Block lower triangular matrices and block upper triangular matrices are popular preconditioners for 2×2 block matrices. In this note we show that a block lower triangular preconditioner gives the same spectrum as a block upper triangular preconditioner and that the eigenvectors of the two preconditioned matrices are related.

In the fuzzy arithmetic, the definitions of addition and multiplication of fuzzy numbers are based on Zadeh’s extension principle. From theoretical and practical points of view, this multiplication of fuzzy numbers owns several unnatural properties. Recently, to avoid this shortcoming, a new multiplicative operation of product type is introduced, the so-called cross-product of fuzzy numbers. Th...

In current study, a particle swarm clustering method is suggested for clustering triangular fuzzy data. This clustering method can find fuzzy cluster centers in the proposed method, where fuzzy cluster centers contain more points from the corresponding cluster, the higher clustering accuracy. Also, triangular fuzzy numbers are utilized to demonstrate uncertain data. To compare triangular fuzzy ...

Ultrathin plasmonic metasurfaces have proven their ability to control and manipulate light at unprecedented levels, leading to exciting optical functionalities and applications. Although to date metasurfaces have mainly been investigated from an electromagnetic perspective, their ultrathin nature may also provide novel and useful mechanical properties. Here we propose a thin piezoelectric plasm...

We report the thermal annealing-induced formation of ring-like structure of Zn(II)-doped magnetite from iron alkoxide leaf-like nanoplate precusor. The phase, structure and morphology of magnetite nanorings were comprehensively characterized by powder X-ray diffraction, X-ray photoelectron spectroscopy, atomic force microscope, scanning electron microscope, and transmission electron microscope....

The preparation of thermochromic vanadium dioxide (VO2) films in an economical way is of interest to realizing the application of smart windows. Here, we reported a successful preparation of self-assembly VO2 nanoplate films on TiO2-buffered glass by a facile hydrothermal process. The VO2 films composed of triangle-shaped plates standing on substrates exhibit a self-generated porous structure, ...

Ultraviolet (UV) photodetectors are important tools in the fields of optical imaging, environmental monitoring, and air and water sterilization, as well as flame sensing and early rocket plume detection. Herein, hexagonal-like Nb₂O₅ nanoplates are synthesized using a facile solvothermal method. UV photodetectors based on single Nb₂O₅ nanoplates are constructed and the optoelectronic properties ...