qq-Trinomial identities

We obtain connection coefficients between $q$-binomial and $q$-trinomial coefficients. Using these, one can transform $q$-binomial identities into a $q$-trinomial identities and back again. To demonstrate the usefulness of this procedure we rederive some known trinomial identities related to partition theory and prove many of the conjectures of Berkovich, McCoy and Pearce, which have recently arisen in their study of the $\phi_{2,1}$ and $\phi_{1,5}$ perturbations of minimal conformal field theory.Comment: 21 pages, AMSLate

Vertex-algebraic structure of the principal subspaces of certain A_1^(1)-modules, I: level one case

This is the first in a series of papers in which we study vertex-algebraic structure of Feigin-Stoyanovsky's principal subspaces associated to standard modules for both untwisted and twisted affine Lie algebras. A key idea is to prove suitable presentations of principal subspaces, without using bases or even ``small'' spanning sets of these spaces. In this paper we prove presentations of the principal subspaces of the basic A_1^(1)-modules. These convenient presentations were previously used in work of Capparelli-Lepowsky-Milas for the purpose of obtaining the classical Rogers-Ramanujan recursion for the graded dimensions of the principal subspaces.Comment: 20 pages. To appear in International J. of Mat

Analytical two-center integrals over Slater geminal functions

We present analytical formulas for the calculation of the two-center two-electron integrals in the basis of Slater geminals and products of Slater orbitals. Our derivation starts with establishing a inhomogeneous fourth-order ordinary differential equation that is obeyed by the master integral, the simplest integral with inverse powers of all interparticle distances. To solve this equation it was necessary to introduce a new family of special functions which are defined through their series expansions around regular singular points of the differential equation. To increase the power of the interparticle distances under the sign of the integral we developed a family of open-ended recursion relations. A handful of special cases of the integrals is also analysed with some remarks on simplifications that occur. Additionally, we present some numerical examples of the master integral that validate the usefulness and correctness of the key equations derived in this paper. In particular, we compare our results with the calculations based on the series expansion of the exp(-\gamma r12) term in the master integral.Comment: 28 pages, 0 figures, 7 table

Preparing the foundations for video-based, practice-placement support: establishing the role from a students’ perspective

Currently, many placement-based health programme students within the UK are supported through face-to-face visits from university staff. Whilst cited in literature as being of value, the face-to-face nature of this contact is not supported. Alternatives including video-based communications methods offer the potential for cost effective, environmentally responsible support. However, in order to establish the fitness for purpose of alternative approaches, the content and purpose of current support needs to be understood. This project aimed to investigate student perceptions of the ideal content and purpose of clinical support visits, and alternatives to the current face-to-face approach. Fifty-six Physiotherapy undergraduate students responded to questionnaires with a further nine participating in a follow-up focus group. Participants emphasised the value of the visit in guiding learning, ensuring progression and resolving arising issues, and highlighted concerns over alternative approaches. Focus group participants discussed the importance of personal and professional confidence in directing requirements for support, and went on to propose a menu of options for methods of communication.Whilst limited in some applications, video technologies may be one of the options. Overall, however, this project supports the need for consideration of individualised learning journeys within curriculum planning

An ALMA 3mm continuum census of Westerlund 1

Context. Massive stars play an important role in both cluster and galactic evolution and the rate at which they lose mass is a key driver of both their own evolution and their interaction with the environment up to and including their terminal SNe explosions. Young massive clusters provide an ideal opportunity to study a co-eval population of massive stars, where both their individual properties and the interaction with their environment can be studied in detail. Aims. We aim to study the constituent stars of the Galactic cluster Westerlund 1 in order to determine mass-loss rates for the diverse post-main sequence population of massive stars. Methods. To accomplish this we made 3mm continuum observations with the Atacama Large Millimetre/submillimetre Array. Results. We detected emission from 50 stars in Westerlund 1, comprising all 21 Wolf-Rayets within the field of view, plus eight cool and 21 OB super-/hypergiants. Emission nebulae were associated with a number of the cool hypergiants while, unexpectedly, a number of hot stars also appear spatially resolved. Conclusions. We were able to measure the mass-loss rates for a unique population of massive post-main sequence stars at every stage of evolution, confirming a significant increase as stars transitioned from OB supergiant to WR states via LBV and/or cool hypergiant phases. Fortuitously, the range of spectral types exhibited by the OB supergiants provides a critical test of radiatively-driven wind theory and in particular the reality of the bi-stability jump. The extreme mass-loss rate inferred for the interacting binary Wd1-9 in comparison to other cluster members confirmed the key role binarity plays in massive stellar evolution. The presence of compact nebulae around a number of OB and WR stars is unexpected; by analogy to the cool super-/hypergiants we attribute this to confinement and sculpting of the stellar wind via interaction with the intra-cluster medium/wind. Given the morphology of core collapse SNe depend on the nature of the pre-explosion circumstellar environment, if this hypothesis is correct then the properties of the explosion depend not just on the progenitor, but also the environment in which it is located

Laser-controlled fluorescence in two-level systems

The ability to modify the character of fluorescent emission by a laser-controlled, optically nonlinear process has recently been shown theoretically feasible, and several possible applications have already been identified. In operation, a pulse of off-resonant probe laser beam, of sufficient intensity, is applied to a system exhibiting fluorescence, during the interval of excited- state decay following the initial excitation. The result is a rate of decay that can be controllably modified, the associated changes in fluorescence behavior affording new, chemically specific information. In this paper, a two-level emission model is employed in the further analysis of this all-optical process; the results should prove especially relevant to the analysis and imaging of physical systems employing fluorescent markers, these ranging from quantum dots to green fluorescence protein. Expressions are presented for the laser-controlled fluorescence anisotropy exhibited by samples in which the fluorophores are randomly oriented. It is also shown that, in systems with suitably configured electronic levels and symmetry properties, fluorescence emission can be produced from energy levels that would normally decay nonradiatively. © 2010 American Chemical Society