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Relating Individual Behaviour to Population Dynamics
No full textHow do the behavioural interactions between individuals in an ecological system produce the global population dynamics of that system? We present a stochastic individual-based model of the reproductive cycle of the mite Varroa jacobsoni, a parasite of honeybees. The model has the interesting property in that its population level behaviour is approximated extremely accurately by the exponential logistic equation or Ricker map. We demonstrated how this approximation is obtained mathematically and how the parameters of the exponential logistic equation can be written in terms of the parameters of the individual-based model. Our procedure demonstrates, in at least one case, how study of animal ecology at an individual level can be used to derive global models which predict population change over time
Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow
Get PDFWe outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of Navier–Stokes equations. We describe a very general strategy for preconditioning that has two basic building blocks; a multigrid V-cycle for the scalar convection–diffusion operator, and a multigrid V-cycle for a pressure Poisson operator. We present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid, robust with respect to the time-step, and only deteriorates very slowly as the Reynolds number is increased
Solving a quadratic matrix equation by Newton's method with exact line searches
Get PDFWe show how to incorporate exact line searches into Newton's method for solving the quadratic matrix equation AX^2 + BX + C = 0, where A, B and C are square matrices. The line searches are relatively inexpensive and improve the global convergence properties of Newton's method in theory and in practice. We also derive a condition number for the problem and show how to compute the backward error of an approximate solution
Granular flow in partially filled slowly rotating drums
Get PDFIn many industrial processes granular materials are mixed together in partially filled slowly rotating drums. In this paper a general theoretical framework is developed for the quasi-two-dimensional motion of granular material in a rotating drum. The key assumption is that the body can be divided into a fluid-like and a solid-like region, that are separated by a non-material singular surface at which discontinuities occur. Experiments show that close to the free surface there is a thin rapidly moving fluid-like avalanche that flows downslope, and beneath it there is a large region of slowly rotating solid-like material. The solid region provides a net transport of material upslope and there is strong mass transfer between the two regions. In the theory the avalanche is treated as a shallow incompressible Mohr–Coulomb or inviscid material sliding on a moving bed at which there is erosion and deposition. The solid is treated as a rigid rotating body, and the two regions are coupled together using a mass jump condition. The theory has the potential to model time-dependent intermittent flow with shock waves, as well as steady-state continuous flow. An exact solution for the case of steady continuous flow is presented. This demonstrates that when the base of the avalanche lies above the axis of revolution a solid core develops in the centre of the drum. Experiments are presented to show how a mono-disperse granular material mixes in the drum, and the results are compared with the predictions using the exact solution
Closed geodesics and periods of automorphic forms
No full textWe study the detailed structure of the distribution of Eichler–Shimura periods of an automorphic form on a compact hyperbolic surface. We show that these periods do not cluster around the asymptotic period over a homology class discovered by Zelditch
Tangential structures on toric manifolds, and connected sums of polytopes
No full textWe extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show
that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of
examples Bi,j, which are toric manifolds over products of simplices, and verify that their natural stably
complex structure is induced by an omniorientation. Studying connected sums of products of the Bi,j allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum # for simple n-dimensional polytopes; when Pn is a product of simplices, we describe Pn#Qn by applying an appropriate sequence of pruning operators, or hyperplane cuts, to Qn
The Bretherton Problem in Elastic-Walled Channels: Finite Reynolds Number Effects
Get PDFThis paper investigates the effect of fluid inertia on the propagation of an air finger into a channel with elastic walls, a problem which can be regarded as a generalisation of the classical Bretherton problem. The study is motivated by the physiological problem of pulmonary airway reopening. Numerical results show that fluid inertia plays a surprisingly important role in this problem: Even for relatively modest ratios of Reynolds and Capillary numbers (Re/Ca approximately 5 - 10), the pressure required to drive the air finger at a given speed increases significantly compared to the zero Reynolds number case. Inertial effects are also shown to be responsible for a noticeable change in the wall deformation ahead of the bubble tip. This is analysed by a Karman-Pohlhausen approximation which yields a linear ODE, the eigenvalues of which determine the wavelength and decay rate of the oscillatory wall displacement field in this region
Milnor attractors and topological attractors of a piecewise linear map
Get PDFA very simple two-dimensional map is discussed. It is shown that for appropriate values of the parameters there is a two dimensional subset of the plane on which the dynamics is transitive and periodic orbits are dense, but that this topological attractor contains a one dimensional set which attracts almost all points (i.e. it is a Milnor attractor). This arises naturally as a precursor to a blowout bifurcation to on-off intermittency in this system, and confirms a conjecture due to Pikovsky and Grassberger
Spin-up of homogeneous and stratified fluids
Get PDFWe consider the manner in which a container filled with viscous fluid adjusts to changes in its rotation rate. We begin with homogeneous flows involving small departures in rotation rate from an initial state of solid-body rotation in an axisymmetric container. This is followed by a summary of other more recent developments, including weakly and fully nonlinear calculations and comparison with experiment and the question of spin-down. The question of "spin-over" is addressed, followed by a brief synopsis of free-surface effects, and a discussion of nonaxisymmetric spin-up. The second part of the review focuses on the effects of stratification on the spin-up process. Linearized (low Rossby number) spin-up within a cylindrical container is described. Thereafter, both experimental and nonlinear computational results are described and compared. The final section focuses on stratified spin-up and spin-down in conical geometries, and a number of comparisons between theory and experiment are given
The Whitney Reduction Network: A Method for Computing Autoassociative Graphs
No full textThis article introduces a new architecture and associated algorithms ideal for implementing the dimensionality reduction of an m-dimensional manifold initially residing in an n-dimensional Euclidean space where n>>m. Motivated by Whitney's embedding theorem, the network is capable of training the identity mapping employing the idea of the graph of a function. In theory, a reduction to a dimension d that retains the differential structure of the original data may be achieved for some d<=2m+1. To implement this network, we propose the idea of a good-projection, which enhances the generalization capabilities of the network, and an adaptive secant basis algorithm to achieve it. The effect of noise on this procedure is also considered. The approach is illustrated with several examples