Inertial levitation

We consider the steady levitation of a rigid plate on a thin air cushion with prescribed injection velocity. This injection velocity is assumed to be much larger than that in a conventional Prandtl boundary layer, so that inertial effects dominate. After applying the classical ‘blowhard’ theory of Cole & Aroesty (1968) to the two-dimensional version of the problem, it is shown that in three dimensions the flow may be foliated into streamline surfaces using Lagrangian variables. An example is given of how this may be exploited to solve the three-dimensional problem when the injection pressure distribution is known

Oscillations of dark solitons in trapped Bose-Einstein condensates

We consider a one-dimensional defocusing Gross--Pitaevskii equation with a parabolic potential. Dark solitons oscillate near the center of the potential trap and their amplitude decays due to radiative losses (sound emission). We develop a systematic asymptotic multi-scale expansion method in the limit when the potential trap is flat. The first-order approximation predicts a uniform frequency of oscillations for the dark soliton of arbitrary amplitude. The second-order approximation predicts the nonlinear growth rate of the oscillation amplitude, which results in decay of the dark soliton. The results are compared with the previous publications and numerical computations.Comment: 13 pages, 3 figure

A model for the break-up of a tuft of fibers

A simple model for the forces acting on a single fiber as it is withdrawn from a tangled fiber assembly is proposed. Particular emphasis is placed on understanding the dynamics of the reptating fiber with respect to the entanglement of fibers within the tuft. The resulting two-parameter model captures the qualitative features of experimental simulation. The model is extended to describe the break-up of a tuft. The results show good agreement with experiment and indicate where a fiber is most likely to fracture based on the density of fiber end-points

Macroscopic models for superconductivity

This paper reviews the derivation of some macroscopic models for superconductivity and also some of the mathematical challenges posed by these models. The paper begins by exploring certain analogies between phase changes in superconductors and those in solidification and melting. However, it is soon found that there are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models

Partial differential equations for self-organization in cellular and developmental biology

Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated

Analysis of the archetypal functional equation in the non-critical case

We study the archetypal functional equation of the form $y(x)=\iint_{R^2} y(a(x-b))\,\mu(da,db)$ ($x\in R$), where $\mu$ is a probability measure on $R^2$; equivalently, $y(x)=E\{y(\alpha (x-\beta))\}$, where $E$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{R^2}\ln |a| \mu(da,db) =E \{\ln |\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $P(\alpha0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrow\alpha (x-\beta)$

A national-scale assessment of climate change impacts on species: assessing the balance of risks and opportunities for multiple taxa

It is important for conservationists to be able to assess the risks that climate change poses to species, in order to inform decision making. Using standardised and repeatable methods, we present a national-scale assessment of the risks of range loss and opportunities for range expansion, that climate change could pose for over 3,000 plants and animals that occur in England. A basic risk assessment that compared projected future changes in potential range with recently observed changes classified 21% of species as being at high risk and 6% at medium risk of range loss under a B1 climate change scenario. A greater number of species were classified as having a medium (16%) or high (38%) opportunity to potentially expand their distribution. A more comprehensive assessment, incorporating additional ecological information, including potentially confounding and exacerbating factors, was applied to 402 species, of which 35 % were at risk of range loss and 42 % may expand their range extent. This study covers a temperate region with a significant proportion of species at their poleward range limit. The balance of risks and opportunities from climate change may be different elsewhere. The outcome of both risk assessments varied between taxonomic groups, with bryophytes and vascular plants containing the greatest proportion of species at risk from climate change. Upland habitats contained more species at risk than other habitats. Whilst the overall pattern was clear, confidence was generally low for individual assessments, with the exception of well-studied taxa such as birds. In response to climate change, nature conservation needs to plan for changing species distributions and increasing uncertainty of the future

Recent Shift in Climate Relationship Enables Prediction of the Timing of Bird Breeding

Large-scale climate processes influence many aspects of ecology including breeding phenology, reproductive success and survival across a wide range of taxa. Some effects are direct, for example, in temperate-zone birds, ambient temperature is an important cue enabling breeding effort to coincide with maximum food availability, and earlier breeding in response to warmer springs has been documented in many species. In other cases, time-lags of up to several years in ecological responses have been reported, with effects mediated through biotic mechanisms such as growth rates or abundance of food supplies. Here we use 23 years of data for a temperate woodland bird species, the great tit (Parus major), breeding in deciduous woodland in eastern England to demonstrate a time-lagged linear relationship between the on-set of egg laying and the winter index of the North Atlantic Oscillation such that timing can be predicted from the winter index for the previous year. Thus the timing of bird breeding (and, by inference, the timing of spring events in general) can be predicted one year in advance. We also show that the relationship with the winter index appears to arise through an abiotic time-lag with local spring warmth in our study area. Examining this link between local conditions and larger-scale processes in the longer-term showed that, in the past, significant relationships with the immediately preceding winter index were more common than those with the time-lagged index, and especially so from the late 1930s to the early 1970s. However, from the mid 1970s onwards, the time-lagged relationship has become the most significant, suggesting a recent change in climate patterns. The strength of the current time-lagged relationship suggests that it might have relevance for other temperature-dependent ecological relationships