Twisted factorization of a banded matrix

The twisted factorization of a tridiagonal matrix T plays an important role in inverse iteration as featured in the MRRR algorithm. The twisted structure simplifies the computation of the eigenvector approximation and can also improve the accuracy. A tridiagonal twisted factorization is given by T=M _k Δ _k N _k where Δ _k is diagonal, M _k ,N _k have unit diagonals, and the k-th column of M _k and the k-th row of N _k correspond to the k-th column and row of the identity, that is . This paper gives a constructive proof for the existence of the twisted factorizations of a general banded matrix A. We show that for a given twist index k, there actually are two such factorizations. We also investigate the implications on inverse iteration and discuss the role of pivoting.      

On Tutte polynomial uniqueness of twisted wheels

A graph G is called T-unique if any other graph having the same Tutte polynomial as G is isomorphic to G. Recently, there has been much interest in determining T-unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105-119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57-65] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all T-unique. In this paper, we prove that the twisted wheels are also T-unique.     

Type-II Red Phosphorus: Wavy Packing of Twisted Pentagonal Tubes

Elemental phosphorus exhibits fascinating structural varieties and versatile properties. The unique nature of phosphorus bonds can lead to the formation of extremely complex structures, and detailed structural information on some phosphorus polymorphs is yet to be investigated. In this study, we investigated an unidentified crystalline phase of phosphorus, type-II red phosphorus (RP), by combining state-of-the-art structural characterization techniques. Electron diffraction tomography, atomic-resolution scanning transmission electron microscopy (STEM), powder X-ray diffraction, and Raman spectroscopy were concurrently used to elucidate the hidden structural motifs and their packing in type-II RP. Electron diffraction tomography, performed using individual crystalline nanowires, was used to identify a triclinic unit cell with volume of 5330 ų, which is the largest unit cell for elemental phosphorus crystals up to now and contains approximately 250 phosphorus atoms. Atomic-resolution STEM imaging, which was performed along different crystal-zone axes, confirmed that the twisted wavy tubular motif is the basic building block of type-II RP. Our study discovered and presented a new variation of building blocks in phosphorus, and it provides insights to clarify the complexities observed in phosphorus as well as other relevant systems.     

Highly Twisted Thermally Activated Delayed Fluorescence (TADF) Molecules and Their Applications in Organic Light-Emitting Diodes (OLEDs)

Thermally activated delayed fluorescence (TADF) materials have attracted great potential in the field of organic light-emitting diodes (OLEDs). Among thousands of TADF materials, highly twisted TADF emitters have become a hotspot in recent years. Compared with traditional TADF materials, highly twisted TADF emitters tend to show multi-channel charge-transfer characters and form rigid molecular structures. This is advantageous for TADF materials, as non-radiative decay processes can be suppressed to facilitate efficient exciton utilization. Accordingly, OLEDs with excellent device performances have also been reported. In this Review, we have summarized recent progress in highly twisted TADF materials and related devices, and give an overview of the molecular design strategies, photophysical studies, and the performances of OLED devices. In addition, the challenges and perspectives of highly twisted TADF molecules and the related OLEDs are also discussed.