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In recent decades, nano/microelectromechanical systems (N/MEMS) have garnered significant attention due to their appealing characteristics, such as compact size, batch fabrication capabilities, high reliability and low power consumption. However, these vibratory systems often present challenges, including zero conditions at the initial time, involving zero velocity and zero displacement, which complicates the solution process. Nonetheless, the theory of geometric potential offers insights into various phenomena in nanoscience and nanotechnology. In this paper, we implement the theory of geometric potential to develop an N/MEMS model. We then analyze the periodicity property of the nonlinear system using a novel method based on Sturm’s algorithm. Our analysis reveals that the model with zero initial conditions exhibits periodic solution under specific conditions on lumped parameter. Finally, we validate our findings by comparing them with numerically achieved results.

In this study, we considered skew curvatures of the surfaces to generate their geometric potentials. The method depends essentially on the mean and Gaussian curvatures and their principal curvatures. In quantum mechanics in the study of the dynamics of massive particle with mass $m$ constrained to move on a surface. In such a case, the difference function of the squared mean curvature with the Gaussian curvature induces a geometric (scalar) potential. This potential appears in the Schröndiger-type equations. Considering the skew curvatures of the rotational surfaces, some results on the meridian curves are obtained. Furthermore, the geometric potentials of level surfaces and generalized helicoidal surfaces are calculated. Finally, we discuss the some applications of these types of surfaces in quantum mechanics.

Interest on (2+1)-dimensional electron systems has increased considerably after the realization of novel properties of graphene sheets, in which the behavior of electrons is effectively described by relativistic equations. Having this fact in mind, the following problem is studied in this work: when a spin-1/2 particle is constrained to move on a curved surface, is it possible to describe this particle without giving reference to the dimensions external to the surface? As a special case of this, a relativistic spin-1/2 particle which is constrained to move on a (2+1)-dimensional hypersurface of the (3+1)-dimensional Minkowskian spacetime is considered, and an effective Dirac equation for this particle is derived using the so-called thin layer method. Some of the results are compared with those obtained in a previous work by Burgess and Jensen.

Electrospinning is an amazing process that enables the production of primarily nanofiber membranes. Nevertheless, the mechanism underlying the formation of two-dimensional (2D) nonwoven structure remains a fascinating enigma. This paper presents a novel morphology of nanofiber membranes with a cluster structure that bears resemblance to the solar nebula. The mechanism of nanofiber cluster structures in the electrospinning process is elucidated through the geometric potential theory, which begins with a zero-dimensional receptor, progresses to a one-dimensional (1D) receptor, a 2D receptor, and ultimately culminates in a fractal-like lattice receptor. Moreover, a three-dimensional (3D) printing-like novel approach is presented for the manipulation of nanofiber membrane’s shape through the use of a lattice-structured receptor. The fabrication process of the cluster structure nanofiber membranes by electrospinning is presented, and their remarkable potential applications are discussed.

In recent years, there has been a considerable increase of interest in the sixth-generation (6G) communications. The advent of 6G wireless communications will facilitate a comprehensive upgrade and transformation of the industry, enabling higher data rates, lower latency, and ubiquitous connectivity. Research on microelectromechanical systems (MEMS) is a crucial element in the advancement of 6G wireless communications. In this paper, we examine the impact of geometric potential on MEMS within the context of fractal space, where a fractal geometric potential MEMS model is proposed and the critical conditions that govern the pull-in phenomenon are elucidated. Moreover, a periodic solution of the model is proposed and compared with other numerical methods.