This special issue of the European Journal of Applied Mathematics is devoted to the ‘Liquid Crystal Theory and Modelling: Discussion Meeting’ that was held in Oxford on the 29th-30th October 2009. This international, interdisciplinary meeting was organised jointly by Dr Apala Majumdar (University of Oxford) and Prof. Nigel Mottram (University of Strathclyde) and was funded by OCCAM (Oxford Centre for Collaborative Applied Mathematics), the London Mathematical Society, Taylor and Francis and Cambridge University Press.

We consider a Josephson junction system installed with a finite length inhomogeneity, either of micro-resistor or micro-resonator type. The system can be modelled by a sine-Gordon equation with a piecewise-constant function to represent the varying Josephson tunneling critical current. The existence of pinned fluxons depends on the length of the inhomogeneity, the variation in the Josephson tunneling critical current and the applied bias current. We establish that a system may either not be able to sustain a pinned fluxon, or – for instance by varying the length of the inhomogeneity – may exhibit various different types of pinned fluxons. Our stability analysis shows that changes of stability can only occur at critical points of the length of the inhomogeneity as a function of the (Hamiltonian) energy density inside the inhomogeneity – a relation we determine explicitly. In combination with continuation arguments and Sturm–Liouville theory, we determine the stability of all constructed pinned fluxons. It follows that if a given system is able to sustain at least one pinned fluxon, a microresistor has exactly one pinned fluxon, i.e. the system selects one unique pinned stable pinned configuration, and a microresonator has at least one stable pinned configuration. Moreover, it is shown that both for micro-resistors and micro-resonators this stable pinned configuration may be non-monotonic – something which is not possible in the homogeneous case. Finally, it is shown that results in the literature on localised inhomogeneities can be recovered as limits of our results on micro-resonators.

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systems where h, K, Ψ and ξ are analytic functions defined in the neighbourhood of O with K(O) ≠ 0 or Ψ(O) ≠ 0 and n ≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the effect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves. J. Fluid Mech.567, 299–326, the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

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.

We study the stochastic effect of resource exploration in dynamic Cournot models of exhaustible resources, such as oil. We firstly treat the case of a monopolist who may undertake costly exploration to replenish his diminishing reserves. We then consider a stochastic game between such an exhaustible producer and a ‘green’ producer that has access to an inexhaustible but relatively expensive source, such as solar power. The effort control variable is taken to be either continuous or discrete (switching control). In both settings, we assume that new discoveries occur according to a jump process with intensity given by the exploration effort. This leads to a study of systems of non-linear first-order delay ordinary differential equations (ODEs). We derive asymptotic expansions for the case of a small-exploration success rate and present some numerical investigations.

We present a new asymptotic formula for the maximum static voltage in a simplified model for on-chip power distribution networks of array bonded integrated circuits. In this model the voltage is the solution of the Poisson's equation in an infinite planar domain whose boundary is an array of circular pads of radius ϵ, and we deal with the singular limit ϵ → 0 case. In comparison with approximations that appear in the electronics engineering literature, our formula is more complete, since we have obtained terms up to order ϵ15. A procedure will be presented to compute all the successive terms, which can be interpreted by using multipole solutions of equations involving spatial derivatives of δ-functions. To deduce the formula, we use the method of matched asymptotic expansions. Our results are completely analytical and we make an extensive use of special functions and the Gauss constant G.

This paper is concerned with the asymptotic spreading of competition diffusion systems, with the purpose of formulating the propagation modes of a co-invasion–coexistence process of two competitors. Using the comparison principle for competitive systems, some results on asymptotic spreading are obtained. Our conclusions imply that the interspecific competitions slow the invasion of one species and decrease the population densities in the coexistence domain. Therefore, the interspecific competitions play a negative role in the evolution of competitive communities.

