Music as Embodied Mathematics: A Study of a Mutually Informing Affinity

The argument examined in this paper is that music – when approached through making and responding to coherent musical structures,facilitated by multiple, intuitively accessible representations – can become a learning context in which basic mathematical ideas can be elicited and perceived as relevant and important. Students' inquiry into the bases for their perceptions of musical coherence provides a path into the mathematics of ratio,proportion, fractions, and common multiples. Ina similar manner, we conjecture that other topics in mathematics – patterns of change,transformations and invariants – might also expose, illuminate and account for more general organizing structures in music. Drawing on experience with 11–12 year old students working in a software music/math environment, we illustrate the role of multiple representations, multi-media, and the use of multiple sensory modalities in eliciting and developing students' initially implicit knowledge of music and its inherent mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.     

Why mechanics should be integral to secondary school mathematics

Mechanics has never been the most popular subject in A-levelmathematics, the UK’s public examination for 16–18-yearolds, either with students, teachers or educators. The attemptsto popularize mechanics have failed and it is conceivable thatthe subject will be dropped from the A-level syllabus in theforeseeable future. This article argues the importance of mechanicsand why it should be integral to secondary school mathematics:Mechanics is the exemplar of mathematical modelling, is thelogical point of entry for the enculturation into scientificthinking and provides the means to develop an understandingof the relationship between mathematics, the theoretical objectsof science and the way science and mathematics speak of theworld. It enables learners across the ‘ability range’to think in the abstract and as such should be taught priorto the 6th form, that is, prior to the UK’s post-compulsorylevel of education.     

Promoting student engagement with mathematics support

This article reports the findings of qualitative research undertakento seek to identify the key reasons why some students are notengaging with mathematics support provided by Loughborough University.The research involved a number of focus groups and ‘onthe spot’ interviews with ‘non-users’ fromacross the campus. Barriers identified include a lack of awarenessof the location of support and a fear of embarrassment. Furtherinterviews were conducted with regular users of the supportin an attempt to understand how some of these barriers to usagemight be overcome. The article will discuss actions that maybe taken to improve student engagement with mathematics supportand the issue of how student motivation may affect such action.     

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.     

Alan Breach Tayler

Alan Breach Tayler, CBE, Director of the Oxford Centre for Industrialand Applied Mathematics, died on 28 January 1995, aged 63. Alan went up to Oxford in 1951 to read Mathematics at BrasenoseCollege. He obtained a first, and after a brief excursion tothe Bristol Aircraft Company, he returned to work for a DPhilwith George Temple. His thesis, completed in 1959 and entitled‘Problems in compressible flow’, contained a mixtureof analytic, approximate and numerical solutions which foreshadowedthe new practical applied mathematics that he embraced later.He became University Lecturer and Tutorial Fellow of St Catherine'sSociety in 1959. During the next twenty-five years, Alan Tayler brought a newethos to applied mathematics. This change came about throughhis recognition that the status quo in the 1960s, which compriseda delicate balance between theory and practice in the area ofapplied mechanics, was capable of far-reaching generalisation;indeed, he saw that such a development was essential since thefollowing decades were to be dominated by computers and an ever-increasingneed for mathematical modelling. In 1967, with Leslie Fox, heinitiated the mathematical Study Groups with Industry, whereinacademic and industrial researchers interact in week-long workshops.These were an immediate success: (1) with industry, who foundnew insights into their problems and new recruiting possibilities;(2) with students, whose enthusiasm to use their theoreticalknowledge soon led to the highly popular MSc in MathematicalModelling and Numerical Analysis; and (3) with faculty, bothpure and applied, who found an undreamed of source of fascinatingnew theoretical problems. For example, one intellectual consequencewas the use of industrial case studies to uncover the new fieldof ‘free boundary problems’, on which several thousandlearned articles have appeared since 1970.     

