Dead loading of a unit cube of compressible isotropic elastic material

A unit cube of compressible isotropic elastic material undergoes a homogeneous dilatation by dead loading forces applied to its faces. Conditions are obtained for stability of the resulting equilibrium state. The physical nature of these conditions is described and the results are illustrated for a compressible Blatz-Ko foamed rubber material.     

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.     

Reduced-basis output bound methods for parabolic problems

^** Email: rovas{at}uiuc.edu^*** Email: luc_machiels{at}mckinsey.com^**** Corresponding author. Email: maday{at}ann.jussieu.fr In this paper, we extend reduced-basis output bound methodsdeveloped earlier for elliptic problems, to problems describedby ‘parameterized parabolic’ partial differentialequations. The essential new ingredient and the novelty of thispaper consist in the presence of time in the formulation andsolution of the problem. First, without assuming a time discretization,a reduced-basis procedure is presented to ‘efficiently’compute accurate approximations to the solution of the parabolicproblem and ‘relevant’ outputs of interest. In addition,we develop an error estimation procedure to ‘a posteriorivalidate’ the accuracy of our output predictions. Second,using the discontinuous Galerkin method for the temporal discretization,the reduced-basis method and the output bound procedure areanalysed for the semi-discrete case. In both cases the reduced-basisis constructed by taking ‘snapshots’ of the solutionboth in time and in the parameters: in that sense the methodis close to Proper Orthogonal Decomposition (POD).     

Ordered C*-Modules

In this first part of a study of ordered operator spaces, wedevelop the basic theory of ‘ordered C^*-bimodules’.A crucial role is played by ‘open tripotents’, aJB^*-triple variant of Akemann's notion of open projection. 2000Mathematics Subject Classification 46L08, 47L07 (primary), 46L07,47B60, 47L05 (secondary).     

Large-amplitude love waves

In the context of the finite elasticity theory, we considera model for compressible solids called ‘compressible neo-Hookeanmaterial’. We show how finite-amplitude inhomogeneousplane wave solutions and finite-amplitude unattenuated solutionscan combine to form a finite-amplitude Love wave. We take alayer of finite thickness overlying a solid half-space, bothmade of different prestressed compressible neo-Hookean materials.We derive an exact solution of the equations of motion and boundaryconditions and also obtain results for the energy density andthe energy flux of the waves. Finally, we investigate the specialcase when the interface between the layer and the substrateis in a principal plane of the prestrain. A numerical exampleis given.     

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’.     

A Generic Identification Theorem for Groups of Finite Morley Rank

The paper contains a final identification theorem for the ‘generic’K^*-groups of finite Morley rank.     

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     

The Kissing Numbers of Convex Bodies -- A Brief Survey

Nearly four hundred years ago, the cubic close-packing of equalspheres in R³ was discovered by Kepler [21], in which each spheretouches 12 others. In 1694, Gregory and Newton discussed thefollowing thirteen spheres problem. Can a rigid material spherebe brought into contact with 13 other such spheres of the samesize? Gregory believed ‘yes’, while Newton thought‘no’. 1991 Mathematics Subject Classication 11H31,52C17.     

Modelling the growth of soil-borne fungi in response to carbon and nitrogen

Email: Angelique.Lamour{at}medew.fyto.wau.nl Growth of soil-borne fungi is poorly described and understood,largely because non-destructive observations on hyphae in soilare difficult to make. Mathematical modelling can help in theunderstanding of fungal growth. Except for a model by Paustian& Schnürer (1987a), fungal growth models do not considercarbon and nitrogen contents of the supplied substrate, althoughthese nutrients have considerable effects on hyphal extensionin soil. We introduce a fungal growth model in relation to soilorganic matter decomposition dealing with the detailed dynamicsof carbon and nitrogen. Substrate with a certain carbon: nitrogenratio is supplied at a constant rate, broken down and then takenup by fungal mycelium. The nutrients are first stored internallyin metabolic pools and then incorporated into structural fungalbiomass. Standard mathematical procedures were used to obtainoverall-steady states of the variables (implicitly from a cubicequation) and the conditions for existence. Numerical computationsfor a wide range of parameter combinations show that at mostone solution for the steady state is biologically meaningful,specified by the conditions for existence. These conditionsspecify a constraint, namely that the ‘energy’ (interms of carbon) invested in breakdown of substrate should beless than the ‘energy’ resulting from breakdownof substrate, leading to a positive carbon balance. The biologicalinterpretation of the conditions for existence is that for growththe ‘energy’ necessary for production of structuralfungal biomass and for maintenance should be less than the mentionedpositive carbon balance in the situation where all substrateis colonized. In summary, the analysis of this complicated fungalgrowth model gave results with a clear biological interpretation.     

An Extension of the Ham Sandwich Theorem

It is shown that both the ‘ham sandwich theorem’and Richard Rado's theorem on general measure (see [6]), whichis known to be a measure theoretic equivalent of E. Helly'stheorem on convex sets, belong to the same family of resultsabout geometric, extremal properties of measures which are definedon Borel sets in Rⁿ.     

Global structure stability of impact-induced tensile waves in a rubber-like material

^** Corresponding author. Email: makong{at}cityu.edu.hk This paper concerns the global structure stability of impact-generatedtensile waves in a 1D bar made of a rubber-like material. Becausethe stress–strain curve changes from concave to convexas the strain increases, the governing quasi-linear system ofpartial differential equations, though hyperbolic, fails tobe ‘genuinely non-linear’ so that the standard formof the initial-boundary value problem corresponding to impactis not well-posed at all levels of loading. However, Knowles(2002, SIAM J. Appl. Math., 62, 1153–1175) constructedthe solutions of the initial-boundary value problem correspondingto impact. Based on this, in this paper we prove the globalstructure stability of the impact-generated tensile waves constructedby Knowles. The method of the proof is constructive.     

On the numerical solution of involutive ordinary differential systems

We consider a Galerkin finite element method that uses piecewisebilinears on a modified Shishkin mesh for a model singularlyperturbed convection–diffusion problem on the unit square.The method is shown to be convergent, uniformly in the perturbationparameter , of order N^–1in a global energy norm, providedonly that N^–1, where O(N²)mesh points are used. Thuson the new mesh the method yields more accurate results thanon Shishkin’s original piecewise uniform mesh, whereit is convergent of order N^–1lnN. Numerical experiments support our theoretical results. Received 14 September, 1998. Revised 24 September, 1999.     

Some new aspects of the transient analysis of discrete-parameter Markov models with an application to the evaluation of repair events in power transmission

In Markov reliability modelling, a partitioned state space isused to describe the behaviour of a system each state of whichis associated with the system either being functional or underrepair. Such a system alternates between working and repairperiods indefinitely. Recent research results on the distributionof the sequences of the lengths of working and repair periodsafford the reliability analyst a set of system characteristicswhich can be used in addition to the traditional ones (reliability,point availability, etc.) to describe the system‘s transientbehaviour. In this paper, we present a concise derivation ofclosed-form expressions for the probability mass function andthe factorial moments of the total cumulative ’time‘spent in a subset of the state space by an irreducible or absorbingdiscrete-parameter Markov chain during the first n time instances.This result is then applied to analyse the sequence of repairevents categorized as ’minor‘ and ’major‘of a Markov model of a power transmission system. The numericalimplementation using the Macintosh version of MatLab is alsodiscussed.     

Contrary Statements About Mathematics

The preceding paper ‘Strong statements of analysis’by A. R. D. Mathias defends a so-called full-blooded set theorywithout full detail [3]. He again objects to a weak set theorywhich he calls ‘Mac’, in which the usual Zermelo–Fraenkelseparation scheme is required only for formulas with suitably‘restricted’ quantifiers. I had proposed that suchseparation is adequate for all standard uses of set theory inmathematics. But Mathias has not produced any counter examplesof actual mathematics which requires the use of a stronger separation.     

