The tensor structure on the representation category of the Wp\mathcal{W}_p triplet algebra

We study the braided monoidal structure that the fusion product induces on the abelian category $\mathcal{W}_p$-mod, the category of representations of the triplet $W$-algebra $\mathcal{W}_p$. The $\mathcal{W}_p$-algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalise the methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch, that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We apply these methods to the braided monoidal structure of $\mathcal{W}_p$-mod, previously constructed by Huang, Lepowsky and Zhang, to prove that this braided monoidal structure is rigid. The rigidity of $\mathcal{W}_p$-mod allows us to prove explicit formulae for the fusion product on the set of all simple and all projective $\mathcal{W}_p$-modules, which were first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and Runkel.Comment: 58 pages; edit: added references and revisions according to referee reports. Version to appear on J. Phys.

Microstructure control during twin roll casting of an AZ31 magnesium alloy

The existing twin roll casting technique for magnesium alloys suffers heterogeneity in both microstructure and chemistry and downstream processing is required to improve the strip quality, resulting in cost rise. In the present work, twin roll casting was carried out using an AZ31 magnesium alloy, with the application of intensive shearing melt conditioning prior to casting. The effect of process parameters such as pouring temperature and casting speed on microstructure control during casting and subsequent downstream processing was studied. Experimental results showed that the melt conditioning treatment allowed the production of AZ31 strips with uniform and refined microstructure free of centreline segregations. It was also shown that an optimized combination of pouring temperature and casting speed, in conjunction with a strip thickness control operation, resulted in uniformly distributed stored energies due to enhanced plastic deformation, which promoted recrystallization during casting and subsequent heat treatment. Strips prepared by twin roll casting and homogenization developed similar microstructural features to those prepared by twin roll casting followed by lengthy downstream processing by homogenization, hot rolling and annealing and displayed a weaker basal texture, exhibiting a potentially better formability.The EPSRC (UK

Kaon-nucleon interaction in the extended chiral SU(3) quark model

The chiral SU(3) quark model is extended to include the coupling between the quark and vector chiral fields. The one-gluon exchange (OGE) which dominantly governs the short-range quark-quark interaction in the original chiral SU(3) quark model is now nearly replaced by the vector-meson exchange. Using this model, the isospin I=0 and I=1 kaon-nucleon S, P, D, F wave phase shifts are dynamically studied by solving the resonating group method (RGM) equation. Similar to those given by the original chiral SU(3) quark model, the calculated results for many partial waves are consistent with the experiment, while there is no improvement in this new approach for the P_{13} and D_{15} channels, of which the theoretical phase shifts are too much repulsive and attractive respectively when the laboratory momentum of the kaon meson is greater than 300 MeV.Comment: 19 pages, 16 figures. Accepted by Phys. Rev.

The effects of protected beams and their connections on the fire resistance of composite buildings

According to full-scale fire tests, it is noticed that tensile membrane action within the concrete floor slabs plays an important role in affecting the fire resistance of composite buildings. It is well known that the development of tensile membrane actions relies on the vertical support along the edges of the slab panel. However, there is at present a lack of research into the influence of vertical supports on the tensile membrane actions of the floor slabs. In this paper, the performances of a generic three dimensional 45m x 45m composite floor subjected to ISO834 Fire and Natural Fire are investigated. Different vertical support conditions and three steel meshes are applied in order to assess the impact of vertical supports on tensile membrane action of floor slabs. Unlike other existing large scale modelling which assumes the connections behave as pinned or rigid for simplicity, two robust 2-node connection element models developed by the authors are used to model the behaviour of end-plate and partial end-plate connections of composite structures under fire conditions. The impact of connections on the 3D behaviour of composite floor is taken into consideration. The load-transfer mechanisms of composite floor when connections fail due to axial tension, vertical shear and bending are investigated. Based on the results obtained, some design recommendations are proposed to enhance the fire resistance of composite buildings

