We consider time-inhomogeneous ordinary differential equations (ODEs) whose parameters are governed by an underlying ergodic Markov process. When this underlying process is accelerated by a factor $\varepsilon^{-1}$, an averaging phenomenon occurs and the solution of the ODE converges to a deterministic ODE as $\varepsilon$ vanishes. We are interested in cases where this averaged flow is globally attracted to a point. In that case, the equilibrium distribution of the solution of the ODE converges to a Dirac mass at this point. We prove an asymptotic expansion in terms of $\varepsilon$ for this convergence, with a somewhat explicit formula for the first-order term. The results are applied in three contexts: linear Markov-modulated ODEs, randomized splitting schemes, and Lotka–Volterra models in a random environment. In particular, as a corollary, we prove the existence of two matrices whose convex combinations are all stable but are such that, for a suitable jump rate, the top Lyapunov exponent of a Markov-modulated linear ODE switching between these two matrices is positive.

Three experiments investigated individuals’ preferences and affective reactions to negative life experiences. Participants had a more intense negative affective reaction when they were exposed to a highly negative life experience than when they were exposed to two negative events: a highly negative and a mildly negative life event. Participants also chose the situation containing two versus one negative event. Thus, “more negative events were better” when the events had different affective intensities. When participants were exposed to events having similar affective intensities, however, two negative events produced a more intense negative affective reaction. In addition, participants chose the situation having one versus two negative life experiences. Thus, “more negative events were worse” when the events had similar affective intensities. These results are consistent with an averaging/summation (A/S) model and delineate situations when “more” negative life events are “better” and when “more” negative life events are “worse.” Results also ruled out several alternative interpretations including the peak-end rule and mental accounting interpretations.

This paper investigates the boundaries of the recent result that eliciting more than one estimate from the same person and averaging these can lead to accuracy gains in judgment tasks. It first examines its generality, analysing whether the kind of question being asked has an effect on the size of potential gains. Experimental results show that the question type matters. Previous results reporting potential accuracy gains are reproduced for year-estimation questions, and extended to questions about percentage shares. On the other hand, no gains are found for general numerical questions. The second part of the paper tests repeated judgment sampling’s practical applicability by asking judges to provide a third and final answer on the basis of their first two estimates. In an experiment, the majority of judges do not consistently average their first two answers. As a result, they do not realise the potential accuracy gains from averaging.

A new two-phase model for concentrated suspensions is derived that incorporates a constitutive law combining the rheology for non-Brownian suspension and granular flow. The resulting model exhibits a yield-stress behaviour for the solid phase depending on the collision pressure. This property is investigated for the simple geometry of plane Poiseuille flow, where an unyielded or jammed zone of finite width arises in the centre of the channel. For the steady states of this problem, the governing equations are reduced to a boundary value problem for a system of ordinary differential equations and the conditions for existence of solutions with jammed regions are investigated using phase-space methods. For the general time-dependent case a new drift-flux model is derived using matched asymptotic expansions that takes into account the boundary layers at the walls and the interface between the yielded and unyielded region. The drift-flux model is used to numerically study the dynamic behaviour of the suspension flow, including the appearance and evolution of an unyielded or jammed regions.

Asymptotic homogenisation via the method of multiple scales is considered for problems in which the microstructure comprises inclusions of one material embedded in a matrix formed from another. In particular, problems are considered in which the interface conditions include a global balance law in the form of an integral constraint; this may be zero net charge on the inclusion, for example. It is shown that for such problems care must be taken in determining the precise location of the interface; a naive approach leads to an incorrect homogenised model. The method is applied to the problems of perfectly dielectric inclusions in an insulator, and acoustic wave propagation through a bubbly fluid in which the gas density is taken to be negligible.

In this paper we introduce discrete-time semi-Markov random evolutions (DTSMREs) and study asymptotic properties, namely, averaging, diffusion approximation, and diffusion approximation with equilibrium by the martingale weak convergence method. The controlled DTSMREs are introduced and Hamilton–Jacobi–Bellman equations are derived for them. The applications here concern the additive functionals (AFs), geometric Markov renewal chains (GMRCs), and dynamical systems (DSs) in discrete time. The rates of convergence in the limit theorems for DTSMREs and AFs, GMRCs, and DSs are also presented.

Motivated by a problem in neural encoding, we introduce an adaptive (or real-time) parameter estimation algorithm driven by a counting process. Despite the long history of adaptive algorithms, this kind of algorithm is relatively new. We develop a finite-time averaging analysis which is nonstandard partly because of the point process setting and partly because we have sought to avoid requiring mixing conditions. This is significant since mixing conditions often place restrictive history-dependent requirements on algorithm convergence.

With the advent of dense sensor arrays (64–256 channels) in electroencephalography and magnetoencephalography studies, the probability increases that some recording channels are contaminated by artifact. If all channels are required to be artifact free, the number of acceptable trials may be unacceptably low. Precise artifact screening is necessary for accurate spatial mapping, for current density measures, for source analysis, and for accurate temporal analysis based on single-trial methods. Precise screening presents a number of problems given the large datasets. We propose a procedure for statistical correction of artifacts in dense array studies (SCADS), which (1) detects individual channel artifacts using the recording reference, (2) detects global artifacts using the average reference, (3) replaces artifact-contaminated sensors with spherical interpolation statistically weighted on the basis of all sensors, and (4) computes the variance of the signal across trials to document the stability of the averaged waveform. Examples from 128-channel recordings and from numerical simulations illustrate the importance of careful artifact review in the avoidance of analysis errors.

A stochastic gradient descent method is combined with a consistent auxiliary estimate to achieve global convergence of the recursion. Using step lengths converging to zero slower than 1/n and averaging the trajectories, yields the optimal convergence rate of 1/√n and the optimal variance of the asymptotic distribution. Possible applications can be found in maximum likelihood estimation, regression analysis, training of artificial neural networks, and stochastic optimization.