Lotka-Volterra systems have been used extensively in modelling population dynamics. In this paper, it is shown that chaotic behaviour in the sense of Smale can exist in timeperiodically perturbed systems of Lotka-Volterra equations. First, a slowly varying threedimensional perturbed Lotka-Volterra system is considered and the corresponding unperturbed system is shown to possess a heteroclinic cycle. By using Melnikov's method, sufficient conditions are obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Then a special case involving a reduction to a two-dimensional Lotka-Volterra system is examined, leading finally to an application with a model for the self-organisation of macromolecules.

The new optima and equilibria discussed in the preceding two papers are compared with the results of bargaining experiments between two and three players performed by Fouraker and Siegel. Experiments where players have complete or incomplete information are considered. There is clear evidence that the new optima are operating, and that traditional optima–Cournot, Pareto and competitive (threat)–are less satisfactory in explaining the course of the whole bargaining process.

We define and solve classes of sparse matrix problems that arise in multilevel modelling and data analysis. The classes are indexed by the number of nested units, with two-level problems corresponding to the common situation, in which data on level-1 units are grouped within a two-level structure. We provide full solutions for two-level and three-level problems, and their derivations provide blueprints for the challenging, albeit rarer in applications, higher-level versions of the problem. While our linear system solutions are a concise recasting of existing results, our matrix inverse sub-block results are novel and facilitate streamlined computation of standard errors in frequentist inference as well as allowing streamlined mean field variational Bayesian inference for models containing higher-level random effects.

In this paper a feasible direction method is presented to find all efficient extreme points for a special class of multiple objective linear fractional programming problems, when all denominators are equal. This method is based on the conjugate gradient projection method, so that we start with a feasible point and then a sequence of feasible directions towards all efficient adjacent extremes of the problem can be generated. Since methods based on vertex information may encounter difficulties as the problem size increases, we expect that this method will be less sensitive to problem size. A simple production example is given to illustrate this method.

We have finally obtained for each of the 6 Painlevés an expression of z, w, w′ that behaves as 1/(z − Z0) + O(1) at each kind of movable singular point. This expression is polynomial in w′ (at most quadratic), and rational in w and z. After it is integrated and exponentiated it yields a function that has a simple zero at each of the singular points.

In this paper we present an Extended Linear-Quadratic Programming method for the minimax problem. We show that the Extended Linear-Quadratic Programming method for the minimax problem is equivalent to the Josephy-Newton method for generalized equation, and establish the local convergence result. Furthermore, we obtain the global convergence result for the minimax problem by means of the equivalence relation between the generalized equation and the normal equation.

A humanoid robot system may be viewed as a collection of segments coupled at rotational joints which geometrically represent constrained rotational Lie groups. This allows a study of the dynamics of the motion of a humanoid robot. Several formulations are possible. In this paper, dual invariant topological structures are constructed and analyzed on the finite-dimensional manifolds associated with the humanoid motion. Both cohomology and homology structures are examined on the tangent (Lagrangian) as well as on the cotangent (Hamiltonian) bundles on the manifold of the humanoid motion configuration. represented by the toral Lie group. It is established all four topological structures give in essence the same description of humanoid dynamics. Practically this means that whichever of these approaches we use, ultimately we obtain the same mathematical results.

We discuss simulation of sensitivities or Greeks of multi-asset European style options under a special Lévy process model: that is, the subordinated Brownian motion model. The Malliavin calculus method combined with Monte Carlo and quasi-Monte Carlo methods is used in the simulations. Greeks are expressed in terms of the expectations of the option payoff functions multiplied by the weights involving Malliavin derivatives for multi-asset options. Numerical results show that the Malliavin calculus method is usually more efficient than the finite difference method for options with nonsmooth payoffs. The superiority of the former method over the latter is even more significant when both are combined with quasi-Monte Carlo methods.

Microbial competition for nutrients is a common phenomenon that occurs between species inhabiting the same environment. Bioreactors are often used for the study of microbial competition since the number and type of microbial species can be controlled, and the system can be isolated from other interactions that may occur between the competing species. A common type of competition is the so-called “pure and simple” competition, where the microbial populations interact in no other way except the competition for a single rate-limiting nutrient that affects their growth rates. The issue of whether pure and simple competition under time-invariant conditions can give rise to chaotic behaviour has been unresolved for decades. The third author recently showed, for the first time, that chaos can theoretically occur in these systems by analysing the dynamics of a model where both competing species grow following the biomass-dependent Contois model while the yield coefficients associated with the two species are substrate-dependent. In this paper we show that chaotic behaviour can occur in a much simpler model of pure and simple competition. We examine the case where only one species grows following the Contois model with variable yield coefficient while the other species is allowed to grow following the simple Monod model with constant yield. We show that while the static behaviour of the proposed model is quite simple, the dynamic behaviour is complex and involves period doubling culminating in chaos. The proposed model could serve as a basis to re-examine the importance of Contois kinetics in predicting complex behaviour in microbial competition.

In this survey we consider a regularized Newton method for the approximate solution of the inverse problem to determine the shape of an obstacle from a knowledge of the far field pattern for the scattering of time-harmonic acoustic or electromagnetic plane waves. Our analysis is in two dimensions and the numerical scheme is based on the solution of boundary integral equations by a Nyström method. We include an example of the reconstruction of a planar domain with a corner both to illustrate the feasibility of the use of radial basis functions for the reconstruction of boundary curves with local features and to connect the presentation to some of the research work of Professor David Elliott.

