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Supramolecular nanocages with inner cavities have attracted increasing attention due to their fascinating molecular aesthetics and vast number of potential applications. Even though a wide array of discrete supramolecular cages with precisely designed sizes and shapes have been established, the controlled assembly of higher-order supramolecular frameworks from discrete molecular entities still represents a formidable challenge. In this work, a novel metallo-organic cage [Zn12L4] was assembled based on a triphenylene-cored hexapod terpyridine ligand. Synchotron X-ray analysis revealed a pair of enantiomeric cages in the crystal with flexible ligands twisted clockwise or anticlockwise due to steric hindrance in the structure. Interestingly, due to the strong π–π intermolecular interaction between triphenylene units, a controlled hierarchical packing of sphere-like cages in the crystal was established having a sparse packing mode with huge channels of around 3.6 nm diameter. This research sheds light on the design of strong π–π interactions in supramolecular hierarchical packing and materials science. 

Abstract Magic-angle twisted bilayer graphene has recently become a thriving material platform realizing correlated electron phenomena taking place within its topological flat bands. Several numerical and analytical methods have been applied to understand the correlated phases therein, revealing some similarity with the quantum Hall physics. In this work, we provide a Mott-Hubbard perspective for the TBG system. Employing the large-scale density matrix renormalization group on the lattice model containing the projected Coulomb interactions only, we identify a first-order quantum phase transition between the insulating stripe phase and the quantum anomalous Hall state with the Chern number of ±1. Our results not only shed light on the mechanism of the quantum anomalous Hall state discovered at three-quarters filling, but also provide an example of the topological Mott insulator, i.e., the quantum anomalous Hall state in the strong coupling limit. 

Estimations and applications of factor models often rely on the crucial condition that the number of latent factors is consistently estimated, which in turn also requires that factors be relatively strong, data are stationary and weakly serially dependent, and the sample size be fairly large, although in practical applications, one or several of these conditions may fail. In these cases, it is difficult to analyze the eigenvectors of the data matrix. To address this issue, we propose simple estimators of the latent factors using cross-sectional projections of the panel data, by weighted averages with predetermined weights. These weights are chosen to diversify away the idiosyncratic components, resulting in “diversified factors.” Because the projections are conducted cross-sectionally, they are robust to serial conditions, easy to analyze and work even for finite length of time series. We formally prove that this procedure is robust to over-estimating the number of factors, and illustrate it in several applications, including post-selection inference, big data forecasts, large covariance estimation, and factor specification tests. We also recommend several choices for the diversified weights.