Theory and Calibration of Swap Market Models

This paper introduces a general framework for market models, named Market Model Approach, through the concept of admissible sets of for-ward swap rates spanning a given tenor structure. We relate this concept to results in graph theory by showing that a set is admissible if and only if the associated graph is a tree. This connection enables us to enumerate all admissible models for a given tenor structure. Three main classes are identified within this framework, and correspond to the co-terminal, co-initial, and co-sliding model. We prove that the LIBOR market model is the only admissible model of a co-sliding type. By focusing on the co-terminal model in a lognormal setting, we develop and compare several approximating analytical formulae for caplets, while swaptions can be priced by a simple Black-type formula. A novel calibration technique is introduced to allow simultaneous calibration to caplet and swaption prices. Empirical calibration of the co-terminal model is shown to be faster, more robust and more efficient than the same procedure applied to the LIBOR market model. We then argue that the co-terminal approach is the simplest and most convenient market model for pricing and hedging a large variety of exotic interest-rate derivatives.Swap Market Model, Cap, Swaption, Calibration, Graph Theory

Gamma-Ray Burst Afterglows from Realistic Fireballs

A GRB afterglow has been commonly thought to be due to continuous deceleration of a postburst fireball. Many analytical models have made simplifications for deceleration dynamics of the fireball and its radiation property, although they are successful at explaining the overall features of the observed afterglows. We here propose a model for a GRB afterglow in which the evolution of a postburst fireball is in an intermediate case between the adiabatic and highly radiative expansion. In our model, the afterglow is both due to the contribution of the adiabatic electrons behind the external blastwave of the fireball and due to the contribution of the radiative electrons. In addition, this model can describe evolution of the fireball from the extremely relativistic phase to the non-relativistic phase. Our calculations show that the fireball will go to the adiabatic expansion phase after about a day if the accelerated electrons are assumed to occupy the total internal energy. In all cases considered, the fireball will go to the mildly relativistic phase about $10^4$ seconds later, and to the non-relativistic phase after several days. These results imply that the relativistic adiabatic model cannot describe the deceleration dynamics of the several-days-later fireball. The comparison of the calculated light curves with the observed results at late times may imply the presence of impulsive events or energy injection with much longer durations.Comment: 18 pages, 10 figures, plain latex file, submitted to Ap

Gamma-ray bursts: postburst evolution of fireballs

The postburst evolution of fireballs that produce $\gamma$-ray bursts is studied, assuming the expansion of fireballs to be adiabatic and relativistic. Numerical results as well as an approximate analytic solution for the evolution are presented. Due to adoption of a new relation among $t$, $R$ and $\gamma$ (see the text), our results differ markedly from the previous studies. Synchrotron radiation from the shocked interstellar medium is attentively calculated, using a convenient set of equations. The observed X-ray flux of GRB afterglows can be reproduced easily. Although the optical afterglows seem much more complicated, our results can still present a rather satisfactory approach to observations. It is also found that the expansion will no longer be highly relativistic about 4 days after the main GRB. We thus suggest that the marginally relativistic phase of the expansion should be investigated so as to check the afterglows observed a week or more later.Comment: 17 pages, 4 figures, MNRAS in pres

N K and Delta K states in the chiral SU(3) quark model

The isospin I=0 and I=1 kaon-nucleon $S$, $P$, $D$, $F$ wave phase shifts are studied in the chiral SU(3) quark model by solving the resonating group method (RGM) equation. The calculated phase shifts for different partial waves are in agreement with the experimental data. Furthermore, the structures of the $\Delta K$ states with L=0, I=1 and I=2 are investigated. We find that the interaction between $\Delta$ and $K$ in the case of L=0, I=1 is attractive, which is not like the situation of the $NK$ system, where the $S$-wave interactions between $N$ and $K$ for both I=0 and I=1 are repulsive. Our numerical results also show that when the model parameters are taken to be the same as in our previous $NN$ and $YN$ scattering calculations, the $\Delta K$ state with L=0 and I=1 is a weakly bound state with about 2 MeV binding energy, while the one with I=2 is unbound in the present one-channel calculation.Comment: 14 pages, 6 figures. PRC70,064004(2004

Open-closed field algebras

We introduce the notions of open-closed field algebra and open-closed field algebra over a vertex operator algebra V. In the case that V satisfies certain finiteness and reductivity conditions, we show that an open-closed field algebra over V canonically gives an algebra over a \C-extension of the Swiss-cheese partial operad. We also give a tensor categorical formulation and categorical constructions of open-closed field algebras over V.Comment: 55 pages, largely revised, an old subsection is deleted, a few references are adde