The present implementations reduce the "barrier to entry" by providing an easy and free way for the community to begin using ellipsoidal harmonics in actual research. We demonstrate our implementation using the specific and physiologically crucial problem of how charged proteins interact with their environment, and ask: what other analytical tools await re-discovery in an era of inexpensive computation?Comment: 25 pages, 3 figure

Sommerfeld's image method in the calculation of van der Waals forces

We show how the image method can be used together with a recent method developed by C. Eberlein and R. Zietal to obtain the dispersive van der Waals interaction between an atom and a perfectly conducting surface of arbitrary shape. We discuss in detail the case of an atom and a semi- infinite conducting plane. In order to employ the above procedure to this problem it is necessary to use the ingenious image method introduced by Sommerfeld more than one century ago, which is a generalization of the standard procedure. Finally, we briefly discuss other interesting situations that can also be treated by the joint use of Sommerfeld's image technique and Eberlein-Zietal method.Comment: To appear in the proceedings of Conference on Quantum Field Theory under the Influence of External Conditions (QFEXT11

Alternative Fourier Expansions for Inverse Square Law Forces

Few-body problems involving Coulomb or gravitational interactions between pairs of particles, whether in classical or quantum physics, are generally handled through a standard multipole expansion of the two-body potentials. We discuss an alternative based on a compact, cylindrical Green's function expansion that should have wide applicability throughout physics. Two-electron "direct" and "exchange" integrals in many-electron quantum systems are evaluated to illustrate the procedure which is more compact than the standard one using Wigner coefficients and Slater integrals.Comment: 10 pages, latex/Revtex4, 1 figure

Teacher fabrication as an impediment to professional learning and development: the external mentor antidote

This paper reports findings from a study of the work of 'external mentors' associated with three programmes of support for the professional learning and development (PLD) of secondary science teachers in England. Focusing on outcomes from analyses of data derived from interviews with 47 mentees and 19 mentors, the paper supports and extends existing research on the construction and maintenance of fabrications in schools, and identifies omissions in the evidence base relating to teacher PLD. It is argued that the kinds of fabrications revealed by the teachers interviewed for this research present a serious impediment to their opportunities for school-based PLD, and that the deployment of external mentors (i.e. those not based in the same schools as the teachers they support) can provide a potentially powerful antidote to this. A number of implications for policy and practice in teacher professional learning and development are discussed. Amongst these, it is argued that more teachers should have the opportunity to access external support for their PLD, and that policy makers and head teachers should seek to reduce the degree to which teachers' 'performance' is observed, inspected and assessed

Weyssenhoff fluid dynamics in general relativity using a 1+3 covariant approach

The Weyssenhoff fluid is a perfect fluid with spin where the spin of the matter fields is the source of torsion in an Einstein-Cartan framework. Obukhov and Korotky showed that this fluid can be described as an effective fluid with spin in general relativity. A dynamical analysis of such a fluid is performed in a gauge invariant manner using the 1+3 covariant approach. This yields the propagation and constraint equations for the set of dynamical variables. A verification of these equations is performed for the special case of irrotational flow with zero peculiar acceleration by evolving the constraints.Comment: 20 page

In-service Initial Teacher Education in the Learning and Skills Sector in England: Integrating Course and Workplace Learning

