Materials, such as vehicle lightweighting, intelligent materials, and aviation damping material, are frequently subjected to prolonged loading conditions. During the service life of materials, micro-damage will inevitably occur. The ultrasonic wave-mixing method is an emerging technique in detecting subtle damage in materials. Investigating the interaction between two waves in nonlinear materials guides the non-destructive detection of defects. In the ultrasonic wave-mixing technique, the resonance condition is commonly employed for the selection of the two primary wave frequencies. However, resonance conditions are often not strictly satisfied in practical applications. The mixing wave still contains important information that requires attention in detection. The theoretical solution of the mixing wave interacted by two-way collinear waves with arbitrary frequencies is derived. The results advance us to understand the intrinsic properties of wave interaction.

The elaborate design of inexpensive, high-performance electrocatalysts from earth-abundant elements toward oxygen evolution reaction (OER) is critical in various (electro)chemical processes. Herein, a novel binder-free catalyst of Ce-coupled Ni3S2/NiS supported on Ni foam (Ce-Ni3S2/NiS@NF) is successfully synthesized via a facile one-step hydrothermal method that enables practical feasibility with a significant enhancement of OER activity through anchoring Ce dopants on an Ni3S2/NiS nanobud host. Ce species coupling can modulate electronic structure, which reduces the reaction energy barrier and optimizes OER catalytic activity. More profoundly, the superhydrophilic and superaerophobic properties of the Ce-Ni3S2/NiS@NF electrode further promote mass transfer. As a result, the Ce-Ni3S2/NiS@NF electrode exhibits excellent OER activity with a low overpotential of 236 and 350 mV to achieve current densities of 10 and 100 mA cm−2, respectively, and long-term durability for 24 h in alkaline medium. These results could supply valuable guidelines for the design of other OER catalysts and beyond.

Ultrasonic wave mixing is a powerful new nondestructive evaluation technique for accessing hidden subtle imperfections in materials. First, this paper derives the resonance conditions of one-way collinear mixing with cubic nonlinearity. The results show that the mixing of two collinear longitudinal waves can generate a resonant longitudinal wave, the mixing of two transverse waves can generate a resonant transverse wave, and the resonant longitudinal wave and transverse wave can be generated, respectively, when two primary longitudinal and transverse waves satisfy different resonance conditions. This is different from the case with quadratic nonlinearity. Furthermore, we obtain the analytical solutions of resonant waves generated by the mixing of harmonic longitudinal and transverse pulses. The waveforms are approximate hexagonal shapes according to the analysis of the envelopes. Finally, the numerical simulation results conducted on material aluminum prove the correctness of the analytical solution. Compared with quadratic nonlinearity, the resonant wave is more sensitive to material constants for some materials when considering one-way collinear mixing of primary longitudinal and transverse pulses based on cubic nonlinearity. The results provide new opportunities on mixing wave applications in nondestructive evaluation.

The resonance conditions of two-way collinear mixing are derived when a longitudinal wave and a transverse wave are propagating in the opposite directions based on cubic nonlinearity. The analytical solutions for mixing waves are obtained. An interesting result is that the resonant longitudinal wave and the resonant transverse wave can be generated, respectively, when considering different resonance conditions. This is different from the case with quadratic nonlinearity that only resonant transverse wave can be generated. For the convenience of applications, the wave mixing of a time-harmonic longitudinal pulse and a time-harmonic transverse pulse is also investigated. A clear mixing process is presented by discussing the parameters introduced in the derivation. Numerical calculation is conducted on material Al, the hexagonal waveforms of the resonant transverse waves are depicted when considering different cases. The results could provide new opportunities for mixing wave applications in nondestructive evaluation.

Linear algebra is not only a powerful tool in dealing with the problem of multi-variables, but also strongly logical. Students always feel that linear algebra is abstract, boring, and difficult to understand. For the teaching of linear algebra, the instructional design of linear algebra that combining the geometry intuition and practical application is proposed to help students understand the abstract knowledge. As an example in our teaching process, the geometric interpretation of matrix, similar matrices, eigenvalues and eigenvectors are given in turn. This teaching method aims to help students shift perception from visual to abstract and thus improve the teaching efficiency of linear algebra. The practical application of eigenvalue in image compression, i.e. Karhunen-Loeve transform, is presented. It is advantages to promote students' motivation in learning and cultivate their abilities in using mathematics to solve practical problems.

Quantum coherence and quantum correlation, as a key feature, have proven essential to our understanding of quantum critical phenomena. In this study, we contribute to characterize the quantum phase transition in the Ising model in a transverse field by using l1 norm coherence and quantum discord. The unitary operator is enforced in the model so that we extend the static problem to dynamics case. For quantum coherence, we show that it oscillates with the increase of magnetic field g. The extent of oscillation decreases when the size of the spin chain increase. Meanwhile, however, the oscillating frequency will increase. For quantum discord, it will decrease with the increase of quantum renormalization group and that result totally different with quantum coherence. We demonstrate that the classical correlation have similar behavior like quantum coherence. Furthermore, we show that quantum coherence also can detect the quantum critical point even when the intrinsic decoherence are enforced in the model.

The effect of non-zero initial displacement value ondisplacement jump and wave form based on the cubic nonlinear spring interface model between two semi-infinite plates is studied. The wave forms of reflected and transmitted pulses are sensitive to the initial value. The phenonmenon for non-zero initial displacement valueis much different from the counterpart for zero initial displacement value.First, the displacement jump is larger in the case of linear spring than that of nonlinear spring, but the displacement jump goes across the curve for linear spring with the initial displacement value increasing. Second, when the initial displacement value is not zero, for reflected pulse, the curve for nonlinear spring interface is on the top of the curve for linear spring interface, but for transmitted wave, the curve for linear spring interface is on the top of the curve for nonlinear spring interface. This is opposite for zero initial displacement. The initial displacement value plays a great role in elastic wave reflecting. The results in this paper will be of help to nondestructive evaluation of the interface.

Quantum entanglement represents a fundamental feature of quantum many-body systems. We combine tripartite entanglement with quantum renormalization group theory to study the quantum critical phenomena. The Ising model and the Heisenberg X X Z model in the presence of the Dzyaloshinskii–Moriya interaction are adopted as the research objects. We identify that the tripartite entanglement can signal the critical point. The derivative of tripartite entanglement shows singularity as the spin chain size increases. Furthermore, the intuitive scaling behavior of the system selected is studied and the result allows us to precisely quantify the correlation exponent by utilizing the power law.