We describe the slow evolution of the wave speed and reaction temperature in a model of filtration combustion. In the counterflow configuration of the process, a porous solid matrix is converted to a porous solid product by injecting an oxidizing gas at high pressure into one end of a fresh sample of the solid while igniting it at the other end. The solid and gas react exothermically at high activation energy and, under favourable conditions, a self-sustaining combustion wave travels along the sample, converting reactants to product. Since the reaction rate depends on the gas pressure p in the pores, small gradients in p cause variations in the conditions of combustion, which, in turn, cause inhomogeneities in the physical properties of the product. We determine the slow evolution of the wave speed, the reaction temperature, and the mass flux of the gas downstream of the reaction zone. In the absence of a pressure gradient, there is a branch of steadily propagating solutions which has a fold. For planar disturbances on the slow time scale, we show that the middle part of the branch is unstable, with the change of stability occurring at the turning points of the branch. When the pressure gradient is nonzero, there are no steadily propagating solutions and the wave continually evolves. Conditions on the state of the gas at the inlet are described such that the variation in the wave speed and reaction temperature throughout the process can be minimized.

Non-smooth approximations of steep sigmoidal switching networks, such as those used as qualitative models of gene regulation, lead to analytic and computational challenges that arise as a result of the discontinuities in the vector fields. In order to highlight the need for care in dealing with such systems, several particular phenomena are presented here through illustrative examples, including ‘Zeno breaking’, or computing beyond the finite time convergence of an infinite sequence of threshold transitions; the ‘Contact’ effect, in which in the discontinuous limit, trajectories can pass through a ‘saddle point’ without stopping, though these solutions are not unique and other solutions stop for arbitrary time intervals; and sensitive behaviour that arises from exotic dynamics within switching regions.

A two-dimensional computational study is performed on the texturing of fibre-filled nematic liquid crystals using the Landau-de Gennes model describing the spatio-temporal evolution of the second moment of the orientation distribution function or quadrupolar tensor order parameter. The investigation is performed on a consistent computational domain comprising a square array of four circular fibres embedded within a unit square containing a uniaxial low molar mass calamitic liquid crystal. Interest is focused on the role of temperature, boundary conditions and their effect on the nucleation and evolution of defect structures. Thermal effects are characterised below and above the temperature at which the nematic state is stable. Simulations in the stable nematic state serves as a scenario for investigating the effect of imposing different external boundary conditions, namely periodic and Dirichlet; the former describes a square lattice array of fibres embedded in a nematic liquid crystal, and the latterdescribes a four-fibre arrangement in an aligned nematic material. In each case, the influence of temperature is characterised, with defect structures forming and either remaining or splitting into lower strength defects. For fibre lattices, splitting transitions of defects at the centre of the domain occur at a critical temperature, but for the four-fibre arrangement, defect transitions occur continuously over a temperature range. The discontinuous defect splitting transition in fibre arrays occurs at lower temperatures than the continuous defect transformation in the four-fibre arrangement. At sufficiently low temperatures, the four-fibre arrangement and the fibre lattice give the same texture consisting of two disclination lines close to each fibre. The evolution of the texture with respect to temperature can be characterised as a change from single-fibre mode at low temperature to a collective mode with a centre-located heterogeneity at higher temperature. At higher temperatures, in the stable isotropic state, it is shown that surface-induced ordering arising from the fibre/liquid crystal interaction propagates into the bulk forming thin disclination lattices around the four-fibre configuration.

For the Cauchy problem for the nonlinear infiltration equation $$\left\{\begin{array}{@{}l@{\qquad}l} u_{t}=\frac{1}{m}(u^{m})_{xx},&x\in{\mathbb{R}}, t>0,m\geq{}1,\\[3pt] u|_{t=0}=u_{0}(x),&x\in{\mathbb{R}}, \end{array} \right.$$ we use its linear solution $u(x,t,1)$ to approach the nonlinear solution $u(x,t,m)$, and obtain the explicit estimate: $$\int_{0}^{T}\int_{\mathbb{R}}|u(x,t,m)-u(x,t,1)|^{2}\,dx\,dt{} \leq{}(C^{\ast}(m-1))^{2},$$ where $C^{\ast}=O(T^{\gamma})$ and $\gamma=\frac{1+m-\alpha}{2(1+m)}$ for any $0<\alpha<1$.

Using the two-scale convergence approach, we derive equations which govern transversal time-harmonic waves through a layered medium taking the form of a poroelastic composite saturated with a viscous fluid. To improve convergence, we construct a corrector. We study how wave speed and attenuation time depend on porosity and frequency. We prove that the Darcy permeability and the acoustic permeability in the Biot equations do not coincide.

