and the associated line bundles

Let be a compact semi-simple Lie group, and let be a maximal unipotent subgroup of the complexified group . In this paper, we classify all the -invariant Kaehler structures on . For each Kaehler structure , let be the line bundle with connection whose curvature is . We then study the holomorphic sections of , which constitute a -representation space.

     

Method for Visible Light‐Induced Photocatalytic Degradation of Methylparaben in Water Using Nanostructured Ag/AgBr@m‐WO3

An efficient method of photocatalytic degradation of methylparaben in water using Ag nanoparticles (NPs) loaded AgBr‐mesoporous‐WO₃ composite photocatalyst (Ag/AgBr@m‐WO₃), under visible light is presented. In this process, quantification of methylparaben in water was carried out by high‐performance liquid chromatography (HPLC) and the HPLC results showed a significant reduction of methylparaben in water due to the enhanced of photocatalytic degradation efficiency of Ag/AgBr@m‐WO₃. For the material synthesis, highly ordered mesoporous‐WO₃ (m‐WO₃) was initially synthesized by sol–gel method and AgBr nanoparticles (NPs) were subsequently introduced in the pores of m‐WO₃, and finally, the Ag nanoparticles were introduced by light irradiation. The enhanced photocatalytic degradation of methylparaben in water is attributed to the formation of surface plasmonic resonance (SPR) due to the introduction of Ag NPs on the surface of the catalyst. Also, the formation of heterojunction between AgBr and mesoporous‐WO₃ in Ag/AgBr@m‐WO₃ significantly inhibited the recombination of light‐induced electron‐hole pairs in the semiconductor composite. The morphological and optical characterizations of the synthesized photocatalysts (Ag/AgBr@m‐WO₃) were carried out using SEM, TEM, XDR, N₂ adsorption–desorption, UV‐VIS diffuse reflectance spectroscopy (DRS). Also, the photocatalytic studies using radical scavengers were carried out and the results indicated that ${\text{O}}_2^{·-}$ is the main reactive species.     

Development and validation of a multiplex real-time PCR method to simultaneously detect 47 targets for the identification of genetically modified organisms

Considering the increase of the total cultivated land area dedicated to genetically modified organisms (GMO), the consumers’ perception toward GMO and the need to comply with various local GMO legislations, efficient and accurate analytical methods are needed for their detection and identification. Considered as the gold standard for GMO analysis, the real-time polymerase chain reaction (RTi-PCR) technology was optimised to produce a high-throughput GMO screening method. Based on simultaneous 24 multiplex RTi-PCR running on a ready-to-use 384-well plate, this new procedure allows the detection and identification of 47 targets on seven samples in duplicate. To comply with GMO analytical quality requirements, a negative and a positive control were analysed in parallel. In addition, an internal positive control was also included in each reaction well for the detection of potential PCR inhibition. Tested on non-GM materials, on different GM events and on proficiency test samples, the method offered high specificity and sensitivity with an absolute limit of detection between 1 and 16 copies depending on the target. Easy to use, fast and cost efficient, this multiplex approach fits the purpose of GMO testing laboratories.

Silk‐Pectin Hydrogel with Superior Mechanical Properties,Biodegradability, and Biocompatibility

A new method is developed to prepare silk hydrogels and silk‐pectin hydrogels via dialysis against methanol to obtain hydrogels with high concentrations of silk fibroin. The relationship between the mechanical and biological properties and the structure of the silk‐pectin hydrogels is subsequently evaluated. The present results suggest that pectin associates with silk molecules when the silk concentration exceeds 15 wt%, suggesting that a silk concentration of over 15 wt% is critical to construct interacting silk‐pectin networks. The silk‐pectin hydrogel reported here is composed of a heterogeneous network, which is different from fiber‐reinforced, interpenetrated networks and double‐network hydrogels, as well as high‐stiffness hydrogels (elastic modulus of 4.7 ± 0.9 MPa, elastic stress limit of 3.9 ± 0.1 MPa, and elastic strain limit of 48.4 ± 0.5%) with regard to biocompatibility and biodegradability.     

ACQ-to-AIE Transformation: Tuning Molecular Packing by Regioisomerization for Two-Photon NIR Bioimaging

The traditional design strategies for highly bright solid-state luminescent materials rely on weakening the intermolecular π–π interactions, which may limit diversity when developing new materials. Herein, we propose a strategy of tuning the molecular packing mode by regioisomerization to regulate the solid-state fluorescence. TBP-e-TPA with a molecular rotor in the end position of a planar core adopts a long-range cofacial packing mode, which in the solid state is almost non-emissive. By shifting molecular rotors to the bay position, the resultant TBP-b-TPA possesses a discrete cross packing mode, giving a quantum yield of 15.6±0.2 %. These results demonstrate the relationship between the solid-state fluorescence efficiency and the molecule's packing mode. Thanks to the good photophysical properties, TBP-b-TPA nanoparticles were used for two-photon deep brain imaging. This molecular design philosophy provides a new way of designing highly bright solid-state fluorophores.     

Al-free Zr-zeolite beta as a regioselective catalyst in the Meerwein-Ponndorf-Verley reaction

Al-free Zr-zeolite beta catalyzes the transfer reduction of ketones to the corresponding alcohols in high yield and with exceptional stereocontrol. Notably, the catalyst is very robust and gives good results even with 10% water content in the reaction mixture.     

Kahler Structures and Weighted Actions on the Complex Torus

Let T be the compact real torus, and T_C its complexification.Fix an integral weight , and consider the -weighted T-actionon T_C. If is a T-invariant Kähler form on T_C, it correspondsto a pre-quantum line bundle L over T_C. Let H be the square-integrableholomorphic sections of L. The weighted T-action lifts to aunitary T-representation on the Hilbert space H, and the multiplicityof its irreducible sub-representations is considered. It isshown that this is controlled by the image of the moment map,as well as the principle that ‘quantization commutes withreduction’.     

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

Partially harmonic forms and models of H-series

Let G   be a real reductive Lie group, let H=TA$H=TA$ be the identity component of a Cartan subgroup, and let h$\mathfrak{h}$ be the corresponding Cartan subalgebra. This leads to a parabolic subgroup of G whose identity component is MAN. The unitary G-representations induced by MAN are known as the H  -series. We study symplectic geometry of G×h$G\times \mathfrak{h}$ and apply geometric quantization to construct unitary G-representations by partially harmonic forms. They are direct integrals of the H-series, indexed by the image of the moment map. We also perform symplectic reduction and symplectic induction, and consider their analogues in representation theory via geometric quantization.