Spectral analysis and computation for the Kuramoto-Sakaguchi integroparabolic equation

A spectral method is developed to numerically solve the so-calledKuramoto–Sakaguchi equation, which is a nonlinear integro-differentialequation of the parabolic type, governing the dynamical statisticalbehaviour of certain populations of nonlinearly coupled randomoscillators. The approach rests on explicit bounds for the spacederivatives of solutions, obtained via energy-like estimates.Bounds for the numerical approximations of solutions are given,and improved (sometimes appreciably) by means of an ‘aposteriori error analysis’. Plots are shown to illustratethe performance of the method, and comparison with a finitedifference approach is also made.     

On the Existence and Stability of Spatially Structured Solutions of the Reaction-Diffusion Equations

The reaction-diffusion equations for the well-known ‘Brusselator’chemical kinetic model are investigated when the model is madeconsistent with the principle of detailed balance. In contrastto the original model, the corrected one does not show solutionswith ‘spatial structure’ i.e. solutions with multipleinternal maxima and multiple internal global minima in bothdependent variables. Sufficient conditions for stability ofthe solutions are given in terms of a Rayleigh quotient forgeneral boundary conditions for nonlinear reaction-diffusionequations in general. For the particular case of the ‘Brusselator’it is shown that conditions for a change of stability are muchmore unlikely to be attained as a result of the restrictionsof detailed balancing. The detailed sufficiency condition forthe stability of any steady-state solution and for no branchingfrom the ‘equilibrium’ branch solution depends onwhether the solution has global extrema inside the region, onits boundary, or both     

On the Inherent Integration Structure of Nonlinear Systems

In this paper we characterize the inherent integration structureof affine nonlinear systems through a set of indices called—inanalogy with existing terminology for linear systems—theorders of the zeros at infinity. We show that our definitionencompasses earlier characterizations due to Hirschorn and Isidori.The discussion is entirely local in nature, so that we are ableto use recent results in the ‘geometric approach’to nonlinear system theory.     

Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems

Let X and Y be reflexive Banach spaces with strictly convexduals, and let T be a compact linear map from X to Y. It isshown that a certain nonlinear equation, involving T and itsadjoint, has a normalised solution (an ‘eigenvector’)corresponding to an ‘eigenvalue’, and that the sameis true for each member of a countable family of similar equationsinvolving the restrictions of T to certain subspaces of X. Theaction of T can be described in terms of these ‘eigenvectors’.There are applications to the p-Laplacian, the p-biharmonicoperator and integral operators of Hardy type.     

The flat-maximum effect and generic linear scoring models: a test

Received on 1 July 1991. Predicting human behaviour patterns with linear correlationmodels has absorbed researchers for the past five decades. Althoughmost observers generally concede that humans are inferior tosuch models in combining information, linear scoring modelsare unfortunately, plagued by the flat-maximum effect or the‘curse of insensitivity’. As Lovie & Lovie(1986)observe: ‘The predictive ability of linear models is insensitiveto large variations in the size of regression weights and tothe number of predictors.’ In essence, seemingly differentscoringmodels tend to produce indistinguishable predictive outcomes. Since its demonstration by Dawes & Corrigan (1974), observershave cast the flat maximum in a decidedly negative light. Incontrast, Lovie & Lovie (1986) present a provocatively contrarianview of the flat maximum‘s positive potential. In thissame vein, we examine the predictive power of a generic credit-scoringmodel versus individual empirically derived systems. If, asWainer (1976) noted in regard to the flat maximum, ’itdon‘t make no nevermind’, generic credit-scoringmodels could provide cheaper alternatives to individual empiricallyderived models. During the period 1984–8, a series of linear credit-scoringmodels were developed for ten Southeastern U.S. credit unions.For each credit union, stepwise multiple regression was employedto select a subset of explanatory variables to be used in adiscriminant analysis. A generic credit-scoring equation wasdeveloped from the resulting discriminant analyses using weightedaverage coefficients from five systems. The predictive powerof the generic model was compared to the predictive power ofholdout sample of the five remaining credit-scoring models. In all cases, the generic model's performance was very closeto that of the empirically derived models. Thus, our findingssupport Lovie & Lovie's (1986) challenge to the conventionalwisdom that the flat maximum casts a pall on the successfulmodelling of judgement processes. Indeed, the flat maximum impliesa positive role for simpler, and hence cheaper, generic models.Although further research is needed, it should be possible todevelop hybrid models with generic cores that perform as wellas empirically derived linear models.     

Order conditions for linearly implicit fractional step Runge Kutta methods

^{In this paper, we study the consistency of a variant of fractionalstep Runge–Kutta methods. These methods are designed tointegrate efficiently semi-linear multidimensional parabolicproblems by means of linearly implicit time integration processes.Such time discretization procedures are also related to a splittingof the space differential operator (or the spatial discretizationof it) as a sum of ‘simpler’ linear differentialoperators and a nonlinear term.}     

A comment on 'Stabilization and optimal control for nonlinear systems'

This note presents a counter-example, to demonstrate that thebasis for the proposed solutions to the nonlinear quadraticregulator problem given in the paper ‘Stabilization andoptimal control for nonlinear systems’ is not true ingeneral.     

Radial Limits of Bloch Functions in the unit Disc

Although a function in the Bloch space may have no radial limits,it is shown that there exist bounded linear functionals whichgive ‘average radial limits’ over an interval onthe boundary. An ‘abelian–tauberian’ theoremis proved, characterizing the existence of a radial limit ata given boundary point in terms of these functionals.     

The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Theory

The aim of this paper is to develop a straightforward analysisof the Galerkin method for two-dimensional boundary integralequations of the first kind with logarithmic kernels. A distinctivefeature of the analysis is that no appeal is made to ‘coercivity’,as a result of which some existence questions cannot be answereddirectly. In return, however, the analysis has no special difficultyin handling corners, cusps, or open arcs. Instead of coercivity,the central feature of the analysis is the positive-definiteproperty of the integral operator for small enough contours.Rates of convergence are predicted theoretically and, in particular,certain linear functionals are shown to exhibit ‘superconvergence’.Numerical results supporting the theory are given in the companionpaper Sloan & Spence (1987) for problems on both open andclosed polygonal arcs.     

Quadratisation de Certaines Structures de Poisson

On sait associer à certaines structures de Poisson surRⁿ, de 1-jet nul en 0, des actions de R² sur Rⁿ, donnéespar le ‘rotationnel’ de leur partie quadratiqueet un autre champ de vecteurs. Lorsque ces actions sont ‘nonrésonantes’ et ‘hyperboliques’, onmontre que ces structures sont ‘quadratisables’,en ce sens qu'il existe des coordonnées dans lesquelles,elles sont quadratiques. Dans le cas de la dimension 3, nosrésultats mènent à la ‘non-dégénérescence’générique des structures de Poisson quadratiquesà rotationnels inversibles. We can associate with some Poisson structures defined on Rⁿwith a zero 1-jet at zero, actions from R² on Rⁿ, given by the‘curl’ of their quadratic part and another vectorfield. Assuming that those actions are ‘hyperbolics’and without ‘resonances’, we give a normal formfor those structures. On R³, we prove that every quadratic Poissonstructure with invertible curl, is generically ‘non degenerate’.     

A queueing model of activities in an intensive care unit

^** Email: griffiths{at}cardiff.ac.uk Activities in an intensive care unit (ICU) at a major teachinghospital are modelled by means of a queue-theoretic approach.Using data supplied by the ICU relating to the admissions process,the bed availability and the length of stay of patients, itwas possible to fit theoretical distributions to the observed‘arrival’ and ‘service’ distributions.Queueing equations relevant to a multi-channel system havingrandom arrivals and hyper-exponential service times for eachchannel are set up, and solved iteratively. Results obtainedmatch well with observations, and the model is then utilisedto investigate several ‘what if? ’ scenarios. Referenceis made to a simulation model developed in conjunction withthe queueing model.     

