Invariants and K-spectrums of local theta lifts

Let $(G,G')$ be a type I irreducible reductive dual pair in $\mathrm{Sp}(W_{\mathbb{R}})$. We assume that $(G,G')$ is in the stable range where $G$ is the smaller member. Let $K$ and $K'$ be maximal compact subgroups of $G$ and $G'$ respectively. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ and $\mathfrak{g}' = \mathfrak{k}' \oplus \mathfrak{p}'$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G'$ respectively. Let ${\widetilde{K}}$ and ${\widetilde{K}}'$ be the inverse images of $K$ and $K'$ in the metaplectic double cover $\widetilde{\mathrm{Sp}}(W_\mathbb{R})$ of ${\mathrm{Sp}}(W_\mathbb{R})$. Let $\rho$ be a genuine irreducible $(\mathfrak{g},{\widetilde{K}})$-module. Our first main result is that if $\rho$ is unitarizable, then except for one special case, the full local theta lift $\rho' = \Theta(\rho)$ is equal to the local theta lift $\theta(\rho)$. Thus excluding the special case, the full theta lift $\rho'$ is an irreducible and unitarizable $(\mathfrak{g}',{\widetilde{K}}')$-module. Our second main result is that the associated variety and the associated cycle of $\rho'$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation $\rho^*$ respectively. Finally we obtain some interesting $(\mathfrak{g},{\widetilde{K}})$-modules whose ${\widetilde{K}}$-spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent $K_\mathbb{C}$-orbits in $\mathfrak{p}^*$

Characterization of Swirling Fluidized Bed

This dissertation is intended to conclude and summarize the overall milestone of Final Year Project, Characterization of Swirling Fluidized Bed. In recent years, the Swirling Fluidized Bed has been regarded as one of the novel designs in fluidization technology. This new technique features an annular-blade distributor which injects the fluidizing gas through a certain inclination, is capable of fluidizing the bed and at the same time causes swirling motion of particles in a circular trajectory. In the present work, the fluidization characteristics and hydrodynamics of a swirling bed are studied using experimental approach. The behavior of gas-particle interaction in a swirling bed in terms of operation regimes, trends of pressure drop across particle bed and hysteresis effects of bed pressure drop with increasing superficial velocity of gas, are explored by varying bed configurations. Three different sizes of spherical Polyvinyl chloride particle, two sizes in irregular shape and two sizes in cylindrical form, are used as bed material by considering four bed weights from 500 g to 2000 g, with increment of 500 g in each step, three blade overlap angles of 9°, 15° and 18°, for air superficial velocities up to approximately 3.5 m/s and two blade inclination of 10° and 15°. In this report, a well-structured review of the literature is constructed to compile the critical and substantive discoveries in the past researches. Furthermore, detailed research methodology and detailed analysis of experiment results are illustrated and expounded. The findings explicitly show that the solid particle size, shape, and bed weight are the major variables that give significant impact on the fluidized bed characteristics, while the blade dimension has relatively smaller effect on the bed behavior. This project has, hopefully, revealed how everything responds in SFB and this correlated relationship could be a precious benchmark in designing a reactor bed. As a conclusion, the research is intended to demonstrate the superiority of SFB over conventional bed. Through this exploration, the author sincerely hopes that this project will become an achievable reference volume for every practitioner in this field, spanning the boundary of various disciplines especially for fluidization engineering

Next-to-leading order QCD corrections to the single top quark production via model-independent t-q-g flavor-changing neutral-current couplings at hadron colliders

We present the calculations of the complete next-to-leading order (NLO) QCD effects on the single top productions induced by model-independent $tqg$ flavor-changing neutral-current couplings at hadron colliders. Our results show that, for the $tcg$ coupling the NLO QCD corrections can enhance the total cross sections by about 60% and 30%, and for the $tug$ coupling by about 50% and 20% at the Tevatron and LHC, respectively, which means that the NLO corrections can increase the experimental sensitivity to the FCNC couplings by about 10%$-$30%. Moreover, the NLO corrections reduce the dependence of the total cross sections on the renormalization or factorization scale significantly, which lead to increased confidence on the theoretical predictions. Besides, we also evaluate the NLO corrections to several important kinematic distributions, and find that for most of them the NLO corrections are almost the same and do not change the shape of the distributions.Comment: minor changes, version published in PR