In this paper, we extend the results in the literature for boundary layer flow over a horizontal plate, by considering the buoyancy force term in the momentum equation. Using a similarity transformation, we transform the partial differential equations of the problem into coupled nonlinear ordinary differential equations. We first analyse several special cases dealing with the properties of the exact and approximate solutions. Then, for the general problem, we construct series solutions for arbitrary values of the physical parameters. Furthermore, we obtain numerical solutions for several sets of values of the parameters. The numerical results thus obtained are presented through graphs and tables and the effects of the physical parameters on the flow and heat transfer characteristics are discussed. The results obtained reveal many interesting behaviours that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.

In this paper we study optimal control computation based on the control parameterization method for a class of optimal control problems involving nonlinear systems with multiple time delays subject to continuous state inequality constraints. Both the state and the control are allowed to have different time delays, and they are uncorrelated in this system. The control of the dynamical system is approximated by a piecewise constant function whose heights are taken as decision vectors. The formulae for computing the gradients of the cost and constraint functions are then derived. Based on this, a computational method for finding the optimal control is developed by utilizing the Sequential Quadratic Programming (SQP) algorithm with an active set strategy. The computational method is applied to an industrial problem arising in the purification process of zinc hydrometallurgy. Numerical simulation shows that the amount of zinc powder that is needed can be decreased significantly, thus avoiding wastage of resources.

This paper shows how to compute the trace of G(T) – G(T0), where G is an infinitely differentiable function with compact support, and where T and T0 are one-dimensional Schrödinger operators on (−∞, ∞) with potentials q and q0. It is assumed that q0 is a simple step potential and that q decays exponentially to q0. The trace is expressed in terms of the reflection and transmission coefficients for the scattering of plane waves by the potential q.

The explicit inverse and determinant of a class of matrices is given. The class is the Hadamard product of two already known classes. Its elements are defined by 3n − 1 parameters, analytical expressions of which compose the Hessenberg form inverse. These expressions enable a recursive formula to be obtained, which gives the inverse in O(n2) multiplications/divisions and O(n) additions/subtractions.

A method for constructing a pair of biorthogonal interpolatory multiscaling functions is given and an explicit formula for constructing the corresponding biorthogonal multiwavelets is obtained. A multiwavelet sampling theorem is also established. In addition, we improve the stability of the biorthogonal interpolatory multiwavelet frame by the linear combination of a pair of biorthogonal interpolatory multiwavelets. Finally, we give an example illustrating how to use our method to construct biorthogonal interpolatory multiscaling functions and corresponding multiwavelets.

In this paper, optimal consumption and investment decisions are studied for an investor who has available a bank account and a stock whose price is a log normal diffusion. The bank pays at an interest rate r(t) for any deposit, and vice takes at a larger rate r′(t) for any loan. Optimal strategies are obtained via Hamilton-Jacobi-Bellman (HJB) equation which is derived from dynamic programming principle. For the specific HARA case, we get the optimal consumption and optimal investment explicitly, which coincides with the classical one under the condition r′(t) ≡ r(t)

A monoenergetic point-source solution of the steady-state cosmic-ray equation of transport for cosmic-rays in the interplanetary region in which monoenergetic particles are released isotropically and continuously from a fixed heliocentric position is derived by Lie Theory. A spherically-symmetric model of the propagation region is assumed incorporating anisotropic diffusion with a diffusion tensor symmetric about the radial direction, and the solar wind velocity is radial and of constant speed V. Because of the point release the solution is non-spherically-symmetric.

Various problems related to the propagation of small amplitude long waves on the surface of superfluid helium (helium II), usually called third sound, are studied on the basis of the appropriate governing equations. The two-fluid continuum model due to Landau is considered, with the effects of healing and relaxation incorporated, and viscosity, heat conduction and compressibility terms retained. The helium vapour is treated as a classical (Newtonian) compressible gas and the exact jump conditions across the liquid/vapour interface are employed. These liquid, vapour and jump equations constitute the exact problem although, in an effort to reduce the complexity of the equations, a simplified set of ‘model’ surface boundaiy conditions is also introduced. This full set of equations is non-dimensionalised taking care that all physical parameters are defined using only the undisturbed depth of the layer as the appropriate length scale. The ratio depth/wave-lenght (δ) is then a separate parameter as is the wave amplitude/depth ratio (ɛ). The limit which corresponds to the wave under discussion is then ɛ, δ → 0 with all the other parameters fixed.

A number of analyses are presented, four of which describe various aspects of the linearised theory and two examine the nature of the far-field nonlinear problem. Using the simplified surface boundary conditions we discuss in turn: the wave motion in the absence of healing: the rôle of a second wave speed leading to a wave hierarchy; and the effects of healing. The final linearised problem makes use of the full vapour model, but again the healing terms are ignored. This latter analysis suggests that if the upper boundary of the vapour is sufficiently close to the liquid surface then third sound is suppressed.

The complexity of the equations, particularly when the nonlinear terms are to be examined, is such that the incompressible limit is now taken in the absence of both healing and relaxation. Imposing the physically realistic limiting process (ɛ, δ → 0) we show that the only equation valid in the far-field is the Burgers equation. However, we also demonstrate that allowing the other parameters to be functions of e (which is not physically realisable in practice) it is easy to derive, for example, the Korteweg-de Vries equation.

Computer or communication networks are so designed that they do not easily get disrupted under external attack. Moreover, they are easily reconstructed when they do get disrupted. These desirable properties of networks can be measured by various parameters, such as connectivity, toughness and scattering number. Among these parameters, the isolated scattering number is a comparatively better parameter to measure the vulnerability of networks. In this paper we first prove that for split graphs, this number can be computed in polynomial time. Then we determine the isolated scattering number of the Cartesian product and the Kronecker product of special graphs and special permutation graphs.

This paper deals with the question of the existence of a solution to the stationary-point problem corresponding to a given nonlinear nondifferentiable program. An existence theorem for the stationary-point problem is presented under some convexity and regularity conditions on the functions involved, which also guarantee an optimal solution to the nonlinear program.