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.

Nanolithography for metallic quasi crystals for nanobio applications

There is currently an urgent need to develop micro and nanotechnique for the fabrications of quasi periodic crystals in a plane for the study and applications of novel optical properties when light propagating in or through such a photonic structures with fold symmetries (10 fold symmetry in this work). It has been clear that quasi periodical crystals in dielectrics with various fold symmetries also exhibits complete photonic band gap (PBG) property as periodical photonic crystals do. However, the novel physical properties related to the interactions of electromagnetic waves with metallic holes arrays in quasi periodical order (metallic quasi crystals) is being discovered both theoretically and experimentally, which demands technical development for the construction of theoretically designed structures. [1] In this work, we report a nanofabrication technique recently developed for the replication of quasi crystal in 100 nm Al on a slab (quartz wafer in this work) by electron beam lithography using chemically amplified resist, UVN-30. A wealth of novel photonic behaviours of lights vertically incident through the q-crystal were observed

A Two-Component Explosion Model for the Giant Flare and Radio Afterglow from SGR1806-20

The brightest giant flare from the soft $\gamma$-ray repeater (SGR) 1806-20 was detected on 2004 December 27. The isotropic-equivalent energy release of this burst is at least one order of magnitude more energetic than those of the two other SGR giant flares. Starting from about one week after the burst, a very bright ($\sim 80$ mJy), fading radio afterglow was detected. Follow-up observations revealed the multi-frequency light curves of the afterglow and the temporal evolution of the source size. Here we show that these observations can be understood in a two-component explosion model. In this model, one component is a relativistic collimated outflow responsible for the initial giant flare and the early afterglow, and another component is a subrelativistic wider outflow responsible for the late afterglow. We also discuss triggering mechanisms of these two components within the framework of the magnetar model.Comment: 7 pages including 3 figures, emulateapj5.sty, accepted for publication in ApJ Letter

Life Tables of Bactrocera cucurbitae (Coquillett) (Diptera: Tephritidae): with a Mathematical Invalidation for Applying the Jackknife Technique to the Net Reproductive Rate

Life table data for the melon fly, Bactrocera cucurbitae (Coquillett), reared on cucumber (Cucumis sativus L.) were collected under laboratory and simulated field conditions. Means and standard errors of life table parameters were estimated for two replicates using the jackknife technique. At 25ºC, the intrinsic rates of increase (_r_) found for the two replicates were 0.1354 and 0.1002 day-1, and the net reproductive rates (_R_~0~) were 206.3 and 66.0 offspring, respectively. When the cucumbers kept under simulated field conditions were covered with leaves, the _r_ and _R_~0~ for the two replicates were 0.0935 and 0.0909 day-1, 17.5 and 11.4 offspring, respectively. However, when similar cucumbers were left uncovered, the _r_ and _R_~0~ for the two replicates were 0.1043 and 0.0904 day-1, and 27.7 and 10.1 offspring, respectively. Our results revealed that considerable variability between replicates in both laboratory and field conditions is possible; this variability should be taken into consideration in data collection and application of life tables. Mathematical analysis has demonstrated that applying the jackknife technique results in unrealistic pseudo-_R_~0~ and overestimation of its variance. We suggest that the jackknife technique should not be used for the estimation of variability of _R_~0~

The Exemption Status of the Bona Fide Pledgee of Unregistered Securities Under the Securities Act of 1933

The number of connections of photovoltaic (PV) to distribution network is increasing. Very few PV connection guidelines that distribution system operators (DSOs) can refer to have been found. This paper deals with network planning guidelines for distribution networks with PV. The paper aims to identify planning rules that are relatively easy to implement.QC 20140625</p