Timeout control scheme for overloaded M/E2/1 queue

A timeout scheme is considered for controlling an infinite ‘firstcome, first served’ overloaded single-server queue. Inthe overload situation, a customer-rejection mechanism is usedfor timing out ‘older’ customers in the queue, i.e.excluding those who have waited longer than a certain time.Applying ‘level-crossing analysis’ to an M/E₂/1queue, exact analytic expressions of performance such as thedensity and distribution functions of waiting time of the customerswho get served, the mean delay of customers, successful throughput,and ‘goodput’ are determined for this queue.     

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domainswith piecewise smooth boundaries is presented. It is intendedfor problems describing the Stokes flow in the situations whereone has corners or cusps formed by parts of the domain boundaryand, due to the nature of the boundary conditions on these partsof the boundary, these regions have a global effect on the shapeof the whole domain and hence have to be resolved with sufficientaccuracy. The algorithm combines the boundary integral equationmethod for the main part of the flow domain and the finite-elementmethod which is used to resolve the corner/cusp regions. Twoparts of the solution are matched along a numerical ‘internalinterface’ or, as a variant, two interfaces, and theyare determined simultaneously by inverting a combined matrixin the course of iterations. The algorithm is illustrated byconsidering the flow configuration of ‘curtain coating’,a flow where a sheet of liquid impinges onto a moving solidsubstrate, which is particularly sensitive to what happens inthe corner region formed, physically, by the free surface andthe solid boundary. The ‘moving contact line problem’is addressed in the framework of an earlier developed interfaceformation model which treats the dynamic contact angle as partof the solution, as opposed to it being a prescribed functionof the contact line speed, as in the so-called ‘slip models’.     

Bifurcations of hyperbolic fixed points for explicit Runge-Kutta methods

Discretization of autonomous ordinary differential equationsby numerical methods might, for certain step sizes, generatesolution sequences not corresponding to the underlying flow—so-called‘spurious solutions’ or ‘ghost solutions’.In this paper we explain this phenomenon for the case of explicitRunge-Kutta methods by application of bifurcation theory fordiscrete dynamical systems. An important tool in our analysisis the domain of absolute stability, resulting from the applicationof the method to a linear test problem. We show that hyperbolicfixed points of the (nonlinear) differential equation are inheritedby the difference scheme induced by the numerical method whilethe stability type of these inherited genuine fixed points iscompletely determined by the method's domain of absolute stability.We prove that, for small step sizes, the inherited fixed pointsexhibit the correct stability type, and we compute the correspondinglimit step size. Moreover, we show in which way the bifurcationsoccurring at the limit step size are connected to the valuesof the stability function on the boundary of the domain of absolutestability, where we pay special attention to bifurcations leadingto spurious solutions. In order to explain a certain kind ofspurious fixed points which are not connected to the set ofgenuine fixed points, we interprete the domain of absolute stabilityas a Mandeibrot set and generalize this approach to nonlinearproblems.     

Kantorovich's hidden duality

A demonstration is given that the ‘resolving multipliers’used by Kantorovich in the solution of a production planningmodel are directly related to dual variables. The interpretationis based on the linear programme formulated by Vajda to representthe Kantorovich model. A numerical example is given.     

Some exact results for nonlinear diffusion with absorption

Simple exact solutions and first integrals are obtained fornonlinear diffusion incorporating absorption. These are obtainedby the standard techniques of separation of variables and theuse of invariant one-parameter group transforma-tions to reducethe governing partial differential equation to various ordinarydifferential equations. For two of the equations so obtained,first integrals are deduced which subsequently give rise toa number of explicit simple solutions. As with all special solutionsof nonlinear partial differential equations, the associatedinitial and boundary conditions are imposed by the particularfunctional form of the solution and irrespective of their physicalapplicability, simple exact solutions are always important,because one of the key features of nonlinearity is the rangeand variety of response which is often bizarre and unexpected,but which is frequently embodied in the simplest of exact solutions.Many of the solutions obtained here are illustrated graphicallywith particular reference to the phenomena of ‘extinction’and ‘blow-up’ and in general demonstrate a widevariety of differing physical response embodied in the disposableconstants, which is characteristic of nonlinear theories.     

