Multiresolution and wavelets

Wavelets may possibly provide a useful basis for the numerical solution of differential or integral equations, in particular when they model processes which operate on very different space- or time-scales (turbulence, boundary layers, shocks, for instance). Our aim in this report is to provide a self-contained introduction to the basic theory of multiresolution and wavelets, accessible to the numerical analyst who knows nothing of signal processing, quantum theory, or any of the other established fields of application

Multiple knot B-spline representation of incompressible flow

An efficient B-spline method for the construction of a piecewise polynomial velocity representation from a given set of normal fluid fluxes is described for steady incompressible flow in three dimensional rectangular regions. The fluxes should be defined across the face-centres of a cartesian tensor product mesh. The proposed spline representation interpolates the given fluxes exactly and also enables the normal fluid velocity to be set identically to zero across or around the surfaces of an arbitrary number of rectangular regions lying in specified planes

Numerical Study of the Spin-Flop Transition in Anisotropic Spin-1/2 Antiferromagnets

Magnetization processes of the spin-1/2 antiferromagnetic $XXZ$ model in two and three spatial dimensions are studied using quantum Monte Carlo method based on stochastic series expansions. Recently developed operator-loop algorithm enables us to show a clear evidence of the first-order phase transition in the presence of an external magnetic field. Phase diagrams of closely related systems, hard core bosons with nearest-neighbor repulsions, are also discussed focusing on possibilities of phase-separated and supersolid phases.Comment: 4 pages, Revtex version 4, with 4 figures embedded, To appear in Phys. Rev.

NMR relaxation rates for the spin-1/2 Heisenberg chain

The spin-lattice relaxation rate $1/T_1$ and the spin echo decay rate $1/T_{2G}$ for the spin-$1\over 2$ antiferromagnetic Heisenberg chain are calculated using quantum Monte Carlo and maximum entropy analytic continuation. The results are compared with recent analytical calculations by Sachdev. If the nuclear hyperfine form factor $A_q$ is strongly peaked around $q=\pi$ the predicted low-temperature behavior [$1/T_1 \sim \ln{^{1/2}(1/T)}$, $1/T_{2G} \sim \ln{^{1/2}(1/T)}/\sqrt{T}$] extends up to temperatures as high as $T/J \approx 0.5$. If $A_q$ has significant weight for $q \approx 0$ there are large contributions from diffusive long-wavelength processes not taken into account in the theory, and very low temperatures are needed in order to observe the asymptotic $T \to 0$ forms.Comment: 9 pages, Revtex 3.0, 5 uuencoded ps figures To appear in Phys. Rev. B, Rapid Com

Confirmatory factor analysis of Clinical Outocmes in Routine Evaluation (CORE-OM) used as a measure of emotional distress in people with tinnitus

BACKGROUND: People with troublesome tinnitus often experience emotional distress. Therefore a psychometrically sound instrument which can evaluate levels of distress and change over time is necessary to understand this experience. Clinical Outcomes in Routine Evaluation (CORE-OM) is a measure of emotional distress which has been widely used in mental health research. Although originally designed as a 4-factor questionnaire, factor analyses have not supported this structure and a number of alternative factor structures have been proposed in different samples. The aims of this study were to test the factor structure of the CORE-OM using a large representative tinnitus sample and to use it to investigate levels of emotional distress amongst people with a range of tinnitus experience. METHODS: The CORE-OM was completed by 342 people experiencing tinnitus who self-rated their tinnitus on a 5-point scale from ‘not a problem’ to ‘a very big problem’. Confirmatory factor analysis was used to test all ten factor models which have been previously derived across a range of population samples. Model fit was assessed using fit criterion and theoretical considerations. Mean scores on the full questionnaire and its subscales were compared between tinnitus problem categories using one-way ANOVA. RESULTS: The best fitting model included 33 of the 34 original items and was divided into three factors: negatively worded items, positively worded items and risk. The full questionnaire and each factor were found to have good internal consistency and factor loadings were high. There was a statistically significant difference in total CORE-OM scores across the five tinnitus problem categories. However there was no significant difference between those who rated their tinnitus ‘not a problem’, and ‘a small problem’ or ‘a moderate problem.’ CONCLUSION: This study found a 3-factor structure for the CORE-OM to be a good fit for a tinnitus population. It also found evidence of a relationship between emotional distress as measured by CORE-OM and perception of tinnitus as a problem. Its use in tinnitus clinics is to be recommended, particularly when emotional distress is a target of therapy

The sign problem in Monte Carlo simulations of frustrated quantum spin systems

We discuss the sign problem arising in Monte Carlo simulations of frustrated quantum spin systems. We show that for a class of ``semi-frustrated'' systems (Heisenberg models with ferromagnetic couplings $J_z(r) < 0$ along the $z$-axis and antiferromagnetic couplings $J_{xy}(r)=-J_z(r)$ in the $xy$-plane, for arbitrary distances $r$) the sign problem present for algorithms operating in the $z$-basis can be solved within a recent ``operator-loop'' formulation of the stochastic series expansion method (a cluster algorithm for sampling the diagonal matrix elements of the power series expansion of ${\rm exp}(-\beta H)$ to all orders). The solution relies on identification of operator-loops which change the configuration sign when updated (``merons'') and is similar to the meron-cluster algorithm recently proposed by Chandrasekharan and Wiese for solving the sign problem for a class of fermion models (Phys. Rev. Lett. {\bf 83}, 3116 (1999)). Some important expectation values, e.g., the internal energy, can be evaluated in the subspace with no merons, where the weight function is positive definite. Calculations of other expectation values require sampling of configurations with only a small number of merons (typically zero or two), with an accompanying sign problem which is not serious. We also discuss problems which arise in applying the meron concept to more general quantum spin models with frustrated interactions.Comment: 13 pages, 16 figure

Reflecting on professional development

This paper describes how a cluster of ten secondary science teachers from six different schools and colleges in the UK designed and undertook small-scale action research projects as an approach to their own Continuing Professional Development (CPD). The participating teachers identified a range of topics for investigation such as student voice, engagement and motivation in STEM (Science Technology Engineering Mathematics) learning and effective practical work. They brought their research into practice either as individuals, pairs or within a team. Central to each project was the use of reflection as a primary tool for ensuring the impact of the professional development and for stimulating a teacher-led process. Reflective practice was developed through the use of three approaches; audio sound bites, reflective blogs, and reflective discussions. The teachers used reflection as a means of self-evaluation as well as evaluation of their action research interventions. The reflective practice which the teachers engaged in enabled them to think carefully about what was taking place in a given situation during their designed interventions, to identify suitable options, and to make conscious choices about what to do in order to make a difference. The participating teachers all concluded that reflective methods that enable self-reflection as well as reflection upon designed classroom interventions are hugely beneficial to effective CPD for teachers