The process of sleep stage identification is a labour-intensive task that involves the specialized interpretation of the polysomnographic signals captured from a patient’s overnight sleep session. Automating this task has proven to be challenging for data mining algorithms because of noise, complexity and the extreme size of data. In this paper we apply nonsmooth optimization to extract key features that lead to better accuracy. We develop a specific procedure for identifying K-complexes, a special type of brain wave crucial for distinguishing sleep stages. The procedure contains two steps. We first extract “easily classified” K-complexes, and then apply nonsmooth optimization methods to extract features from the remaining data and refine the results from the first step. Numerical experiments show that this procedure is efficient for detecting K-complexes. It is also found that most classification methods perform significantly better on the extracted features.

There are many ways to define how long diffusive processes take, and an appropriate “critical time” is highly dependent on the specific application. In particular, we are interested in diffusive processes through multilayered materials, which have applications to a wide range of areas. Here we perform a comprehensive comparison of six critical time definitions, outlining their strengths, weaknesses, and potential applications. A further four definitions are also briefly considered. Equivalences between appropriate definitions are determined in the asymptotic limit as the number of layers becomes large. Relatively simple approximations are obtained for the critical time definitions. The approximations are more accessible than inverting the analytical solution for time, and surprisingly accurate. The key definitions, their behaviour and approximations are summarized in tables.

The effect of initial stresses on incident quasi SV-waves at a plane interface between two dissimilar pre-stressed elastic half-spaces is investigated. The reflection and refraction coefficients of the reflected and refracted qSV- and qP-waves are derived with the help of appropriate boundary conditions. The coefficients are found to be functions of the angle of incidence and the initial stresses and incremental elastic parameters of the pre-stressed elastic half-spaces.

This paper considers singular systems that involve both continuous dynamics and discrete events with the coefficients being modulated by a continuous-time Markov chain. The underlying systems have two distinct characteristics. First, the systems are singular, that is, characterized by a singular coefficient matrix. Second, the Markov chain of the modulating force has a large state space. We focus on stability of such hybrid singular systems. To carry out the analysis, we use a two-time-scale formulation, which is based on the rationale that, in a large-scale system, not all components or subsystems change at the same speed. To highlight the different rates of variation, we introduce a small parameter ε>0. Under suitable conditions, the system has a limit. We then use a perturbed Lyapunov function argument to show that if the limit system is stable then so is the original system in a suitable sense for ε small enough. This result presents a perspective on reduction of complexity from a stability point of view.