We devise schemes for producing, in the least possible time, p identical objects with n agents that work at differing speeds. This involves halting the process to transfer production across agent types. For the case of two types of agent, we construct schemes based on the Euclidean algorithm that seeks to minimize the number of pauses in production.

For microscale heterogeneous partial differential equations (PDEs), this article further develops novel theory and methodology for their macroscale mathematical/asymptotic homogenization. This article specifically encompasses the case of quasi-periodic heterogeneity with finite scale separation: no scale separation limit is required. A key innovation herein is to analyse the ensemble of all phase-shifts of the heterogeneity. Dynamical systems theory then frames the homogenization as a slow manifold of the ensemble. Depending upon any perceived scale separation within the quasi-periodic heterogeneity, the homogenization may be done in either one step or two sequential steps: the results are equivalent. The theory not only assures us of the existence and emergence of an exact homogenization at finite scale separation, it also provides a practical systematic method to construct the homogenization to any specified order. For a class of heterogeneities, we show that the macroscale homogenization is potentially valid down to lengths which are just twice that of the microscale heterogeneity! This methodology complements existing well-established results by providing a new rigorous and flexible approach to homogenization that potentially also provides correct macroscale initial and boundary conditions, treatment of forcing and control, and analysis of uncertainty.

We construct a new stochastic interest rate model with two stochastic factors, by introducing a stochastic long-run equilibrium level into the Vasicek interest rate model which follows another Ornstein–Uhlenbeck process. With the interest rate under the Black–Scholes model being assumed to follow the newly proposed model, a closed-form representation of European option prices is successfully presented, when the analytical characteristic function of the underlying log-price under a forward measure is derived. To assess the model performance, a preliminary empirical study is conducted using S&P 500 index and its options, with the Vasicek model and an alternative two-factor Vasicek model taken as benchmarks.

Clustering is a method of allocating data points in various groups, known as clusters, based on similarity. The notion of expressing similarity mathematically and then maximizing it (minimize dissimilarity) can be formulated as an optimization problem. Spectral clustering is an example of such an approach to clustering, and it has been successfully applied to visualization of clustering and mapping of points into clusters in two and three dimensions. Higher dimension problems remained untouched due to complexity and, most importantly, lack of understanding what “similarity” means in higher dimensions. In this paper, we apply spectral clustering to long timeseries EEG (electroencephalogram) data. We developed several models, based on different similarity functions and different approaches for spectral clustering itself. The results of the numerical experiment demonstrate that the created models are accurate and can be used for timeseries classification.

We find solutions that describe the levelling of a thin fluid film, comprising a non-Newtonian power-law fluid, that coats a substrate and evolves under the influence of surface tension. We consider the evolution from periodic and localized initial conditions as separate cases. The particular (similarity) solutions in each of these two cases exhibit the generic property that the profiles are weakly singular (that is, higher-order derivatives do not exist) at points where the pressure gradient vanishes. Numerical simulations of the thin film equation, with either periodic or localized initial condition, are shown to approach the appropriate particular solution.

Many industrial design problems are characterized by a lack of an analytical expression defining the relationship between design variables and chosen quality metrics. Evaluating the quality of new designs is therefore restricted to running a predetermined process such as physical testing of prototypes. When these processes carry a high cost, choosing how to gather further data can be very challenging, whether the end goal is to accurately predict the quality of future designs or to find an optimal design. In the multi-fidelity setting, one or more approximations of a design’s performance are available at varying costs and accuracies. Surrogate modelling methods have long been applied to problems of this type, combining data from multiple sources into a model which guides further sampling. Many challenges still exist; however, the foremost among them is choosing when and how to rely on available low-fidelity sources. This tutorial-style paper presents an introduction to the field of surrogate modelling for multi-fidelity expensive black-box problems, including classical approaches and open questions in the field. An illustrative example using Australian elevation data is provided to show the potential downfalls in blindly trusting or ignoring low-fidelity sources, a question that has recently gained much interest in the community.