We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multi-scale theory for such systems and apply it to the Kuramoto–Sivashinsky equation as a case study. We first focus on a regime close to the instability onset (primary bifurcation), where the system can be described by a single dominant mode. We show analytically that the resulting noise in the equation describing the amplitude of the dominant mode largely depends on the nature of the stochastic forcing. For a highly degenerate noise, in the sense that it is acting on the first stable mode only, the amplitude equation is dominated by a pure multiplicative noise, which in turn induces the dominant mode to undergo several critical state transitions and complex phenomena, including intermittency and stabilisation, as the noise strength is increased. The intermittent behaviour is characterised by a power-law probability density and the corresponding critical exponent is calculated rigorously by making use of the first-passage properties of the amplitude equation. On the other hand, when the noise is acting on the whole subspace of stable modes, the multiplicative noise is corrected by an additive-like term, with the eventual loss of any stabilised state. We also show that the stochastic forcing has no effect on the dominant mode dynamics when it is acting on the second stable mode. Finally, in a regime which is relatively far from the instability onset so that there are two unstable modes, we observe numerically that when the noise is acting on the first stable mode, both dominant modes show noise-induced complex phenomena similar to the single-mode case.

Secondary flows consisting of two pairs of vortices arise when two fluid streams meet at a confluence, such as in the airways of the human lung during expiration or at the vertebrobasilar junction in the circulatory system, where the left and right vertebral arteries converge. In this paper the decay of these secondary flows is studied by considering a four-vortex perturbation from Poiseuille flow in a straight, three-dimensional pipe. A polynomial eigenvalue problem is formulated and the exact solution for the zero Reynolds number R is derived analytically. This solution is then extended by perturbation analysis to produce an approximation to the eigenvalues for R ≪ 1. The problem is also solved numerically for 0 ≤ R ≤ 2,000 by a spectral method, and the stability of the computed eigenvalues is analysed using pseudospectra. For all Reynolds numbers, the decay rate of the swirling perturbation is found to be governed by complex eigenvalues, with the secondary flows decaying more slowly as R increases. A comparison with results from an existing computational study of merging flows shows that the two models give rise to similar secondary flow decay rates.

An ℝ-linear conjugation problem modelling the process of power fields forming in a heterogeneous infinite planar structure with an elliptical inclusion is considered. Exact analytical solutions are derived in the class of piece-wise meromorphic functions with their principal parts fixed. Cases with internal singularities and with singularities of the given principal parts at the interface are investigated.

Basic methods from bifurcation theory are applied to derive a criterion that predicts when a symmetric tricritical point may occur in a transition between condensed ordered phases described by any finite number of scalar order parameters. At such a point, a change of order takes place in the phase transition, which passes from first to second order, or vice versa.

The excluded volume of a pair of molecules is proportional to the second virial coefficient in hard-core models that represent, for instance, the reference model in perturbation approaches to statistical theories of fluids [see, e.g. Chap. 5 of Kalikmanov, V. (2001), Statistical Physics of Fluids. Texts and Monograph in Physics, Springer, Berlin]. In three space dimensions, there exist exact results for convex molecules and in fact lack of convexity has been a major obstacle in applying the mathematical techniques employed in the convex case. In this paper, we illustrate how a mixed—analytical and numerical—method can be used to obtain exact expressions of the excluded volume for a pair of non-convex molecules conceived as aggregates of hard spheres; these can model van der Waals regions associated to the atoms forming each molecule. To compute the excluded volume for molecules endowed with C2v symmetry, modelled as chains of tangent hard spheres, we adapt a numerical code available to the scientific community. Because the result is a rather cumbersome expression in term of the relative orientation of the interacting molecules, we expand it over a suitable set of symmetry adapted Wigner functions to build up approximate, but faithful expressions, and we also prove analytical results announced elsewhere [Bisi, F., Durand, G., Rosso, R. & Virga, E. (2008), Polar steric interactions for v-shaped molecules. Phys. Rev. E, 78, 011705].