A mathematical model is presented for the washing of a band, or tow, of fibres which is pulled past a submerged obstacle. A lubrication flow between the obstacle and the tow is coupled to a flow through the tow, which is modelled as a porous medium. Coupled first-order ordinary differential equations for the fluid pressure and volume flux are solved numerically and the results are analysed. The washing of the tow induced by the reversal of the viscous boundary layers lying above and below the tow is discussed briefly. In Part 2, the model is extending to allow for fibre bending. The numerical solution of the resulting differential equation shows that at a critical value of the tow porosity, the tow touches the obstacle. An existence proof for a family of touching solutions is obtained. Using the existence of these solutions, a method for constructing a solution with a rubbing region, where the gap width is zero, is outlined. The numerical solutions for these solutions are also given.

The connection between Residual Neural Networks (ResNets) and continuous-time control systems (known as NeurODEs) has led to a mathematical analysis of neural networks, which has provided interesting results of both theoretical and practical significance. However, by construction, NeurODEs have been limited to describing constant-width layers, making them unsuitable for modelling deep learning architectures with layers of variable width. In this paper, we propose a continuous-time Autoencoder, which we call AutoencODE, based on a modification of the controlled field that drives the dynamics. This adaptation enables the extension of the mean-field control framework originally devised for conventional NeurODEs. In this setting, we tackle the case of low Tikhonov regularisation, resulting in potentially non-convex cost landscapes. While the global results obtained for high Tikhonov regularisation may not hold globally, we show that many of them can be recovered in regions where the loss function is locally convex. Inspired by our theoretical findings, we develop a training method tailored to this specific type of Autoencoders with residual connections, and we validate our approach through numerical experiments conducted on various examples.

Within the online media universe, there are many underlying communities. These may be defined, for example, through politics, location, health, occupation, extracurricular interests or retail habits. Government departments, charities and commercial organisations can benefit greatly from insights about the structure of these communities; the move to customer-centred practices requires knowledge of the customer base. Motivated by this issue, we address the fundamental question of whether a sub-network looks like a collection of individuals who have effectively been picked at random from the whole, or instead forms a distinctive community with a new, discernible structure. In the former case, to spread a message to the intended user base it may be best to use traditional broadcast media (TV, billboard), whereas in the latter case a more targeted approach could be more effective. In this work, we therefore formalise a concept of testing for sub-structure and apply it to social interaction data. First, we develop a statistical test to determine whether a given sub-network (induced sub-graph) is likely to have been generated by sampling nodes from the full network uniformly at random. This tackles an interesting inverse alternative to the more widely studied “forward” problem. We then apply the test to a Twitter reciprocated mentions network where a range of brand name based sub-networks are created via tweet content. We correlate the computed results against the independent views of 16 digital marketing professionals. We conclude that there is great potential for social media based analytics to quantify, compare and interpret online brand allegiances systematically, in real time and at large scale.

The behaviour of two-dimensional finite blobs of conducting viscous fluid in a Hele-Shaw cell subject to an electric field is considered. The time-dependent free boundary problem is studied both analytically using the Schwarz function of the free boundary and numerically using a boundary integral method. Various problems are considered, including (i) the behaviour of an initially circular blob of conducting fluid subject to an electric point charge located arbitrarily within the blob, (ii) the delay in cusp formation on the free boundary in sink-driven flow due to a strategically placed electric charge and (iii) the stability of exact steady solutions having both hydrodynamic and electric forcing.

Steady-state diffusion in long axisymmetric structures is considered. The goal is to assess one-dimensional approximations by comparing them with axisymmetric eigenfunction expansions. Two problems are considered in detail: a finite tube with one end that is partly absorbing and partly reflecting; and two finite coaxial tubes with different cross-sectional radii joined together abruptly. Both problems may be modelled using effective boundary conditions, containing a parameter known as the trapping rate. We show that trapping rates depend on the lengths of the finite tubes (and that they decay slowly as these lengths increase) and we show how trapping rates are related to blockage coefficients, which are well known in the context of potential flow along tubes of infinite length.

Short-time existence and uniqueness results in Sobolev spaces are proved for Hele-Shaw flow with kinetic undercooling and for Stokes flow without surface tension. In both cases, the flow is driven by arbitrarily distributed sources and sinks in the interior of the liquid domain. The proofs are based on a general approach consisting of the reformulation of the problem as a Cauchy problem for a nonlinear, nonlocal evolution equation on the unit sphere, quasilinearization by equivariance, investigation of the linearization, and Galerkin approximations. In the situation discussed here, the linearized evolution operator is a first-order differential operator, and thus the evolution equation is of hyperbolic type. Finally, a brief survey of the properties of the evolution equations that arise from Hele-Shaw flow and Stokes flow with and without regularization is given.

