Free surface flows generated by a moving distribution of pressure are considered. The fluid consists of two superposed layers in a two-dimensional channel. The upper layer is inviscid and the lower layer, which is introduced as a damping mechanism, is modelled by the mathematically convenient lubrication equations. Numerical and analytical solutions are presented. Special attention is given to solutions for which there is a train of waves on each side of the distribution of pressure. It is shown that, depending on the values of the parameters, the short waves can appear on either side of the distribution of pressure.

A two component system driven by both interface area and interface curvature is studied with a new phase field model. We show that if the curvature impact in the system is strong enough, there exist bubble profiles. A bubble profile describes a pattern of an inner core of one component surround by an outer membrane of the other component. It is a radial solution to a fourth-order nonlinear partial differential equation. We show the existence of such profiles in all dimensions, although the profile is unstable if the dimension is greater than 2.

Consider the classical Hele–Shaw situation with two parallel planes separated by a narrow gap, and suppose the plan-view of the region occupied by fluid to be confined to an infinite strip by barriers in the form of two infinite parallel lines. With the fluid initially occupying a bounded, simply-connected region that touches both barriers along a single line segment, we seek to predict the evolution of the plan-view as the blob of fluid is driven along the strip by a pressure difference between its two free boundaries. Supposing the relevant free boundary condition to be one of constant pressure (but a different constant pressure on each free boundary), we show that the motion is characterized by (a) the existence of two functions, analytic in disjoint half-planes, that are invariants of the motion and (b) the centre of area of the plan-view of the blob has a component of velocity down the infinite strip that is simply related to the imposed pressure difference. These features allow explicit analytic solutions to be found; generically, the mathematical solution breaks down when cusps appear in the retreating free boundary. A rectangular blob, of course, moves down the strip unchanged, with no breakdown, but if it encounters stationary blobs of fluid placed within the strip then, modulo multiply-connected complications, these are first absorbed into the advancing front of the rectangular blob and then disgorged from its retreating rear, leaving behind stationary blobs of exactly the same form in exactly the same place as those originally present, but consisting of different fluid particles. This soliton-like interaction involves no phase change: with a given pressure difference driving the motion, the rectangular blob is in the same position at a given time after the interaction as it would have been had no intervening blobs been present.

The Hopf bifurcation from spike solutions for the classical Gierer–Meinhardt system in a onedimensional interval is considered. The existence of time-periodic solution near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of the limit cycle are determined, and it is shown that the limit cycle is unstable.

The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson–Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefficients, which are based on cell problems due to diffusivity, permittivity and interface electric flux. The first-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.

This work looks at the asymptotic behaviour of the solution to the Helmholtz equation in a penetrable domain of $\mathbb{R}$3 with a thin layer of thickness δ which tends to 0. We use the method of multi-scale expansion to derive and justify an asymptotic expansion of the solution with respect to the thickness δ up to any order. We then provide approximate transmission conditions of order two defined on an interface located inside the thin layer, with accuracy up to O(δ2), which allow one to take into account the influence of the thin layer.

It is shown that the set of conservation laws for the nonlinear system of equations describing plane steady potential barotropic flow of gas is given by the set of conservation laws for the linear Chaplygin system. All the conservation laws of zero order for the Chaplygin system are found. These include both known and new nonlinear conservation laws. It is found that the number of conservation laws of the first order is not more than three, assuming that the laws do not depend on the velocity potential and are not non-obvious ones. The components of these conservation laws are quadratic with respect to the stream function and its derivatives. All the Chaplygin functions are found, for which the Chaplygin system has three non-obvious conservation laws of the first order that are independent of velocity potential. All such non-obvious first-order conservation laws are found.

Whether enjoying the lucid prose of a favourite author or slogging through some other writer’s cumbersome, heavy-set prattle (full of parentheses, em dashes, compound adjectives, and Oxford commas), readers will notice stylistic signatures not only in word choice and grammar but also in punctuation itself. Indeed, visual sequences of punctuation from different authors produce marvellously different (and visually striking) sequences. Punctuation is a largely overlooked stylistic feature in stylometry, the quantitative analysis of written text. In this paper, we examine punctuation sequences in a corpus of literary documents and ask the following questions: Are the properties of such sequences a distinctive feature of different authors? Is it possible to distinguish literary genres based on their punctuation sequences? Do the punctuation styles of authors evolve over time? Are we on to something interesting in trying to do stylometry without words, or are we full of sound and fury (signifying nothing)?

