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— This paper proposes a method for certifying the local asymptotic stability of a given nonlinear Ordinary Differential Equation (ODE) by using Sum-of-Squares (SOS) programming to search for a partially quadratic Lyapunov Function (LF). The proposed method is particularly well suited to the stability analysis of ODEs with high dimensional state spaces. This is due to the fact that partially quadratic LFs are parametrized by fewer decision variables when compared with general SOS LFs. The main contribution of this paper is using the Center Manifold Theorem to show that partially quadratic LFs that certify the local asymptotic stability of a given ODE exist under certain conditions. 

Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose \begin{document}$ 1 $\end{document}-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.

 

Pure molybdenum disulfide (MoS₂) solid lubricant coatings could attain densities comparable to doped films (and the associated benefits to wear rate and environmental stability) through manipulation of the microstructure via deposition parameters. Unfortunately, pure films can exhibit highly variable microstructures and mechanical properties due to processes that are not controlled during deposition (i.e., batch-to-batch variation). This work focuses on developing a relationship between density, hardness, friction, and wear for pure sputtered MoS₂coatings. Results show that dense films (ρ = 4.5 g/cm³) exhibit a 100 × lower wear rate compared to porous coatings (ρ = 3.04–3.55 g/cm³). The tribological performance of high density pure MoS₂coatings is shown to surpass that of established composite coatings, achieving a wear rate 2 × (k = 5.74 × 10^–8mm³/Nm) lower than composite MoS₂/Sb₂O₃/Au in inert environments.

 

Totaling at 7.4 billion people, the world’s population is rapidly growing, bringing along with it an increase in waste generation. The impact of this exponential increase in waste generation has resulted in the increased formation and utilization of landfills. In the present day, landfills are utilized to dispose of chemical, hazardous, municipal, and electronic wastes. However, despite their convenience, most landfills are improperly managed and face constant changes from the surrounding environment that interfere with their internal landfill processes. The objectives of this mixed review are to highlight the negative impacts landfills have on the environment and public health as well as outline the need for proper management practices to mitigate these effects. Inadequate management of landfills leads to issues concerning leachate collection and landfill gas (LFG) generation, which give rise to groundwater contamination and air pollution. This paper recognizes the disadvantages of utilizing landfills as the main disposal method by focusing on these two primary effects that improper management of landfills has on the environment and human health. Many experts have also reported that communities within close proximity to improperly managed landfills have an increased risk of health issues. Apart from implementing proper landfill management practices, it is important to develop solutions to reduce waste generation altogether. This review discusses some of the innovative methods implemented by other countries to reduce landfill waste and the production of greenhouse gases as well as possible steps individuals can take to minimize their ecological footprints. 

Abstract We report an investigation of the friction mechanisms of MoS 2 thin films under changing environments and contact conditions using a variety of computational and experimental techniques. Molecular dynamics simulations were used to study the effects of water and molecular oxygen on friction and bonding of MoS 2 lamellae during initial sliding. Characterization via photoelectron emission microscopy (PEEM) and Kelvin probe force microscopy (KPFM) were used to determine work function changes in shear modified material within the top few nanometers of MoS 2 wear scars. The work function was shown to change with contact conditions and environment, and shown by density functional theory (DFT) calculations and literature reports to be correlated with lamellae size and thickness of the basally oriented surface layer. Results from nanoscale simulations and macroscale experiments suggest that the evolution of the friction behavior of MoS 2 is linked primarily to the formation or inhibition of a basally oriented, molecularly thin surface film with long-range order.