A spectral method is developed to study the steady and unsteady flow of fluid into a line sink from a horizontally confined aquifer, and the results are compared to solutions obtained implementing the finite element package COMSOLTM. The aquifer or drain is considered to be confined below so that the solutions are fundamentally unsteady. Comparison is made between the two methods in determining the drawdown of the surface.

This paper addresses a free boundary problem for a steady, uniform patch of vorticity surrounding a single flat plate of zero thickness and finite length. Exact solutions to this problem have previously been found in terms of conformal maps represented by Cauchy-type integrals. Here, however, it is demonstrated how, by considering an associated circular-arc polygon and using ideas from automorphic function theory, these maps can be expressed in a simple non-integral form.

We study a problem related to the spin-coating process in which a fluid coats a rotating surface. Our interest lies in the contact-line region for which we propose a simplified travelling wave approximation. We construct solutions to this problem by a shooting method that matches solution branches in the contact-line region and in the interior of the droplet. Furthermore, we prove uniqueness and qualitative properties of the solution connected to the fourth-order nature of the equation, such as a global maximum in the film height close to the contact line, elevated from the average height of the film.

Boundary value problems for linear transport equations sometimes require the explicit construction of solutions when boundary conditions are prescribed only on parts of the boundary and thus necessitate the construction of half-range expansions. In contrast to the standard eigenfunction expansion, a half-range expansion of a given function must be reconstructed just over half the domain, using just half of the eigenfunctions. The difficulty of such expansions arises because the eigenfunctions are not orthogonal, though they are complete, over half the domain, and there is no obvious method of obtaining the expansion coefficients. Here we use complex variable techniques to find explicit formulas for the coefficients of half-range expansions for regular, negative definite Sturm–Liouville operators. We prove that the half-range expansion formula is unique, and find the corresponding half-range Green functions.

Consensus-based optimisation (CBO) is a versatile multi-particle metaheuristic optimisation method suitable for performing non-convex and non-smooth global optimisations in high dimensions. It has proven effective in various applications while at the same time being amenable to a theoretical convergence analysis. In this paper, we explore a variant of CBO, which incorporates truncated noise in order to enhance the well-behavedness of the statistics of the law of the dynamics. By introducing this additional truncation in the noise term of the CBO dynamics, we achieve that, in contrast to the original version, higher moments of the law of the particle system can be effectively bounded. As a result, our proposed variant exhibits enhanced convergence performance, allowing in particular for wider flexibility in choosing the noise parameter of the method as we confirm experimentally. By analysing the time evolution of the Wasserstein-$2$ distance between the empirical measure of the interacting particle system and the global minimiser of the objective function, we rigorously prove convergence in expectation of the proposed CBO variant requiring only minimal assumptions on the objective function and on the initialisation. Numerical evidences demonstrate the benefit of truncating the noise in CBO.

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

An analysis is made of the damping of sound and structural vibrations by vorticity production in the apertures of a bias flow, perforated elastic plate. Unsteady motion causes vorticity to be generated at the aperture edges; the vorticity and its energy are swept away by the bias flow and result in a net loss of acoustic and vibrational energy. In this paper we investigate the interaction of an arbitrary fluid-structure disturbance with a small circular aperture in the presence of a high Reynolds number, low Mach number bias flow. By considering the limit in which the aperture is small compared to the length scale of the impinging disturbance, it is shown that the effect of the interaction can be represented by a concentrated source in the plate bending wave equation consisting of a delta function and two of its axisymmetric derivatives. A generalized bending wave equation is then formulated for a plate perforated with an homogeneous distribution of small, bias flow circular apertures. This equation is used to predict the attenuation of sound and resonant bending waves by vorticity production. Acoustic damping is found to be significant provided the fluid loading is sufficiently small for the plate to be regarded as rigid (e.g. for an aluminium plate in air when the frequency is not too small). On the other hand, a bending wave is effectively damped only when the fluid loading is large enough for the wave to produce a substantial pressure drop across the plate; when this occurs the predicted attenuations are comparable with those usually achieved by the application of elastomeric damping materials. Numerical predictions are presented for steel and aluminium plates in air and water.

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

Finite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to $\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space $\textbf{H}^s(\text{curl};\,\Omega)$, and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for $\textbf{H}(\text{curl})$-elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the $H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments.