Well-posedness and stationary solutions

In this paper we prove existence and uniqueness of variational inequality solutions for a bistable quasilinear parabolic equation arising in the theory of solid-solid phase transitions and discuss its stationary solutions, which can be discontinuous

Multiplicity of periodic solutions in bistable equations

We study the number of periodic solutions in two first order non-autonomous differential equations both of which have been used to describe, among other things, the mean magnetization of an Ising magnet in the time-varying external magnetic field. When the strength of the external field is varied, the set of periodic solutions undergoes a bifurcation in both equations. We prove that despite profound similarities between the equations, the character of the bifurcation can be very different. This results in a different number of coexisting stable periodic solutions in the vicinity of the bifurcation. As a consequence, in one of the models, the Suzuki-Kubo equation, one can effect a discontinuous change in magnetization by adiabatically varying the strength of the magnetic field.Comment: Fixed typos; added and reordered figures. 18 pages, 6 figures. An animation of orbits is available at http://www.maths.strath.ac.uk/~aas02101/bistable

Characterising submonolayer deposition via visibility graphs

We use visibility graphs as a tool to analyse the results of kinetic Monte Carlo (kMC) simulations of submonolayer deposition in a one-dimensional point island model. We introduce an efficient algorithm for the computation of the visibility graph resulting from a kMC simulation and show that from the properties of the visibility graph one can determine the critical island size, thus demonstrating that the visibility graph approach, which implicitly combines size and spatial data, can provide insights into island nucleation and growth processes

Hysteresis and economics - taking the economic past into account

The goal of this article is to discuss the rationale underlying the application of hysteresis to economic models. In particular, we explain why many aspects of real economic systems are hysteretic is plausible. The aim is to be explicit about the difficulties encountered when trying to incorporate hysteretic effects into models that can be validated and then used as possible tools for macroeconomic control. The growing appreciation of the ways that memory effects influence the functioning of economic systems is a significant advance in economic thought and, by removing distortions that result from oversimplifying specifications of input-output relations in economics, has the potential to narrow the gap between economic modeling and economic reality

Stress Corrosion and Stress Induced Surface Morphology of Epitaxial Films

In addition to several new predictions, the general theory of thermodynamic stability of heterogeneous systems with rearrangement has allowed us to understand the roots of several experimental and theoretical results of the past. One of them is an outstanding paper of Asaro and Tiller on stress corrosion cracking by surface diffusion published two decades ago. We compare results of Asaro and Tiller with conclusions of thermodynamic theory of solids with rearrangement and develop some Asaro-Tiller results in the directions dictated by the needs of thin films technology and experiment. A surface diffusion model in a prestressed elastic solid is studied on the basis of the Onsager approach of irreversible thermodynamics. The master system governing a quasi-static evolution of the surface corrugations is derived in the framework of nonlinear elasticity and for the model of a surface energy incorporating both the Laplace excess pressure under curved interface and the Herring curvature term in the local chemical potential. Then, we derive a dispersion relation of the growth rate of two-dimensional infinitesimal corrugations atop an isotropic uniformly stressed elastic layer clamped to a substrate. The relation predicts different patterns of surface morphology produced by the fastest unstable corrugations. The patterning which develops depends on the applied stresses, thickness and material parameters of the layer and substrate

Convergence in a multidimensional randomized Keynesian beauty contest

We study the asymptotics of a Markovian system of $N \geq 3$ particles in $[0,1]^d$ in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent $U [0,1]^d$ random particle. We show that the limiting configuration contains $N-1$ coincident particles at a random location $\xi_N \in [0,1]^d$. A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d=1 we give additional results on the distribution of the limit $\xi_N$, showing, among other things, that it gives positive probability to any nonempty interval subset of $[0,1]$, and giving a reasonably explicit description in the smallest nontrivial case, N=3.Comment: 26 pages, 4 figure

Pendent Steady Rivulets: From Lubrication to Bifurcation

We consider the shape of the free surface of steady pendent rivulets beneath a planar substrate. We formulate the governing equations in terms of two closely related dynamical systems and use classical phase-plane techniques to develop the bifurcation structure of the problem. Our results explain why lubrication theory is unable to capture this bifurcation structure for pendent rivulets, although it is successful in the related problem of sessile rivulets

Decomposition of the Leinster-Cobbold Diversity Index

The Leinster and Cobbold diversity index possesses a number of merits; in particular, it generalises many existing indices and defines an effective number. We present a scheme to quantify the contribution of richness, evenness, and taxonomic similarity to this index. Compared to the work of van Dam (2019), our approach gives unbiased estimates of both evenness and similarity in a non-homogeneous community. We also introduce a notion of taxonomic tree equilibration which should be of use in the description of community structure.Comment: 10 pages, 1 figur

Minimal travelling wave speed and explicit solutions in monostable reaction-diffusion equations

We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation