Hyers-Ulam stability of an iterative equation for strictly increasing continuous functions

Aequationes mathematicae - The Hyers-Ulam stability of the iterative equation $$f^n=F$$ for continuous functions F was studied under the assumptions that F is a homeomorphism on its range, and the...     

Acetylene-Mediated Borophosphene Dirac Materials as Efficient Anode Materials for Lithium-Ion Batteries

Generally, graphynes have been generated by the insertion of acetylenic content (−C≡C−) in the graphene network in different ratios. Also, several aesthetically pleasing architectures of two-dimensional (2D) flatlands have been reported with the incorporation of acetylenic linkers between the heteroatomic constituents. Prompted by the experimental realization of boron phosphide, which has provided new insights on the boron-pnictogen family, we have modelled novel forms of acetylene-mediated borophosphene nanosheets by joining the orthorhombic borophosphene stripes with different widths and with different atomic constituents using acetylenic linkers. Structural stabilities and properties of these novel forms have been assessed using first-principles calculations. Investigation of electronic band structure elucidates that all the novel forms show the linear band crossing closer to the Fermi level at Dirac point with distorted Dirac cones. The linearity in the hole and electronic bands impose the high Fermi velocity to the charge carriers close to that of graphene. Finally, we have also unravelled the propitious features of acetylene-mediated borophosphene nanosheets as anodes in Li-ion batteries.     

Electrochemically Generated Copper(I)-Catalyzed Click Chemistry: Triazole Synthesis and Insights into Their Photophysical Properties

Herein, we report the electro-click (e-CLICK) reaction via electrochemically generated copper(I) catalytic species. The reaction worked under constant potential electrolysis (−0.25 V) with copper(II) nitrate and 2,2′-bipyridine serving as pre-catalyst and ligand respectively. The reaction accommodates electronically different organic azides and terminal alkynes to afford 1,2,3-triazoles in good synthetic yields with excellent 1,4-regioselectivity. Compared to traditional click chemistry, the developed e-CLICK methodology avoids the use of external reducing agents such as sodium ascorbate. Absorptive properties of selected triazole products were assessed by UV-visible electronic spectra. Triazole ( 3 e ) was further applied for the spectrophotometric detection of Co²⁺ and Cu²⁺ species.