Lie algebra and invariant tensor technology for g2

Proceeding in analogy with su(n) work on lambda matrices and f- and d-tensors, this paper develops the technology of the Lie algebra g2, its seven dimensional defining representation gamma and the full set of invariant tensors that arise in relation thereto. A comprehensive listing of identities involving these tensors is given. This includes identities that depend on use of characteristic equations, especially for gamma, and a good body of results involving the quadratic, sextic and (the non-primitivity of) other Casimir operators of g2.Comment: 29 pages, LaTe

Investigations of Optical Coherence Properties in an Erbium-doped Silicate Fiber for Quantum State Storage

We studied optical coherence properties of the 1.53 $\mu$m telecommunication transition in an Er$^{3+}$-doped silicate optical fiber through spectral holeburning and photon echoes. We find decoherence times of up to 3.8 $\mu$s at a magnetic field of 2.2 Tesla and a temperature of 150 mK. A strong magnetic-field dependent optical dephasing was observed and is believed to arise from an interaction between the electronic Er$^{3+}$ spin and the magnetic moment of tunneling modes in the glass. Furthermore, we observed fine-structure in the Erbium holeburning spectrum originating from superhyperfine interaction with $^{27}$Al host nuclei. Our results show that Er$^{3+}$-doped silicate fibers are promising material candidates for quantum state storage

Simulation and measurement of hts josephson heterodyne oscillator

We report continuing investigations into practical applications of the ac Josephson effect as the basis for a voltage-tunable radio-frequency oscillator. We have previously demonstrated experimentally that useful power levels (10 s of nW) and linewidths of a few kHz can be achieved in the heterodyne output from a High-Temperature-Superconducting Resistive SQUID (HTS-RSQUID) operating in the frequency range 1-50 MHz. Those results were achieved with 2-junction R-SQUIDs incorporating current-biased shunt resistors of a few micro-ohms. We have now modified the fabrication procedures, and adjusted the shunt resistors and bias current values so that higher frequencies can be achieved. The Josephson junctions are of step-edge type, rather than the bi-crystal type used in our earlier work. The step-edge technique permits much more flexibility in the geometrical lay-out and utilizes the more cost-effective single-crystal MgO substrates. In the present paper, we report numerical simulations and experimental measurements on these devices in the frequency range up to 2 GHz

Quantitative Predictive Modelling Approaches to Understanding Rheumatoid Arthritis:A Brief Review

Rheumatoid arthritis is a chronic autoimmune disease that is a major public health challenge. The disease is characterised by inflammation of synovial joints and cartilage erosion, which lead to chronic pain, poor life quality and, in some cases, mortality. Understanding the biological mechanisms behind the progression of the disease, as well as developing new methods for quantitative predictions of disease progression in the presence/absence of various therapies is important for the success of therapeutic approaches. The aim of this study is to review various quantitative predictive modelling approaches for understanding rheumatoid arthritis. To this end, we start by briefly discussing the biology of this disease and some current treatment approaches, as well as emphasising some of the open problems in the field. Then, we review various mathematical mechanistic models derived to address some of these open problems. We discuss models that investigate the biological mechanisms behind the progression of the disease, as well as pharmacokinetic and pharmacodynamic models for various drug therapies. Furthermore, we highlight models aimed at optimising the costs of the treatments while taking into consideration the evolution of the disease and potential complications.Publisher PDFPeer reviewe

Cultivating Talent through a Principal Pipeline

This report, the second in a series, describes early results of Wallace's Principal Pipeline Initiative, a multi-year effort to improve school leadership in six urban school districts. The report describes changes in the six districts' practices to recruit, train and support new principals. It also offers early lessons for other districts considering changes to their own principal pipelines

Districts Taking Charge of the Principal Pipeline

Six urban school districts received support from The Wallace Foundation to address the critical challenge of supplying schools with effective principals. The experiences of these districts may point the way to steps other districts might take toward this same goal. Since 2011, the districts have participated in the Principal Pipeline Initiative, which set forth a comprehensive strategy for strengthening school leadership in four interrelated domains of district policy and practice:Leader standards to which sites align job descriptions, preparation, selection, evaluation, and support.Preservice preparation that includes selective admissions to high-quality programs.Selective hiring, and placement based on a match between the candidate and the school.On-the-job evaluation and support addressing the capacity to improve teaching and learning, with support focused on needs identified by evaluation.The initiative also brought the expectation that district policies and practices related to school leaders would build the district's capacity to advance its educational priorities. The evaluation of the Principal Pipeline Initiative has a dual purpose: to analyze the processes of implementing the required components in the participating districts from 2011 through 2015; and then to assess the results achieved in schools led by principals whose experiences in standards-based preparation, hiring, evaluation, and support have been consistent with the initiative's requirements. This report addresses implementation of all components of the initiative as of 2014, viewing implementation in the context of districts' aims, constraints, and capacity

Regional and seasonal patterns of epipelagic fish assemblages from the central California Current