The aim of the paper is to advance understanding of in-service learning and skills sector trainee teachers’ learning and propose ways of improving their learning. A conceptual framework is developed by extending Billett’s (International Journal of Educational Research 47:232–240, 2008) conceptualisation of workplace learning, as a relationally interdependent process between the opportunities workplaces afford for activities and interactions and how individuals engage with these, to a third base of participation, the affordances of the initial teacher education course. Hager and Hodkinson’s (British Educational Research Journal 35:619–638, 2009) metaphor of ‘learning as becoming’ is used to conceptualise the ways trainees reconstruct learning in a continuous transactional process of boundary crossing between course and workplace. The findings of six longitudinal case studies of trainees’ development, and evidence from other studies, illustrate the complex interrelationships between LSS workplace affordances, course affordances and trainee characteristics and the ways in which trainees reconstruct learning in each setting. The experience of teaching and interacting with learners, interactions with colleagues, and access to workplace resources and training are important workplace affordances for learning. However, some trainees have limited access to these affordances. Teaching observations, course activities and experiences as a learner are significant course affordances. Trainees’ beliefs, prior experiences and dispositions vary and significantly influence their engagement with course and workplace affordances. It is proposed that better integration of course and workplace learning through guided participation in an intentional workplace curriculum and attention to the ways trainees choose to engage with this, together with the use of practical theorising has the potential to improve trainee learning

Non-parametric comparison of histogrammed two-dimensional data distributions using the Energy Test

When monitoring complex experiments, comparison is often made between regularly acquired histograms of data and reference histograms which represent the ideal state of the equipment. With the larger HEP experiments now ramping up, there is a need for automation of this task since the volume of comparisons could overwhelm human operators. However, the two-dimensional histogram comparison tools available in ROOT have been noted in the past to exhibit shortcomings. We discuss a newer comparison test for two-dimensional histograms, based on the Energy Test of Aslan and Zech, which provides more conclusive discrimination between histograms of data coming from different distributions than methods provided in a recent ROOT release.The Science and Technology Facilities Council, U

The Quantum-Classical Correspondence in Polygonal Billiards

We show that wave functions in planar rational polygonal billiards (all angles rationally related to Pi) can be expanded in a basis of quasi-stationary and spatially regular states. Unlike the energy eigenstates, these states are directly related to the classical invariant surfaces in the semiclassical limit. This is illustrated for the barrier billiard. We expect that these states are also present in integrable billiards with point scatterers or magnetic flux lines.Comment: 8 pages, 9 figures (in reduced quality), to appear in PR

Raman effect in AlGaAs waveguides for subpicosecond pulses

The Raman effect in semiconductor waveguides below half‐gap is studied both experimentally and numerically. We report the depolarized Raman gain spectra up to 300 cm−1 in Al0.24Ga0.76As at pump wavelengths of 0.515 and 1.55 μm from the measurement of the absolute Raman scattering cross sections using GaAs as a reference scatterer. In addition, the coupled propagation equations for the AlGaAs waveguides are modified to include the Raman effect. By solving the coupled propagation equations numerically, we verify that the energy transfer between two orthogonally polarized pulses demonstrated in previous pump‐probe experiments [M. N. Islam et al., J. Appl. Phys. 71, 1927 (1992)] is caused by Raman effect. We also show numerically that the Raman effect induces spectral distortions on the pulses, and the energy transfer is inversely proportional to the pulse widths. The energy transfer results in a severe cross‐talk problem for sub‐picosecond pulses in AlGaAs waveguides. For example, the energy exchange is about 30% for 300 fs pulses under π phase shift conditions. Therefore, the Raman effect limits the performance of semiconductor waveguides in optical switching applications for sub‐picosecond pulses. © 1995 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71265/2/JAPIAU-78-4-2198-1.pd

A well-separated pairs decomposition algorithm for k-d trees implemented on multi-core architectures

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Variations of k-d trees represent a fundamental data structure used in Computational Geometry with numerous applications in science. For example particle track tting in the software of the LHC experiments, and in simulations of N-body systems in the study of dynamics of interacting galaxies, particle beam physics, and molecular dynamics in biochemistry. The many-body tree methods devised by Barnes and Hutt in the 1980s and the Fast Multipole Method introduced in 1987 by Greengard and Rokhlin use variants of k-d trees to reduce the computation time upper bounds to O(n log n) and even O(n) from O(n2). We present an algorithm that uses the principle of well-separated pairs decomposition to always produce compressed trees in O(n log n) work. We present and evaluate parallel implementations for the algorithm that can take advantage of multi-core architectures.The Science and Technology Facilities Council, UK