Generalizations of Weighted Trapezoidal Inequality for Monotonic Mappings and Its Applications

In this paper, we establish some generalizations of weighted trapezoid inequality for monotonic mappings, and give several applications for r − moment, the expectation of a continuous random variable and the Beta mapping

A simple proof of inequalities related to means

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Synthesis of Borophene Nanoribbons on Ag(110) Surface

We present the successful synthesis of single-atom-thick borophene nanoribbons (BNRs) by self-assembly of boron on Ag(110) surface. The scanning tunneling microscopy (STM) studies reveal high quality BNRs: all the ribbons are along the [-110] direction of Ag(110), and can run across the steps on the surface. The width of ribbons is distributed in a narrow range around 10.3 nm. High resolution STM images revealed four ordered surface structures in BNRs. Combined with DFT calculations, we found that all the four structures of boron nanoribbons consist of the boron chains with different width, separated by hexagonal hole arrays. The successful synthesis of BNRs enriches the low dimensional allotrope of boron and may promote further applications of boropheneComment: 14 pages, 3 figure