Fermionic solution of the Andrews-Baxter-Forrester model II: proof of Melzer's polynomial identities

We compute the one-dimensional configuration sums of the ABF model using the fermionic technique introduced in part I of this paper. Combined with the results of Andrews, Baxter and Forrester, we find proof of polynomial identities for finitizations of the Virasoro characters $\chi_{b,a}^{(r-1,r)}(q)$ as conjectured by Melzer. In the thermodynamic limit these identities reproduce Rogers--Ramanujan type identities for the unitary minimal Virasoro characters, conjectured by the Stony Brook group. We also present a list of additional Virasoro character identities which follow from our proof of Melzer's identities and application of Bailey's lemma.Comment: 28 pages, Latex, 7 Postscript figure

Modelling the hepatitis B vaccination programme in prisons

A vaccination programme offering hepatitis B (HBV) vaccine at reception into prison has been introduced into selected prisons in England and Wales. Over the coming years it is anticipated this vaccination programme will be extended. A model has been developed to assess the potential impact of the programme on the vaccination coverage of prisoners, ex-prisoners, and injecting drug users (IDUs). Under a range of coverage scenarios, the model predicts the change over time in the vaccination status of new entrants to prison, current prisoners and IDUs in the community. The model predicts that at baseline in 2012 57% of the IDU population will be vaccinated with up to 72% being vaccinated depending on the vaccination scenario implemented. These results are sensitive to the size of the IDU population in England and Wales and the average time served by an IDU during each prison visit. IDUs that do not receive HBV vaccine in the community are at increased risk from HBV infection. The HBV vaccination programme in prisons is an effective way of vaccinating this hard-to-reach population although vaccination coverage on prison reception must be increased to achieve this

Polynomial Identities, Indices, and Duality for the N=1 Superconformal Model SM(2,4\nu)

We prove polynomial identities for the N=1 superconformal model SM(2,4\nu) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q \rightarrow q^{-1} between SM(2,4\nu) and M(2\nu-1,4\nu). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.Comment: 41 pages, harvmac, no figures; new identities, proofs and comments added; misprints eliminate

Rayleigh-Taylor instability-induced mixing: initial conditions modeling, three-dimensional simulations and comparisons with experiment

A spectral/compact finite-difference method with a third-order Adams-Bashforth-Moulton time-evolution scheme is used to perform a direct numerical simulation (DNS) of Rayleigh-Taylor flow. The initial conditions are modeled by parameterizing the multi-mode velocity and density perturbations measured just off of the splitter plate in water channel experiments. Parameters in the DNS are chosen to match the experiment as closely as possible. The early-time transition from a weakly-nonlinear to a strongly-nonlinear state, as well as the onset of turbulence, is examined by comparing the DNS and experimental results. The mixing layer width, molecular mixing parameter, vertical velocity variance, and density variance spectrum obtained from the DNS are shown to be in good agreement with the corresponding experimental values

How to measure the Bogoliubov quasiparticle amplitudes in a trapped condensate

We propose an experiment, based on two consecutive Bragg pulses, to measure the momentum distribution of quasiparticle excitations in a trapped Bose gas at low temperature. With the first pulse one generates a bunch of excitations carrying momentum $q$, whose Doppler line is measured by the second pulse. We show that this experiment can provide direct access to the amplitudes $u_{q}$ and $v_{q}$ characterizing the Bogoliubov transformations from particles to quasiparticles. We simulate the behavior of the nonuniform gas by numerically solving the time dependent Gross-Pitaevskii equation.Comment: 12 pages, 4 figures include

The Excitation Spectrum of a Bose-Einstein Condensate

We report the first measurement of the excitation spectrum and the static structure factor of a Bose-Einstein condensate. The excitation spectrum displays a linear phonon regime, as well as a parabolic single-particle regime. The linear regime provides an upper limit for the superfluid critical velocity, by the Landau criterion. The excitation spectrum agrees well with the Bogoliubov spectrum, in the local density approximation. This agreement continues even for excitations close to the long-wavelength limit of the region of applicability of the approximation. Feynman's relation between the excitation spectrum and the static structure factor is verified, within an overall constant

Analytic Approximation of the Tavis-Cummings Ground State via Projected States

We show that an excellent approximation to the exact quantum solution of the ground state of the Tavis-Cummings model is obtained by means of a semi-classical projected state. This state has an analytical form in terms of the model parameters and, in contrast to the exact quantum state, it allows for an analytical calculation of the expectation values of field and matter observables, entanglement entropy between field and matter, squeezing parameter, and population probability distributions. The fidelity between this projected state and the exact quantum ground state is very close to 1, except for the region of classical phase transitions. We compare the analytical results with those of the exact solution obtained through the direct Hamiltonian diagonalization as a function of the atomic separation energy and the matter-field coupling.Comment: 22 pages, 13 figures, accepted for publication in Physics Script

Experimental characterization of initial conditions and spatio-temporal evolution of a small Atwood number Rayleigh-Taylor mixing layer

The initial multi-mode interfacial velocity and density perturbations present at the onset of a small Atwood number, incompressible, miscible, Rayleigh-Taylor instability-driven mixing layer have been quantified using a combination of experimental techniques. The streamwise interfacial and spanwise interfacial perturbations were measured using high-resolution thermocouples and planar laser-induced fluorescence (PLIF), respectively. The initial multi-mode streamwise velocity perturbations at the two-fluid density interface were measured using particle-image velocimetry (PIV). It was found that the measured initial conditions describe an initially anisotropic state, in which the perturbations in the streamwise and spanwise directions are independent of one another. The evolution of various fluctuating velocity and density statistics, together with velocity and density variance spectra, were measured using PIV and high-resolution thermocouple data. The evolution of the velocity and density statistics is used to investigate the early-time evolution and the onset of strongly-nonlinear, transitional dynamics within the mixing layer. The early-time evolution of the density and vertical velocity variance spectra indicate that velocity fluctuations are the dominant mechanism driving the instability development. The implications of the present experimental measurements on the initialization of Reynolds-averaged turbulent transport and mixing models and of direct and large-eddy simulations of Rayleigh-Taylor instability-induced turbulence are discussed