Early warning for epilepsy patients is crucial for their safety and well being, in particular, to prevent or minimize the severity of seizures. Through the patients’ electroencephalography (EEG) data, we propose a meta learning framework to improve the prediction of early ictal signals. The proposed bilevel optimization framework can help automatically label noisy data at the early ictal stage, as well as optimize the training accuracy of the backbone model. To validate our approach, we conduct a series of experiments to predict seizure onset in various long-term windows, with long short-term memory (LSTM) and ResNet implemented as the baseline models. Our study demonstrates that not only is the ictal prediction accuracy obtained by meta learning significantly improved, but also the resulting model captures some intrinsic patterns of the noisy data that a single backbone model could not learn. As a result, the predicted probability generated by the meta network serves as a highly effective early warning indicator.

We first derive Alber’s equation for the Wigner distribution function using the fourth-order nonlinear Schrödinger equation, and on the basis of this equation we next analyse the stability of the narrowband approximation of the Joint North Sea Wave Project spectrum. Therefore, one interesting result of this study concerns the effect of modulational instability obtained from the fourth-order nonlinear Schrödinger equation. The analysis is restricted to one horizontal direction, parallel to the direction of wave motion, to take advantage of potential flow theory. We find that shear currents considerably modify the instability behaviours of weakly nonlinear waves. The key point of this study is that the present fourth-order analysis shows considerable deviations in the modulational instability properties from the third-order analysis and reduces the growth rate of instability. Moreover, we present here a connection between the random and deterministic properties of a random wavetrain for vanishing spectrum bandwidth.

The hydroelastic interaction between water waves and multiple submerged porous elastic plates of arbitrary lengths in deep water is examined using the Galerkin approximation technique. We observe the influence of flexible porous plates of arbitrary lengths by analysing the reflection coefficient, dissipated energy and wave forces acting on the plates. Results are presented for various values of angle of incidence, separation lengths of plates, porosity levels, submergence depth and flexural rigidity. The convergence and accuracy of the method are verified by comparing the results with existing literature. The significant impact of flexural rigidity in the presence of porosity on wave reflection, dissipated energy and wave forces is demonstrated. Moreover, a notable reduction in wave load is observed with an increase in the number of plates.

Recently, we analysed spontaneous symmetry breaking (SSB) of solitons in linearly coupled dual-core waveguides with fractional diffraction and cubic nonlinearity. In a practical context, the system can serve as a model for optical waveguides with the fractional diffraction or Bose–Einstein condensate of particles with Lévy index $\alpha <2$. In an earlier study, the SSB in the fractional coupler was identified as the bifurcation of subcritical type, becoming extremely subcritical in the limit of $\alpha \rightarrow 1$. There, the moving solitons and collisions between them at low speeds were also explored. In the present paper, we present new numerical results for fast solitons demonstrating restoration of symmetry in post-collision dynamics.

Three-dimensional short-crested water waves are known to host harmonic resonances (HRs). Their existence depends on their sporadicity versus their persistency. Previous studies, using a unique yet hybrid solution, suggested that HRs exhibit sporadic instability, with the domain of instability exhibiting a bubble-like structure which experiences a loss of stability followed by a re-stabilization. Through the calculation of their complete multiple solution structures and normal forms, we discuss the particular harmonic resonance (2,6). The (2,6) resonance was chosen, not only because it is of lower order, and thus more likely to be significant, but also because it is representative of a fully developed three-dimensional water wave field. Its appearance, growth rate and persistency are discussed. On our converged solutions, we show that, at an incidence angle for which HR (2,6) occurs, the associated superharmonic instability is no longer sporadic. It was also found that the multiple solution operates a subcritical pitchfork bifurcation, so regardless of the value of the control parameter, the wave steepness, a stable branch of the solution always exists. As a result, the analysis reveals two competing processes that either provoke and enhance HRs, or inhibit their appearance and development.

The study applies a two-dimensional adaptive mesh refinement (AMR) method to estimate the coordinates of the locations of the centre of vortices in steady, incompressible flow around a square cylinder placed within a channel. The AMR method is robust and low cost, and can be applied to any incompressible fluid flow. The considered channel has a blockage ratio of $1/8$. The AMR is tested on eight cases, considering flows with different Reynolds numbers ($5\le Re\le 50$), and the estimated coordinates of the location of the centres of vortices are reported. For all test cases, the initial coarse meshes are refined four times, and the results are in good agreement with the literature where a very fine mesh was used. Furthermore, this study shows that the AMR method can capture the location of the centre of vortices within the fourth refined cells, and further confirms an improvement in the estimation with more refinements.

