A two-component geodesic equation on a space of constant positive curvature

We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter–Saxton equation.     

Output response in orthogonal directions of a He–Ne laser caused by birefringent-external-cavity feedback

The output responses of phase, amplitude, and polarization of light intensity of a He–Ne laser with the birefringent external cavity are presented. All phenomena are observed in two orthogonal directions related to the birefringent element’s orientation, and are studied under weak optical feedback. Experiments are carried out in five available detecting positions. It is found that laser intensities are modulated in the two orthogonal directions simultaneously, with a phase difference dominated by the phase retardation of the birefringence element. The modulation amplitudes are different in the two directions. A special polarization phenomenon is also observed and discussed. This research may provide a potential novel measuring principle for transparent birefringent materials.     

Finite-time H ∞ control for a class of Markovian jump delayed systems with input saturation

This work considers the finite-time control problem for a class Markovian jump delayed systems with partially known transition rates subject to saturating actuators. By employing local sector conditions and an appropriate Lyapunov–Krasovkii function, an observer-based state feedback controller is designed to guarantee that the resulted closed-loop constrained system is mean-square locally asymptotically finite-time stabilizable. Some sufficient conditions for the solution to this problem are derived in terms of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness of proposed method.     

A one‐pot method for synthesis of reduced graphene oxide‐supported Cu–Cu2O and catalytic application in tandem reaction of halides and sodium azide with terminal alkynes

Reduced graphene oxide (RGO)‐supported Cu–Cu₂O nanocomposite material (Cu‐Cu₂O@RGO) was prepared through a one‐pot reflux synthesis method. The morphology, crystal structure and composition of the prepared Cu‐Cu₂O@RGO were characterized using transmission electron microscopy, X‐ray diffraction, and X‐ray photoelectron, infrared and Raman spectroscopies. Cu‐Cu₂O@RGO as a heterogeneous catalyst was applied to tandem reactions of halides and sodium azide with terminal alkynes to synthesize effectively 1,4‐disubstituted 1,2,3‐triazoles. Moreover, the catalyst showed excellent recyclability performance with very little leaching of the metal. Compared with homogeneous catalysts, Cu‐Cu₂O@RGO as a green and efficient catalyst was recoverable, easy to separate and highly stable in the tandem method for the synthesis of 1,2,3‐triazole compounds.     

Structure‐guided RP‐HPLC chromatography of diastereomeric α‐helical peptide analogs substituted with single amino acid stereoisomers

An α‐helical model peptide (Ac‐EAEKAAKE‐X‐EKAAKEAEK‐amide) was used as a template to examine the efficacy of conventional reversed‐phase high‐performance liquid chromatography (RP‐HPLC) in separating peptide analogs with single substitutions (at position X) of diasteromeric amino acids Ile, allo‐Ile, d ‐Ile and d ‐allo‐Ile. We compared differences in peptide retention behavior on a C₈ column and a C₁₈ column at different temperatures. We demonstrated how subtle differences in peptide secondary structure affected by the different substitutions of amino acids with identical overall hydrophobicity enabled effective resolution of these peptide analogs. We also demonstrated the ability of RP‐HPLC to separate Ile‐ and allo‐Ile‐substituted analogs of a 26‐residue α‐helical antimicrobial peptide (AMP), with the substitution site towards the C‐terminus of the α‐helix. These peptides show different values of antibacterial activity and hemolytic activity, and different selectivity against bacteria and human cells. Our results underline the ability of RP‐HPLC to resolve even difficult diasteromeric peptide mixtures as well as its value in monitoring very subtle hydrophobicity changes in de novo‐designed AMP. Copyright © 2013 John Wiley & Sons, Ltd.