In our investigation, we examine a large corpus of documents from Project Gutenberg (a digital library with many possible editorial influences). We extract punctuation sequences from each document in our corpus and record the number of words that separate punctuation marks. Using such information about punctuation-usage patterns, we attempt both author and genre recognition, and we also examine the evolution of punctuation usage over time. Our efforts at author recognition are particularly successful. Among the features that we consider, the one that seems to carry the most explanatory power is an empirical approximation of the joint probability of the successive occurrence of two punctuation marks. In our conclusions, we suggest several directions for future work, including the application of similar analyses for investigating translations and other types of categorical time series.

Inviscid irrotational flow around a pair of coaxial disks is considered in the limit in which the distance 2h between the disks is small compared to their radius a. The disks have zero thickness and accelerate away from one another along their common axis. The added mass M of each accelerating disk is increased by the presence of the other disk. Analytic predictions are obtained when h/a ≪ 1, with M ~ πa/(8h)-ln(h/a)/2+0.77875+. . . The term O(a/h) can be obtained by means of an inviscid analysis of approximately unidirectional flow within the gap between the disks, but the correction terms have not been reported previously. The irrotational flow problem satisfies Neumann boundary conditions on the surface of the disks, but is otherwise analogous to the Dirichlet problem of the capacitance of a pair of charged disks, which has been the subject of much study and controversy.

Based on the fact that the incubation periods of epidemic disease in asymptomatically infected and infected individuals are inevitable and different, we propose a diffusive susceptible, asymptomatically infected, symptomatically infected and vaccinated (SAIV) epidemic model with delays in this paper. To see whether epidemic disease can propagate spatially with a constant speed, we focus on the travelling wave solution for this model. When the basic reproduction number of the corresponding spatial-homogenous delayed differential system is greater than one and the wave speed is greater than or equal to the critical speed, we prove that this model admits nontrivial positive travelling wave solutions. Our theoretical results are of benefit to the prevention and control of epidemic.

For certain singularly perturbed two-component reaction–diffusion systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behaviour in terms of some parameter in the system. For some such systems, such as the Gray–Scott model, a spike self-replication behaviour is observed as the parameter varies across the saddle-node point. We demonstrate and analyse a qualitatively new type of transition as a parameter is slowly decreased below the saddle node value, which is characterized by a finite-time blow-up of the spike solution. More specifically, we use a blend of asymptotic analysis, linear stability theory, and full numerical computations to analyse a wide variety of dynamical instabilities, and ultimately finite-time blow-up behaviour, for localized spike solutions that occur as a parameter β is slowly ramped in time below various linear stability and existence thresholds associated with steady-state spike solutions. The transition or route to an ultimate finite-time blow-up can include spike nucleation, spike annihilation, or spike amplitude oscillation, depending on the specific parameter regime. Our detailed analysis of the existence and linear stability of multi-spike patterns, through the analysis of an explicitly solvable non-local eigenvalue problem, provides a theoretical guide for predicting which transition will be realized. Finally, we analyse the blow-up profile for a shadow limit of the reaction–diffusion system. For the resulting non-local scalar parabolic problem, we derive an explicit expression for the blow-up rate near the parameter range where blow-up is predicted. This blow-up rate is confirmed with full numerical simulations of the full PDE. Moreover, we analyse the linear stability of this solution that blows up in finite time.

We introduce novel multi-agent interaction models of entropic spatially inhomogeneous evolutionary undisclosed games and their quasi-static limits. These evolutions vastly generalise first- and second-order dynamics. Besides the well-posedness of these novel forms of multi-agent interactions, we are concerned with the learnability of individual payoff functions from observation data. We formulate the payoff learning as a variational problem, minimising the discrepancy between the observations and the predictions by the payoff function. The inferred payoff function can then be used to simulate further evolutions, which are fully data-driven. We prove convergence of minimising solutions obtained from a finite number of observations to a mean-field limit, and the minimal value provides a quantitative error bound on the data-driven evolutions. The abstract framework is fully constructive and numerically implementable. We illustrate this on computational examples where a ground truth payoff function is known and on examples where this is not the case, including a model for pedestrian movement.

This paper proposes a neural network architecture for solving systems of non-linear equations. A back propagation algorithm is applied to solve the problem, using an adaptive learning rate procedure, based on the minimization of the mean squared error function defined by the system, as well as the network activation function, which can be linear or non-linear. The results obtained are compared with some of the standard global optimization techniques that are used for solving non-linear equations systems. The method was tested with some well-known and difficult applications (such as Gauss–Legendre 2-point formula for numerical integration, chemical equilibrium application, kinematic application, neuropsychology application, combustion application and interval arithmetic benchmark) in order to evaluate the performance of the new approach. Empirical results reveal that the proposed method is characterized by fast convergence and is able to deal with high-dimensional equations systems.

