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.     

Asymmetric double-layer composite electrolyte with enhanced ionic conductivity and interface stability for all-solid-state lithium metal batteries

All-solid-state Li metal battery has been regarded as a promising battery technology due to its high energy density based on the high capacity of lithium metal anode and high safety based on the all solid state electrolyte without inflammable solvent.However,challenges still exist mainly in the poor contact and unstable interface between electrolyte and electrodes.Herein,we demonstrate an asymmetric design of the composite polymer electrolyte with two different layers to overcome the interface issues at both the cathode and the anode side simultaneously.At the cathode side,the polypropylene carbonate layer has enough viscosity and flexibility to reduce the inter-facial resistance,while at the Li anode side,the polyethylene oxide layer modified with hexagonal boron nitride has high mechanical strength to suppress the Li dendrite growth.Owing to the synergetic effect between different components,the asprepared double layer composite polymer electrolyte demonstrates a large electrochemical window of5.17 V,a high ionic conductivity of 6.1×10~(-4) S/cm,and a transfe rence number of 0.56,featuring excellent ion transport kinetics and good chemical stability.All-solid-state Li metal battery assembled with LiFePO_4 cathode and Li anode delivers a high capacity of 150.9 mAh/g at 25℃ and 0.1 C-rate,showing great potential for practical applications.