The coastal Pacific Ocean off northern and central California encompasses the strongest seasonal upwelling zone in the California Current ecosystem. Headlands and bays here generate complex circulation features and confer unusual oceanographic complexity. We sampled the coastal epipelagic fish community of this region with a surface trawl in the summer and fall of 2000–05 to assess patterns of spatial and temporal community structure. Fifty-three species of fish were captured in 218 hauls at 34 fixed stations, with clupeiform species dominating. To examine spatial patterns, samples were grouped by location relative to a prominent headland at Point Reyes and the resulting two regions, north coast and Gulf of the Farallones, were plotted by using nonmetric multidimensional scaling. Seasonal and interannual patterns were also examined, and representative species were identified for each distinct community. Seven oceanographic variables measured concurrently with trawling were plotted by principal components analysis and tested for correlation with biotic patterns. We found significant differences in community structure by region, year, and season, but no interaction among main effects. Significant differences in oceanographic conditions mirrored the biotic patterns, and a match between biotic and hydrographic structure was detected. Dissimilarity between assemblages was mostly the result of differences in abundance and frequency of occurrence of about twelve common species. Community patterns were best described by a subset of hydrographic variables, including water depth, distance from shore, and any one of several correlated variables associated with upwelling intensity. Rather than discrete communities with clear borders and distinct member species, we found gradients in community structure and identified stations with similar fish communities by region and by proximity to features such as the San Francisco Bay

Quasideterminant solutions of noncommutative integrable systems

Quasideterminants are a relatively new addition to the field of integrable systems. Their simple structure disguises a wealth of interesting and useful properties, enabling solutions of noncommutative integrable equations to be expressed in a straightforward and aesthetically pleasing manner. This thesis investigates the derivation and quasideterminant solutions of two noncommutative integrable equations - the Davey-Stewartson (DS) and Sasa-Satsuma nonlinear Schrodinger (SSNLS) equations. Chapter 1 provides a brief overview of the various concepts to which we will refer during the course of the thesis. We begin by explaining the notion of an integrable system, although no concrete definition has ever been explicitly stated. We then move on to discuss Lax pairs, and also introduce the Hirota bilinear form of an integrable equation, looking at the Kadomtsev-Petviashvili (KP) equation as an example. Wronskian and Grammian determinants will play an important role in later chapters, albeit in a noncommutative setting, and, as such, we give an account of their widespread use in integrable systems. Chapter 2 provides further background information, now focusing on noncommutativity. We explain how noncommutativity can be defined and implemented, both specifically using a star product formalism, and also in a more general manner. It is this general definition to which we will allude in the remainder of the thesis. We then give the definition of a quasideterminant, introduced by Gel'fand and Retakh in 1991, and provide some examples and properties of these noncommutative determinantal analogues. We also explain how to calculate the derivative of a quasideterminant. The chapter concludes by outlining the motivation for studying our particular choice of noncommutative integrable equations and their quasideterminant solutions. We begin with the DS equations in Chapter 3, and derive a noncommutative version of this integrable system using a Lax pair approach. Quasideterminant solutions arise in a natural way by the implementation of Darboux and binary Darboux transformations, and, after describing these transformations in detail, we obtain two types of quasideterminant solution to our system of noncommutative DS equations - a quasi-Wronskian solution from the application of the ordinary Darboux transformation, and a quasi-Grammian solution by applying the binary transformation. After verification of these solutions, in Chapter 4 we select the quasi-Grammian solution to allow us to determine a particular class of solution to our noncommutative DS equations. These solutions, termed dromions, are lump-like objects decaying exponentially in all directions, and are found at the intersection of two perpendicular plane waves. We extend earlier work of Gilson and Nimmo by obtaining plots of these dromion solutions in a noncommutative setting. The work on the noncommutative DS equations and their dromion solutions constitutes our paper published in 2009. Chapter 5 describes how the well-known Darboux and binary Darboux transformations in (2+1)-dimensions discussed in the previous chapter can be dimensionally-reduced to enable their application to (1+1)-dimensional integrable equations. This reduction was discussed briefly by Gilson, Nimmo and Ohta in reference to the self-dual Yang-Mills (SDYM) equations, however we explain these results in more detail, using a reduction from the DS to the nonlinear Schrodinger (NLS) equation as a specific example. Results stated here are utilised in Chapter 6, where we consider higher-order NLS equations in (1+1)-dimension. We choose to focus on one particular equation, the SSNLS equation, and, after deriving a noncommutative version of this equation in a similar manner to the derivation of our noncommutative DS system in Chapter 3, we apply the dimensionally-reduced Darboux transformation to the noncommutative SSNLS equation. We see that this ordinary Darboux transformation does not preserve the properties of the equation and its Lax pair, and we must therefore look to the dimensionally-reduced binary Darboux transformation to obtain a quasi-Grammian solution. After calculating some essential conditions on various terms appearing in our solution, we are then able to determine and obtain plots of soliton solutions in a noncommutative setting. Chapter 7 seeks to bring together the various results obtained in earlier chapters, and also discusses some open questions arising from our work

Six Districts Begin the Principal Pipeline Initiative

This first report of an ongoing evaluation of The Wallace Foundation's Principal Pipeline Initiative describes the six participating school districts' plans and activities during the first year of their grants. The evaluation, conducted by Policy Studies Associates and the RAND Corporation, isintended to inform policy makers and practitioners about the process of carrying out new policies and practices for school leadership and about the results of investments in the Principal Pipeline Initiative. This report is based on collection and analysis of qualitative data, including the districts' proposals, work plans, and progress reports and semi-structured interviews in spring 2012 with 91 administrators employed by districts and their partner institutions. Leaders in all districts report wanting to enlarge their pools of strong applicants for principal positions and to identify and cultivate leadership talent as early as possible in educators' careers.Districts are actively working on allrequired pipeline components: (1) with stakeholder participation, they have developed standards and identified competencies for principals, which they plan to use to guide principal training, hiring, evaluation, and support; (2) they are initiating or strengthening partnerships with university training programs; (3) for hiring, they have standard performance tasks and are developing systems to capture data on candidates' experience; (4) they have diagnostic evaluation tools and are working to build the capacity of principals' supervisors and mentors to support principals' skill development. In addition, all are also bolstering district-run training programs for graduates of university training programs who aspire to become principals