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In this paper, we consider the Fourier spectral method for numerically solving the modified Swift-Hohenberg equation. The semi-discrete and fully discrete schemes are established. Moreover, the existence, uniqueness and the optimal error bound are also considered.

Fluorescence imaging has currently emerged as one of the most frequently used noninvasive imaging technologies to selectively monitor biological processes in living systems. In past decades, gold nanoclusters (Au NCs) has received increasing attraction because of their intrinsic fluorescence and their inherent biocompatibility. As a stabilizing and reducing agent, an abundant, sustainable, and widely used polypeptide derived drug molecule, aprotinin (Ap), is selected for the synthesis of Au nanoclusters (Ap-Au NCs) due to characteristic bioactivity, excellent biocompatibility, biodegradability, and non-allergenic character. Herein, Ap encapsulated Au NCs with desirable red fluorescence was facilely produced for the first time, which were subsequently used for cell imaging and detection of various analytes. Much interestingly, dynamically subcellular targeting  from the cytoplasm to the nucleus in HeLa cells was observed. Besides, it has shown that, the selective and quantitative detection of trypsin has been established by using Ap-Au NCs. Finally, Ap-Au NCs were readily used for quantitative detection of mercury and copper. The photoluminescence of the Ap-Au NCs was quenched with the addition of the aforementioned analytes. This study not only  discusses a multifunctional nanomaterial  for cell imaging, dynamically nuclear targeting and biosensing, but also opens crucial insights on the integration of funtional biomolecule with metal nanoclusters intended for extensively biomedical applications.

In order to improve the versatility of the control method of the four-rotor aircraft, a general mathematical model, the rectangular four-rotor aircraft, is modeled, and two special cases, square cross structure and square X structure, are deduced. Based on the conventional global sliding mode control, an improved global sliding mode control is proposed to control the position and the attitude of the four-rotor aircraft. The dynamic sliding mode surface of the improved global sliding mode control can evolve into the linear sliding mode surface in a limited time by changing the decay function of the dynamic sliding mode surface. In this way, the controlled system has not only the strong global robustness but also the quick response rate. Simulation results show that the position and the attitude of the four-rotor aircraft can be controlled by the improved global sliding mode control, and the control performances, for instance the response speed, can be improved.

In this paper, for a nonlinear differential equation describing crystal surface growth, the finite element method is presented. A nice order error estimates is derived by means of a finite element projection approximation.