Seiberg-Witten Invariants and Surface Singularities. II: Singularities with Good C*-Action

A previous conjecture is verified for any normal surface singularitywhich admits a good C^*-action. This result connects the Seiberg–Witteninvariant of the link (associated with a certain ‘canonical’spin^c structure) with the geometric genus of the singularity,provided that the link is a rational homology sphere. As an application, a topological interpretation is found ofthe generalized Batyrev stringy invariant (in the sense of Veys)associated with such a singularity. The result is partly based on the computation of the Reidemeister–Turaevsign-refined torsion and the Seiberg–Witten invariant(associated with any spin^c structure) of a Seifert 3-manifoldwith negative orbifold Euler number and genus zero.     

Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem

We consider a Galerkin finite element method that uses piecewisebilinears on a modified Shishkin mesh for a model singularlyperturbed convection–diffusion problem on the unit square.The method is shown to be convergent, uniformly in the perturbationparameter , of order N^–1in a global energy norm, providedonly that N^–1, where O(N²)mesh points are used. Thuson the new mesh the method yields more accurate results thanon Shishkin’s original piecewise uniform mesh, whereit is convergent of order N^–1lnN. Numerical experiments support our theoretical results. Received 14 September, 1998. Revised 24 September, 1999.     

On the Algebra of Operations for Hopf Cohomology

In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liuleviciusdefined Steenrod squaring operations Sq^k on the cohomology ringof any cocommutative Hopf algebra over Z/2. Later, J. P. Mayshowed that these operations satisfy Adem relations, interpretedso that Sq⁰ is not the unit but an independent operation. Thus,these Adem relations are homogeneous of length two in the generators.This paper is concerned with the bigraded algebra B that isgenerated by elements Sq^k and subject to Adem relations; itshows that the Cartan formula gives a well-defined coproducton B. Also, it is shown that B with both multiplication andcomultiplication should be considered neither a Hopf algebranor a bialgebra, but another kind of structure, for which thename ‘algebra with coproducts’ is proposed in thepaper. 2000 Mathematics Subject Classification 55S10 (primary),55T15 (secondary).     

Two-dimensional nonlinear advection-diffusion in a model of surfactant spreading on a thin liquid film

The spreading of a localized monolayer of dilute, insoluble surfactant, discharged from a point source that moves at constant speed over a thin liquid film coating a planar substrate, is described according to lubrication theory by a pair of coupled nonlinear evolution equations for the monolayer concentration and the film depth h. Numerical and asymptotic techniquesare here used to show that the extent and structure of sucha spreading asymmetric monolayer can be well approximated bya single nonlinear advection–diffusion equation involving alone. At large times the solution is composed of three, spatiallydistinct, asymptotic regions: (i) a quasi-steady ‘nose’region (containing the source), in which there is a dominantbalance between two-dimensional nonlinear diffusion and advection;(ii) an ‘advective’ region, in which longitudinaladvection balances transverse diffusion; and (iii) a ‘tail’region, in which unsteady diffusion is dominant. In each region,local similarity solutions are obtained either exactly (inthe advective region) or approximately (elsewhere) by rescalingnumerical solutions of the initial-value problem. If the sourceconcentration decreases with time, it is demonstrated that the monolayer’s width is greatest in the tail region, whereasfor a source of increasing concentration the monolayer is widestin the advective region. For the simpler one-dimensional problemof a monolayer spreading from a line source, the same balanceshold but with transverse diffusion eliminated; here self-similarsolutions are found in all three regions that agree closelywith numerical solutions of the initial-value problem. Received 7 October, 1998. Revised 11 April, 2000. ⁺ antoine@mip.ups-tlse.fr Present address: Division of Theoretical Mechanics, Schoolof Mathematical Sciences, University of Nottingham , UniversityPark, Nottingham NG7 2RD, UK. Oliver.Jensen@nottingham.ac.uk.