In the paper, Bifurcation analysis of the twist-Fréedericksz transition in a nematic liquid-crystal cell with pre-twist boundary conditions (2009 Eur. J. Appl. Math.20, 269–287) by da Costa et al. the twist-Fréedericksz transition in a nematic liquid-crystal one-dimensional cell of lenght L was studied, imposing an antisymmetric net twist Dirichlet condition at the cell boundaries. In the present paper, we extend that study to the more general case of net twist Dirichlet conditions without any kind of symmetry restrictions. We use phase-plane analysis tools and appropriately defined time maps to obtain the bifurcation diagrams of the model when L is the bifurcation parameter, and related these diagrams with the one in the antisymmetric situation. The stability of the bifurcating solutions is investigated by applying the method of Maginu (1978 J. Math. Anal. Appl.63, 224–243).

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

We study the analyticity properties of solutions for a class of non-linear evolutionary pseudo-differential equations possessing global attractors. In order to do this we utilise an analyticity criterion for spatially periodic functions, which involves the rate of growth of a suitable norm of the nth derivative of the solution, with respect to the spatial variable, as n tends to infinity. This criterion can be used to a wide class of dissipative-dispersive partial differential equations, provided they possess global attractors. Using this criterion and the spectral method developed in Akrivis et al. [1] we have improved previous results.

The onset of shear-banding in a deforming elastoplastic solid has been linked to change of type of the governing partial differential equations. If uniform material properties are assumed, then (i) deformations prior to shear-banding are uniform, and (ii) the onset of shear-banding occurs simultaneously at all points in the sample. In this paper we study, in the context of a model for anti-plane shearing of a granular material, the effect of a small variation in material properties (e.g. in yield strength) within the sample. Using matched asymptotic expansions, we find that (i) the deformation is extremely non-uniform in a short time period immediately preceding the formation of shear-bands; and (ii) generically, a shear-band forms at a single location in the sample.

We consider an eigenvalue problem of three-dimensional elasticity for a multi-structure consisting of a finite three-dimensional solid linked with a thin-walled elastic cylinder. An asymptotic method is used to derive the junction conditions and to obtain the skeleton model for the multi-structure. Explicit asymptotic formulae have been obtained for the first six eigen-frequencies.

We derive and analyse an energy to model lipid raft formation on biological membranes involving a coupling between the local mean curvature and the local composition. We apply a perturbation method recently introduced by Fritz, Hobbs and the first author to describe the geometry of the surface as a graph over an undeformed Helfrich energy minimising surface. The result is a surface Cahn–Hilliard functional coupled with a small deformation energy. We show that suitable minimisers of this energy exist and consider a gradient flow with conserved Allen–Cahn dynamics, for which existence and uniqueness results are proven. Finally, numerical simulations show that for the long-time behaviour raft-like structures can emerge and stabilise, and their parameter dependence is further explored.

We model larviciding of mosquitoes taking into account the evolution of resistance to the larvicides, the evolutionary costs of resistance and the implications for malaria control. There is evidence that resistance comes with various costs, one of which is reduced adult longevity for resistant mutants. The mosquito adult lifespan is one of the most crucial parameters in malaria transmission due to a long developmental time for the malaria parasite in the insect. A possible malaria control strategy is therefore to shorten this adult lifespan by larviciding with a potent larvicide to which mosquitoes become resistant. This novel strategy is studied using a mathematical model for the wild type and resistant mutants and by incorporating the malaria disease dynamics using an SEI type model with standard incidence that incorporates the latency period of the parasite in wild-type and resistant mosquitoes. We consider the linear stability of the malaria-free equilibrium in which the resistant strain is dominant and derive a condition for the global eradication of malaria. Numerical simulations are presented, which offer further insights. The parameter to which the analysis is most sensitive is the per-capita death rate of resistant adult mosquitoes. Increasing this parameter dramatically reduces the basic reproduction number R0. However, increasing it too much causes the wild type to outcompete the resistant mutants, and the control strategy fails. Exploitation of costs of resistance to larvicides thus offers a possible malaria control measure if the larvicide is sufficiently potent and costs of resistance are neither too great nor too small.