Numerical analysis of boundary integral solution of the Helmholtz equation in domains with non-smooth boundaries

In the paper we carry out a complete analysis of several efficientnumerical methods for the solution of boundary integral equationsdefined on a non-smooth boundary. In particular the solutionof the Helmholtz equation in the exterior of a closed wedgeis studied. The analytical behaviour of the solution of theresulting boundary integral equation (with a non-compact operator)near the wedge is investigated. Numerical analysis of the collocationand iterated collocation method for the problem is presented.Graded meshes are used to reflect the ‘singular’behaviour of the analytical solution, as well as the degreeof the polynomial approximant, in order to yield results with‘optimal convergence rates’. Finally the convergenceanalysis of some modified two-grid iterative methods for thefast solution of the resulting linear systems is given and numericalresults are presented which agree with the theoretical predictions.     

Tollmien-Schlichting/vortex interactions in compressible boundary-layer flows

A weakly nonlinear interaction of oblique Tollmien-Schlichtingwaves and longitudinal vortices in compressible, high Reynoldsnumber, boundary-layer flow over a flat plate is consideredfor all ranges of the Mach number. The interaction equationsconsist of equations for the vortex which is indirectly forcedby the waves via a boundary condition, whereas a vortex termappears in the amplitude equation for the wave pressure. Thedownstream solution properties of interaction equations arefound to depend on the sign of an interaction coefficient. Thisparticular type of weakly nonlinear interaction was first proposedby Hall & Smith (1989), who considered incompressible flows;however, there are some errors in their formulation. Correctedresults for the incompressible regime are presented for comparisonwith those calculated for compressible flows. Compressibilityis found to have a significant effect on the interaction properties,principally through its impact on the waves and their governingmechanism, the ‘triple-deck’ structure. It is foundthat, in general, the flow quantities will grow slowly withincreasing downstream coordinate. However, for flows with Machnumber values below 2, there exists a small band of wave obliquenessangles for which the solutions terminate in abrupt, finite-distance‘break-ups’.     

On improving the Hashin-Shtrikman bounds for the effective properties of three-phase composite media

For a composite comprising an isotropic mixture of two isotropicdielectric materials, the Hashin-Shtrikman bounds for the overalldielectric constant tensor are attainable and hence are thebest possible. Considering instead a three-phase composite,the Hashin-Shtrikman bounds are the best that are known in termsof volume fractions alone, and yet, in the limit of vanishingvolume fraction of the material of greatest dielectric constant,the three-phase upper bound remains strictly greater than thetwo-phase bound. A similar comment applies to the lower bound,in relation to a small volume fraction of the material withthe smallest dielectric constant. Although this phenomenon mayreflect a limitation of the Hashin-Shtrikman methodology, itremains conceivable that some microgeometries exist for whichall the ‘third’ phase is positioned in regions ofhigh field concentration, so that it always has a large effect.This paper resolves this problem to some extent, by generatinga new upper bound that ranges continuously from the Hashin-Shtrikmantwo-phase bound to the Hashin-Shtrikman three-phase bound asthe volume fraction c₃ of the ‘third’ material increasesfrom zero. The Hashin-Shtrikman three-phase bound thus cannotbe optimal, at least when c₃ is small. The method of derivationof the new bound relies on an application of the theory of functionsof bounded mean oscillation, recently developed in the contextof bounding the behaviour of nonlinear composites.     

Involutions and Anti-Automorphisms of Algebras

This paper concerns involutions and anti-automorphisms on finite-dimensionalassociative algebras over a field. It is based on a survey talkgiven at the workshop on ‘Quadratic forms, algebras withinvolution and algebraic K-theory’ at the University ofNottingham, 13–15 May 2004. This workshop was part ofthe EU network HPRN-CT-2002-00287 on ‘Algebraic K-theory,linear algebraic groups and related structures’ and followedthe London Mathematical Society Midlands Regional Meeting inNottingham on 12 May, 2004. The contents of the article are as follows:

1. Introduction andhistorical remarks;
2. Some general results;
3. Central simplealgebras;
4. Group algebras.

2000 Mathematics Subject Classification16W10 (primary), 16K20 (secondary).