In this paper, we revisit the quantile mechanics approach, which was introduced by Steinbrecher and Shaw (Steinbrecher, G. & Shaw, W. T. (2008) Quantile mechanics. European. J. Appl. Math.19, 87–112). Our objectives are (i) to derive the method of trimmed moments (MTM) estimators for the parameters of gamma and Student's t distributions, and (ii) to examine their large- and small-sample statistical properties. Since trimmed moments are defined through the quantile function of the distribution, quantile mechanics seems like a natural approach for achieving objective (i). To accomplish the second goal, we rely on the general large sample results for MTMs, which were established by Brazauskas et al. (Brazauskas, V., Jones, B. L. & Zitikis, R. (2009) Robust fitting of claim severity distributions and the method of trimmed moments. J. Stat. Plan. Inference139, 2028–2043), and then use Monte Carlo simulations to investigate small-sample behaviour of the newly derived estimators. We find that, unlike the maximum likelihood method, which usually yields fully efficient but non-robust estimators, the MTM estimators are robust and offer competitive trade-offs between robustness and efficiency. These properties are essential when one employs gamma or Student's t distributions in such outlier-prone areas as insurance and finance.

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.

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.

In this paper, we will show how to obtain asymptotic solutions for the problem of pricing Asian options. Under the assumption that the underlying follows geometric Brownian motion, we will derive Taylor expansion series for the fixed and floating strike Asian options. While there will be no analytical formulae for calculating expansion coefficients, we will provide relatively simple algorithms for calculating them. The methodology is particularly effective for the case of continuously sampled fixed-strike Asian calls where it takes only seconds to obtain constants for the Taylor expansion series that can converge beyond 10 significant digits. It is needless to say that we need to calculate Taylor expansion constants only once and the option price would be an analytical expression constructed from a cumulative normal distribution function, an exponential function and finite sums.

For a sensor network, a tractable spatially dependent node deployment model is presented with the property that the density is inversely proportional to the sink distance. A stochastic model is formulated to examine message advancements under greedy routing in such a sensor network. The aim of this work is to demonstrate that an inhomogeneous Poisson process can be used to model a sensor network with spatially dependent node density. Symmetric elliptic integrals and asymptotic approximations are used to describe the random behaviour of hops. Types of dependence that affect hop advancements are examined. We observe that the dependence between successive jumps in a multi-hop path is captured by including only the previous forwarding node location. We include a simple uncoordinated sleep scheme, and observe that the complexity of the model is reduced when sufficiently many nodes are asleep. All expressions involving multi-dimensional integrals are derived and evaluated with quasi-Monte Carlo integration methods based on Halton sequences and recently developed lattice rules. An importance sampling function is derived to speed up the quasi-Monte Carlo methods. The ensuing results agree extremely well with simulations.

We develop a theory to represent dislocations and disclinations in single crystals at the continuum (or mesoscopic) scale by directly modelling the defect densities as concentrated effects governed by the distribution theory. The displacement and rotation multi-valuedness is resolved by introducing the intrinsic and single-valued Frank and Burgers tensors from the distributional gradients of the strain field. Our approach provides a new understanding of the theory of line defects as developed by Kröner [10] and other authors [6, 9]. The fundamental identity relating the incompatibility tensor to the Frank and Burgers vectors (and which is a cornerstone of the theory of dislocations in single crystals) is proved in the 2D case under appropriate assumptions on the strain and strain curl growth in the vicinity of the assumed isolated defect lines. In general, our theory provides a rigorous framework for the treatment of crystal line defects at the mesoscopic scale and a basis to strengthen the theory of homogenisation from mesoscopic to macroscopic scale.

We study the radial-hedgehog solution in a three-dimensional spherical droplet, with homeotropic boundary conditions, within the Landau–de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a global Landau–de Gennes minimiser in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. The static properties of the radial-hedgehog solution are governed by a non-linear singular ordinary differential equation. We study the analogies between Ginzburg–Landau vortices and the radial-hedgehog solution and demonstrate a Ginzburg–Landau limit for the Landau–de Gennes theory. We prove that the radial-hedgehog solution is not the global Landau–de Gennes minimiser for droplets of finite radius and sufficiently low temperatures and prove the stability of the radial-hedgehog solution in other parameter regimes. These results contain quantitative information about the effect of geometry and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities.