Assignment flows denote a class of dynamical models for contextual data labelling (classification) on graphs. We derive a novel parametrisation of assignment flows that reveals how the underlying information geometry induces two processes for assignment regularisation and for gradually enforcing unambiguous decisions, respectively, that seamlessly interact when solving for the flow. Our result enables to characterise the dominant part of the assignment flow as a Riemannian gradient flow with respect to the underlying information geometry. We consider a continuous-domain formulation of the corresponding potential and develop a novel algorithm in terms of solving a sequence of linear elliptic partial differential equations (PDEs) subject to a simple convex constraint. Our result provides a basis for addressing learning problems by controlling such PDEs in future work.

The probability density for the solution yn of a stochastic difference equation is considered. Following Knessl et al. [1], it is shown to satisfy a master equation, which is solved asymptotically for large values of the index n. The method is illustrated by deriving the large deviation results for a sum of independent identically distributed random variables and for the joint density of two dependent sums. Then it is applied to a difference approximation to the Helmholtz equation in a random medium. A large deviation result is obtained for the probability density of the decay rate of a solution of this equation. Both the exponent and the pre-exponential factor are determined.

We consider a one-dimensional stochastic model of sediment deposition in which the complete time history of sedimentation is the sum of a linear trend and a fractional Brownian motion wH(t) with self-similarity parameter H ∈ (0, 1). The thickness of the sedimentary layer as a function of time, d(t), looks like the Cantor staircase. The Hausdorff dimension of the points of growth of d(t) is found. We obtain the statistical distribution of gaps in the sedimentary record, periods of time during which the sediments have been eroded. These gaps define sedimentary unconformities. In the case H = 1/2 we obtain the statistical distribution of layer thicknesses between unconformities and investigate the multifractality of d(t). We show that the multifractal structures of d(t) and the local time function of Brownian motion are identical; hence d(t) is not a standard multifractal object. It follows that natural statistics based on local estimates of the sedimentation rate produce contradictory estimates of the range of local dimension for d(t). The physical object d(t) is interesting in that it involves the above anomalies, and also in its mechanism of fractality generation, which is different from the traditional multiplicative process.

Three-dimensional nonstationary flow of a viscous incompressible liquid is investigated in a layer, driven by a nonuniform distribution of temperature on its free boundaries. If the temperature given on the layer boundaries is quadratically dependent on horizontal coordinates, external mass forces are absent, and the motion starts from rest then the free boundary problem for the Navier–Stokes equations has an ‘exact’ solution in terms of two independent variables. Here the free boundaries of the layer remain parallel planes and the distance between them must be also determined. In present paper, we formulate conditions for both the unique solvability of the reduced problem globally in time and the collapse of the solution in finite time. We further study qualitative properties of the solution such as its behaviour for large time (in the case of global solvability of the problem), and the asymptotics of the solution near the collapse moment in the opposite case.

We study a mean-field model of superconducting vortices in one and two dimensions. The existence of a weak solution and a steady-state solution of the model are proved. A special case of the steady-state problem is shown to be of the form of a free boundary problem. The solutions of this free boundary problem are investigated. It is also shown that the weak solution of the one-dimensional model is unique and satisfies an entropy inequality.

Diffusion-driven instability in systems of reaction-diffusion equations is a commonly used model for pattern formation in both embryology and ecology. In ecological applications, model parameters tend to oscillate in time, because of either daily or seasonal fluctuations in the environment. I investigate the effects of such fluctuations on diffusion-driven instability by considering analytically the possibility of Turing bifurcations when the parameter values (diffusion coefficients and kinetic parameters) oscillate in time between two sets of constant values, with a period that is either very short or very long compared to the time scale of the growth and predation kinetics. I show that oscillations in the kinetics can have quite different effects from oscillations in the dispersal terms. I also discuss the comparison between the solution forms predicted by linear theory and the numerical solutions of a simple nonlinear predator-prey model.

A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

We describe and analyze a model of filtration combustion, in which a gas is forced at high pressure into a porous solid matrix so that after ignition and under favourable conditions a combustion wave can propagate through the medium. We consider the case of counter flow, where the gas is forced into the reaction zone through the unreacted part of the porous solid. Relations are derived for the steady state propagation of a planar combustion wave or front in the limit of high-activation energy, from which the propagation speed, reaction temperature, and reacted mass fraction of the solid product can be found in terms of the mass flux of the injected gas, the gas pressure and mass flux on exit from the front, and other physico-chemical parameters describing the system. Two distinct modes of combustion are discussed, corresponding to the reaction being driven to completion by exhaustion of either the gaseous or the solid component, these being referred to as the gas-deficient and solid-deficient modes of burning respectively. For both homogeneous and heterogeneous forms of the reaction rate we find that there is a critical inlet mass flux for the incoming gas below which steady state solutions no longer exist and that there are parameter values for which multiple steady states occur.

We study the singular limit of the dimensionless phase-field equations

We consider two cases: either the space dimension is 1 and then ɛ tends to zero; or the solutions are radially symmetric and then both ɛ and α tend to zero. It turns out that, in the first case, the limiting functions solve the Stefan problem with kinetic undercooling, provided the initial temperature is small compared to the surface tension and the latent heat. In the second case, the limiting functions satisfy the Stefan problem coupled with the Gibbs–Thomson law for the melting temperature. We show, in addition, that the multiplicity of the interface is always one, in a sense to be explained at the end of § 1. As main tool we use energy type estimates, and prove that the formal first-order asymptotic expansion with respect to ɛ in fact gives an approximation of the exact solution. Our results hold without smoothness assumptions on the limiting Stefan problem.

Schölte waves, waves bound to the interface between a fluid and an elastic half-space, are, for many material combinations, evanescent; as they propagate, they are damped due to radiation. A representation of the general evanescent Schölte wave is here obtained in terms of a solution to the membrane equation with complex speed, linked, at each instant, to a complex-valued harmonic function in a half-space. This derivation generalises one obtained recently for (non-evanescent) Rayleigh, Stoneley and Schölte waves. An alternative description is also obtained, in which the time-evolution of the normal displacement of the interface satisfies a first-order, complex-valued, non-local evolution equation. Amongst some explicit solutions obtained are decaying solutions allied to a general solution to the Helmholtz equation, and a solution closely related to a Gaussian beam. In the plane–strain case, the general Schölte wave splits into two disturbances, one right-travelling and one left-travelling, each being described at all times in terms of a harmonic function in a half-plane, decaying with depth yet having arbitrary boundary values. This representation highlights the dual elliptic–hyperbolic nature typical of guided waves and gives a surprisingly compact representation for the two-dimensional case.

We construct an asymptotic representation for the solution $u(x,t)$ of a singularly-perturbed linear fifth-order evolution equation which accounts for the relevant exponentially-small terms in all regions of the complex $x$ plane. The particular equation that we study is chosen in part to highlight the complexities that arise in high-order examples, resulting in particular from the non-existence of a suitable (steady-state) heteroclinic connection. Key points of this calculation are the identification, location and evolution of the active (in the sense that non-zero, though exponentially-small, terms are switched on across them) Stokes lines, and of the higher-order Stokes lines across which these can be activated or inactivated. In doing so, we need in particular to analyse two ‘levels’ of higher-order Stokes lines and to present the associated mechanisms by which they can themselves be activated or inactivated. By piecing together the information concerning which Stokes lines (both ordinary and higher-order) are active, we are able to deduce systematically which of the competing exponentials that can potentially arise within the asymptotic solution are actually present in each region of the complex plane.

We consider the one-dimensional and two-dimensional filtration-absorption equation ut = uΔu−(c−1)(∇u)2. The one-dimensional case was considered previously by Barenblatt et al. [4], where a special class of self-similar solutions was introduced. By the analogy with the 1D case we construct a family of axisymmetric solutions in 2D. We demonstrate numerically that the self-similar solutions obtained attract the solutions of non-self-similar Cauchy problems having the initial condition of compact support. The main analytical result we provide is the linear stability of the above self-similar solutions both in the 1D case and in the 2D case.

We investigate the local behaviour at the boundary singularity for the following moving boundary problems: (i) Hele-Shaw flows in which the interface is initially non-analytic; (ii) power-law Hele-Shaw flows in which the interface contains a corner; (iii) Stefan problems in which the interface contains a corner. Both well-posed (‘injection’) and ill-posed (‘suction’) problems are considered. Related results for corner development in the presence of an impermeable boundary are also noted.

The motion of interfaces for a mass-conserving Allen–Cahn equation that are attached to the boundary of a two-dimensional domain is studied. In the limit of thin interfaces, the interface motion for this problem is known to be governed by an area-preserving mean curvature flow. A numerical front-tracking method, that allows for a numerical solution of this type of curvature flow, is used to compute the motion of interfaces that are attached orthogonally to the boundary. Results obtained from these computations are favourably compared with a previously-derived asymptotic result for the motion of attached interfaces that enclose a small area. The area-preserving mean curvature flow predicts that a semi-circular interface is stationary when it is attached to a flat segment of the boundary. For this case, the interface motion is shown to be metastable and an explicit characterization of the metastability is given.

An equation governing the evolution of a Hele-Shaw free boundary flow in the presence of an arbitrary external potential – generalised Hele-Shaw flow – is derived in terms of the Schwarz function g(z, t) of the free boundary. This generalises the well-known equation ∂g/∂t = 2∂w/∂z, where w is the complex potential, which has been successfully employed in constructing many exact solutions in the absence of external potentials. The new equation is used to re-derive some known explicit solutions for equilibrium and time-dependent free boundary flows in the presence of external potentials, including those with singular potential fields, uniform gravity and centrifugal forces. Some new solutions are also constructed that variously describe equilibrium flows with higher order hydrodynamic singularities in the presence of electric point sources and an unsteady solution describing bubbles under the combined influence of strain and centrifugal potential.

We consider the problem of neutral inclusions for two-dimensional conductivity and anti-plane elasticity. The neutral inclusion, when inserted in a matrix having a uniform field, does not disturb the field outside the inclusion. The inclusion consists of a core and a shell. We show that if the inclusion is neutral to two linearly independent fields, then the core and the shell are confocal ellipses.

This work investigates analytically, the use of piezoelectric tiles placed on stairways for vibrational energy harvesting – harnessing electrical power from natural vibrational phenomena – from pedestrian footfalls. While energy harvesting from pedestrian traffic along flat pathways has been studied in the linear regime and realised in practical applications, the greater amounts of energy naturally expended in traversing stairways suggest better prospects for harvesting. Considering the characteristics of two types of commercially available piezoelectric tiles – Navy Type III and Navy Type V – analytical models for the coupled electromechanical system are formulated. The harvesting potential of the tiles is then studied under conditions of both deterministic and carefully developed random excitation profiles for three distinct cases: linear, monostable nonlinear and an array of monostable nonlinear tiles on adjacent steps with linear coupling between them. The results indicate enhanced power output when the tiles are: (1) placed on stairways, (2) uncoupled and (3) subjected to excitation profiles with stochastic frequency. In addition, the Navy Type V tiles are seen to outperform the Navy Type III tiles. Finally, the strongly nonlinear regime outperforms the linear one suggesting that the realisation of commercially available piezoelectric tiles with appropriately tailored nonlinear characteristics will likely have a significant impact on energy harvesting from pedestrian traffic.

In this note we modify and extend the work of Howison & King [12] to describe the situation of flow around a wedge of arbitrary angle in a Hele-Shaw cell. An ingenious complex-variable method due to Polubarinova-Kochina is used to construct an explicit solution to the zero-surface tension problem.

We develop and test an intuitively simple dynamic network model to describe the type of time-varying connectivity structure present in many technological settings. The model assumes that nodes have an inherent hierarchy governing the emergence of new connections. This idea draws on newly established concepts in online human behaviour concerning the existence of discussion catalysts, who initiate long threads, and online leaders, who trigger feedback. We show that the model captures an important property found in e-mail and voice call data – ‘dynamic communicators’ with sufficient foresight or impact to generate effective links and having an influence that is grossly underestimated by static measures based on snaphots or aggregated data.

The work presents the analysis of the free boundary value problem related to the one-dimensional invasion model of new species in biofilm reactors. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of non-linear hyperbolic partial differential equations governing the microbial species growth and a system of semi-linear elliptic partial differential equations describing the substrate trends. The model is completed with a system of elliptic partial differential equations governing the diffusion and reaction of planktonic cells, which are able to switch their mode of growth from planktonic to sessile when specific environmental conditions are found. Two systems of non-linear differential equations for the substrate and planktonic cells mass balance within the bulk liquid are also considered. The free boundary evolution is governed by a differential equation that accounts for detachment. The qualitative analysis is performed and a uniqueness and existence result is presented. Furthermore, two special models of biological and engineering interest are discussed numerically. The invasion of Anammox bacteria in a constituted biofilm inhabiting the deammonification units of the wastewater treatment plants is simulated. Numerical simulations are run to evaluate the influence of the colonization process on biofilm structure and activity.