Abstract In this paper, we consider a reaction-diffusion equation that models the time-almost periodic response to climate change within a straight, infinite cylindrical domain. The shifting edge of the habitat is characterised by a time-almost periodic function, reflecting the varying pace of environmental changes. Note that the principal spectral theory is an important role to study the dynamics of reaction-diffusion equations in time heterogeneous environment. Initially, for time-almost periodic parabolic equations in finite cylindrical domains, we develop the principal spectral theory of such equations with mixed Dirichlet–Neumann boundary conditions. Subsequently, we demonstrate that the approximate principal Lyapunov exponent serves as a definitive threshold for species persistence versus extinction. Then, the existence, exponential decay and stability of the forced wave solutions $U(t,x_{1},y)=V\left (t,x_{1}-\int ^{t}_{0}c(s)ds,y\right )$ are established. Additionally, we analyse how fluctuations in the shifting speed affect the approximate top Lyapunov exponent.

Abstract We extend a classical model of continuous opinion formation to explicitly include an age-structured population. We begin by considering a stochastic differential equation model which incorporates ageing dynamics and birth/death processes, in a bounded confidence type opinion formation model. We then derive and analyse the corresponding mean field partial differential equation and compare the complex dynamics on the microscopic and macroscopic levels using numerical simulations. We rigorously prove the existence of stationary states in the mean field model, but also demonstrate that these stationary states are not necessarily unique. Finally, we establish connections between this and other existing models in various scenarios.

Abstract Due to the widespread availability of effective antiretroviral therapy regimens, average lifespans of persons with HIV (PWH) in the United States have increased significantly in recent decades. In turn, the demographic profile of PWH has shifted. Older persons comprise an ever-increasing percentage of PWH, with this percentage expected to further increase in the coming years. This has profound implications for HIV treatment and care, as significant resources are required not only to manage HIV itself, but also associated age-related comorbidities and health conditions that occur in ageing PWH. Effective management of these challenges in the coming years requires accurate modelling of the PWH age structure. In the present work, we introduce several novel mathematical approaches related to this problem. We present a workflow combining a PDE model for the PWH population age structure, where publicly available HIV surveillance data are assimilated using the Ensemble Kalman Inversion algorithm. This procedure allows us to rigorously reconstruct the age-dependent mortality trends for PWH over the last several decades. To project future trends, we introduce and analyse a novel variant of the dynamic mode decomposition (DMD), nonnegative DMD. We show that nonnegative DMD provides physically consistent projections of mortality and HIV diagnosis while remaining purely data-driven, and not requiring additional assumptions. We then combine these elements to provide forecasts for future trends in PWDH mortality and demographic evolution in the coming years.

Abstract In recent years, there has been significant interest in the effect of different types of adversarial perturbations in data classification problems. Many of these models incorporate the adversarial power, which is an important parameter with an associated trade-off between accuracy and robustness. This work considers a general framework for adversarially perturbed classification problems, in a large data or population-level limit. In such a regime, we demonstrate that as adversarial strength goes to zero that optimal classifiers converge to the Bayes classifier in the Hausdorff distance. This significantly strengthens previous results, which generally focus on $L^1$ -type convergence. The main argument relies upon direct geometric comparisons and is inspired by techniques from geometric measure theory.

Abstract The unprecedented success of deep learning (DL) makes it unchallenged when it comes to classification problems. However, it is well established that the current DL methodology produces universally unstable neural networks (NNs). The instability problem has caused a substantial research effort – with a vast literature on so-called adversarial attacks – yet there has been no solution to the problem. Our paper addresses why there has been no solution to the problem, as we prove the following: any training procedure based on training rectified linear unit (ReLU) neural networks for classification problems with a fixed architecture will yield neural networks that are either inaccurate or unstable (if accurate) – despite the provable existence of both accurate and stable neural networks for the same classification problems. The key is that the stable and accurate neural networks must have variable dimensions depending on the input, in particular, variable dimensions is a necessary condition for stability. Our result points towards the paradox that accurate and stable neural networks exist; however, modern algorithms do not compute them. This yields the question: if the existence of neural networks with desirable properties can be proven, can one also find algorithms that compute them? There are cases in mathematics where provable existence implies computability, but will this be the case for neural networks? The contrary is true, as we demonstrate how neural networks can provably exist as approximate minimisers to standard optimisation problems with standard cost functions; however, no randomised algorithm can compute them with probability better than $1/2$ .

Abstract The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

Abstract We develop a high-order asymptotic expansion for the mean first passage time (MFPT) of the capture of Brownian particles by a small elliptical trap in a bounded two-dimensional region. This new result describes the effect that trap orientation plays on the capture rate and extends existing results that give information only on the role of trap position on the capture rate. Our results are validated against numerical simulations that confirm the accuracy of the asymptotic approximation. In the case of the unit disk domain, we identify a bifurcation such that the high-order correction to the global MFPT (GMFPT) is minimized when the trap is orientated in the radial direction for traps centred at $0\lt r\lt r_c :=\sqrt {2-\sqrt {2}}$ . When centred at position $r_c\lt r\lt 1$ , the GMFPT correction is minimized by orientating the trap in the angular direction. In the scenario of a general two-dimensional geometry, we identify the orientation that minimizes the GMFPT in terms of the regular part of the Neumann Green’s function. This theory is demonstrated on several regular domains such as disks, ellipses and rectangles.

Abstract In this paper we study the class of optimal entropy-transport problems introduced by Liero, Mielke and Savaré in Inventiones Mathematicae 211 in 2018. This class of unbalanced transport metrics allows for transport between measures of different total mass, unlike classical optimal transport where both measures must have the same total mass. In particular, we develop the theory for the important subclass of semi-discrete unbalanced transport problems, where one of the measures is diffuse (absolutely continuous with respect to the Lebesgue measure) and the other is discrete (a sum of Dirac masses). We characterize the optimal solutions and show they can be written in terms of generalized Laguerre diagrams. We use this to develop an efficient method for solving the semi-discrete unbalanced transport problem numerically. As an application, we study the unbalanced quantization problem, where one looks for the best approximation of a diffuse measure by a discrete measure with respect to an unbalanced transport metric. We prove a type of crystallization result in two dimensions – optimality of a locally triangular lattice with spatially varying density – and compute the asymptotic quantization error as the number of Dirac masses tends to infinity.