Systems with Time Delay in the Calculus of Variations: The Method of Steps

In a recent article (see [8]), we derived necessary and sufficientconditions for minima for the fixed-endpoint problem in thecalculus of variations involving a constant delay in the phasecoordinates. These conditions are expressed, explicitly, interms of the first and second variations. The vanishing of thefirst variation is characterized in terms of an extended Euler'sequation, just as for delay-free problems, but the characterizationof the conditions on the second variation remained unsolved.In this paper we convert, through the ‘method of steps’,the delay problem into one without delay. Although this problemwill not have fixed-endpoint constraints, it allows us to introduce,in a natural way, the concept of ‘conjugate sequence’;this solves the main difficulty for delay problems, namely,to give initial conditions for existence and uniqueness of solutionsof the Hamiltonian system (which is a difference–differentialsystem with both advanced and retarded arguments). The conditionson the second variation are then characterized by an extra conditionwhich is based exclusively on a solution of a given matrix Riccatiequation.     

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.     

Free Groups from Fields

In the nineteenth century, field theory brilliantly resolveda number of questions that had taxed mathematicians for centuries;for example, ‘The circle cannot be squared’ by straightedge and compass, and solving polynomial equations by radicalsis not always possible. These successes have continued to beheld up as superb examples of the power of mathematical thought,and are demonstrated at an undergraduate level. The purposeof this article is to provide another such natural example whichleads to a concrete realisation of the free group on 2 generators.     

An economic production lot-size model with shortages and time-dependent demand

In the present article, a production-inventory model is developedover an infinite plan ning horizon where the demand varies linearlywith time, unit production cost is taken as a function of theproduction rate, and shortages in inventory are permitted andare fully back-ordered. The machine production rate, which isassumed to be flexible, is treated as a decision variable. Theassociated nonlinear programming problem is modified by usingthe barrier-function method, and then a search technique isused to find the solution numerically. The analysis of the presentmodel of the production system points to optimality under conditionsthat are commonly recognized as ‘just in time’.     

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’.     

Bounds and Self-Consistent Estimates for the Overall Properties of Nonlinear Composites

Variational ‘self-consistent’ estimates for nonlinearproblems are formulated, building on a variational formulationpreviously developed by the authors. The formulation employsa linear ‘comparison medium’ for whose propertiessome ‘self-consistent’ choice is made. In contrastto linear problems, three possible self-consistent choices presentthemselves. The results that they give are analysed for twoparticular systems (a nonlinear dielectric and a nonlinear lossycomposite) for which bounds are already available. Estimatesbased on self-consistent embedding of a single inclusion ina homogeneous matrix composed of ‘comparison material’are also developed.     

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.     

Monodromy and Connectedness

The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is ‘mobile’ (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from ‘relative’ or total-space irreducibilityto ‘absolute’ or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05.     

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).     

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     

Nathan Jacobson (1910-1999)

Nathan Jacobson, who died on 5 December 1999, was an outstandingalgebraist, whose work on almost all aspects of algebra wasof fundamental importance, and whose writings will exercisea lasting influence. He had been an honorary member of the Societysince 1972. Nathan Jacobson (later known as ‘Jake’ to his friends)was born in Warsaw (in what he describes as the ‘Jewishghetto’) on 5 October 1910 (through an error some documentshave the date 8 September); he was the second son of CharlesJacobson (as he would be known later) and his wife Pauline,née Rosenberg. His family emigrated to the USA duringthe First World War, first to Nashville, Tennessee, where hisfather owned a small grocery store, but they then settled inBirmingham, Alabama, where Nathan received most of his schooling.Later the family moved to Columbus, Mississippi, but the youngNathan entered the University of Alabama in 1926 and graduatedin 1930. His initial aim was to follow an uncle and obtain adegree in law, but at the same time he took all the (not verynumerous) mathematics courses, in which he did so well thathe was offered a teaching assistantship in mathematics in hisjunior (3rd) year. This marked a turning point; he now decidedto major in mathematics and pursue this study beyond College.During his final year at Alabama he applied for admission andfinancial aid to three top graduate schools in the country:Princeton, Harvard and Chicago. He was awarded a research assistantshipat Princeton; after the first year he was appointed a part-timeinstructor for two years, and during his fourth year he wasappointed a Procter Fellow. The stipend was enough to enablehim to make a grand tour of Europe by car in 1935, in the companyof two Princeton fellow-students at the time: H. F. Bohnenblustand Robert J. Walker.     

The Mysterious Mr. Ammann

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community ” is the broadest. We include “schools ” of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.     

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.     

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.