Complicated option pricing models attract much attention in financial industries, as they produce relatively better accurate values by taking into account more realistic assumptions such as market liquidity, uncertain volatility and so forth. We propose a new hybrid method to accurately explore the behaviour of the nonlinear pricing model in illiquid markets, which is important in financial risk management. Our method is based on the Newton iteration technique and the Fréchet derivative to linearize the model. The linearized equation is then discretized by a differential quadrature method in space and a quadratic trapezoid rule in time. It is observed through computations that the accurate solutions for the model emerge using very few grid points and time elements, compared with the finite difference method in the literature. Furthermore, this method also helps to avoid consideration of the convergence issues of the Newton approach applied to the nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization. It is important to note that the Fréchet derivative supports to enhance the convergence order of the proposed iterative scheme.

The pth ($p\geq 1$) moment exponential stability, almost surely exponential stability and stability in distribution for stochastic McKean–Vlasov equation are derived based on some distribution-dependent Lyapunov function techniques.

We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite-volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.

Assuming linear theory, the phenomenon of scattering of waves by a circular arc shaped barrier with nonuniform porosity is studied. The water region is considered to be of infinite or finite depth. Based on a judicious application of Green’s integral theorem, the corresponding boundary value problem is reduced to a hypersingular integral equation of second kind. The boundary element method and the collocation method are adopted to solve the hypersingular integral equation, and we ensure a good matching of the solutions obtained by the two methods. The reflection coefficient and energy dissipation are evaluated by using the solution of the integral equation which is then studied graphically. Different choices of distributions of pores on the barrier are considered, and we observe that the nonuniform porosity of the barrier has significant effect on the reflected wave and the energy dissipation.

The behaviour of an axisymmetric bubble in a pure liquid forced by an acoustic pressure field is analysed. The bubble is assumed to have a sharp deformable interface, which is subject both to surface tension and to Rayleigh viscosity damping. Two modelling regimes are considered. The first is a linearized solution, based on the assumption of small axisymmetric deformations to an otherwise spherical bubble. The second involves a semi-numerical solution of the fully nonlinear problem, using a novel spectral method of high accuracy. For large-amplitude nonspherical bubble oscillations, the fully nonlinear solutions show that a complicated resonance structure is possible and that curvature singularities may occur at the interface, even in the presence of surface tension. Rayleigh viscosity at the interface prevents singularity formation, but eventually causes the bubble to become purely spherical unless shape-mode resonances occur. An extended analysis is also presented for purely spherical bubbles, which allows for a more detailed study of the effects of resonance and the Rayleigh viscosity at the bubble surface.

We propose a novel time-asymptotically stable, implicit–explicit, adaptive, time integration method (denoted by the $\theta $-method) for the solution of the fractional advection–diffusion-reaction (FADR) equations. The spectral analysis of the method (involving the group velocity and the phase speed) indicates a region of favourable dispersion for a limited range of Péclet number. The numerical inversion of the coefficient matrix is avoided by exploiting the sparse structure of the matrix in the iterative solver for the Poisson equation. The accuracy and the efficacy of the method is benchmarked using (a) the two-dimensional fractional diffusion equation, originally proposed by researchers earlier, and (b) the incompressible, subdiffusive dynamics of a planar viscoelastic channel flow of the Rouse chain melts (FADR equation with fractional time-derivative of order ) and the Zimm chain solution (). Numerical simulations of the viscoelastic channel flow effectively capture the nonhomogeneous regions of high viscosity at low fluid inertia (or the so-called “spatiotemporal macrostructures”), experimentally observed in the flow-instability transition of subdiffusive flows.

We study the influence of a low-frequency harmonic vibration on the formation of the two-dimensional rolling solitary waves in vertically co-flowing two-layer liquid films. The system consists of two adjacent layers of immiscible fluids with the first layer being sandwiched between a vertical solid plate and the second fluid layer. The solid plate oscillates harmonically in the horizontal direction inducing Faraday waves at the liquid–liquid and liquid–air interfaces. We use a reduced hydrodynamic model derived from the Navier–Stokes equations in the long-wave approximation. Linear stability of the base flow in a flat two-layer film is determined semi-analytically using Floquet theory. We consider sub-millimetre-thick films and focus on the competition between the long-wavelength gravity-driven and finite wavelength Faraday instabilities. In the linear regime, the range of unstable wave vectors associated with the gravity-driven instability broadens at low and shrinks at high vibration frequencies. In nonlinear regimes, we find multiple metastable states characterized by solitary-like travelling waves and short pulsating waves. In particular, we find the range of the vibration parameters at which the system is multistable. In this regime, depending on the initial conditions, the long-time dynamics is dominated either by the fully developed solitary-like waves or by the shorter pulsating Faraday waves.

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.