The sound of music and its link with mathematics

This article discusses a relatively unnoticed application ofmathematics by describing its connection with an aspect of music,in particular, the musical scales. Stemming from a problem foundin a Year 9 mathematics textbook commonly used in Singapore,the article illustrates the role of mathematics in musical scalesby first considering the frequency ratios of consecutive musicalnotes in the ‘just scale’ and secondly explaininghow an anomaly in the ‘just scale’ caused by theuneven frequency ratios is resolved with the help of mathematics,thereby leading to the development of the well-tempered scale.The article ends with an exploration of the frequency ratiosof consecutive musical notes in the well-tempered scale. Notonly does the article aim to broaden the teachers’ horizonswith such an introduction to the mathematical aspect of music,it also hopes to enrich their mathematical experiences as well.     

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.     

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.     

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.     

关于理工科高等数学研究型教学与大学生创新意识培养研究的构想

由于高等数学是高校理工科一门重要的基础课,因此进行高等数学研究型教学与大学生创新意识培养研究是当今高校理工科专业高等数学教学中进行素质教育的必然要求.本文论述了其研究的目的、理论依据和意义,同时阐述了研究型教学与创新意识的培养问题研究的现状和趋势.最后提出了理工科高等数学研究型教学与大学生创新意识培养研究的实施方案和计划、实施细则、效益分析和研究环境.     

The hidden mathematics of the mars exploration rover mission

Conclusions  A lot of the mathematics of MER is hidden and not only from the public but even from the applied scientists working on the mission. As briefly sketched above, for the scientists, this could be disastrous in a worst-case scenario. The hiding of mathematics, both in our everyday life and within science itself, is a matter not often discussed in public —which in itself is a disaster, taking into account the consequences the hiding of mathematics might have for the public. We like to think that this article may help let in some light. Another question raised by our work is that of beliefs in mathematics. Only occasionally are the beliefs of mathematicians discussed. We found repeatedly that mathematical elements of MER are not actually considered to be mathematics among the applied scientists themselves, not on first hand anyway. Is this due to the fundamentally different views of what mathematics is between applied scientists (including engineers) and pure scientists of the 20th century? We do not know. Finally, we comment on the nature of the mathematics involved in MER. Because of the extreme nature of a Mars mission, one might expect “extreme” mathematics, mathematics developed for the sole purpose of this mission.     

About traveling salesmen and telephone networks—Combinatorial optimization problems at high school

This article introduces an investigation dealing with the question of what role the mathematical discipline “combinatorial optimization” can play in mathematics and computer science education at high school. Combinatorial optimization is a lively field of applied mathematics and computer science that has developed very fast through the last decades.     

Finite-length mask effects in the isolation oxidation of silicon

Author to whom all correspondence should be addressed A moving-boundary problem modelling the two-dimensional isolationoxidation of silicon is analysed in the limit of reaction-controlledoxidation for a finite-length nitride mask. Encroachment underthe mask caused by silicon oxidation then occurs from both sidesto produce two ‘bird's beaks’, and it is the interactionbetween these beaks on which attention is focused. This effect,termed ‘bird's beak punchthrough’, is currentlyof interest in submicron silicon-isolation technologies.     

A quote a day educates

Conclusion  I often ponder on my duties as a teacher of the subject I love. I feel I am responsible for more than simply transmitting knowledge. I wish I could help my students see mathematics from various vantage points. One of these should be from a point high enough to afford a full, sweeping view of the mathematical valley below—maybe missing the details we strive to convey in class-but seeing thelandscape of mathematics. Claude Bragdon said, “Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself.” I want my students to consider that such a bold statement might actually be true.     

Metaheuristics can solve sudoku puzzles

In this paper we present, to our knowledge, the first application of a metaheuristic technique to the very popular and NP-complete puzzle known as ‘sudoku’. We see that this stochastic search-based algorithm, which uses simulated annealing, is able to complete logic-solvable puzzle-instances that feature daily in many of the UK’s national newspapers. We also introduce a new method for producing sudoku problem instances (that are not necessarily logic-solvable) and use this together with the proposed SA algorithm to try and discover for what types of instances this algorithm is best suited. Consequently we notice the presence of an ‘easy-hard-easy’ style phase-transition similar to other problems encountered in operational research.     

Increasing Student Access to Qualified Science and Mathematics Teachers Through an Urban School‐University Partnership

Urban schools across the United States face a pervasive problem in their science and mathematics programs — a disproportionate number of the teachers in these classrooms are not certified, thus making them underqualified to teach these subject areas. Furthermore, urban schools deal with teacher shortages and attrition in these critical areas. The situation was found to be particularly severe in the Detroit Public School District. In response, Wayne State University and Detroit Public Schools embarked on a school‐university partnership program to prepare teachers in science and mathematics through an alternative pathway to teacher certification program. This partnership program has proven to be successful in recruiting, preparing, and retaining a significant number of qualified minority science and mathematics teachers to serve the students in Detroit schools.     

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 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.     

A changing trajectory of proof learning in the geometry inquiry classroom

The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’.     

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.     

Metaphors Reveal Preservice Elementary Teachers' Views of Mathematics and Science Teaching

Typically, experiences shape one's attitudes toward the “thing” that is being experienced, whether it is a person, pet, movie, etc. Classroom experiences also shape one's attitudes toward a subject, teacher, learning, and so forth. Studies have shown that attitudes become more negative as students move from elementary to secondary schools. Elementary preservice teachers have had many classroom experiences by the time they reach their methods courses. These varied experiences have shaped their attitudes about the learning and teaching of mathematics and science.     

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.     

A Short and Partial History of Probabilistic Normed Spaces

This contribution aims at presenting a survey of a portion of the theory of Probabilistic Normed spaces. No result will be proved, so that the reader is referred to the original sources for the proofs. The theory of PN spaces has many facets and touches on many branches of mathematics, for instance, geometry, functional analysis, topology, probability. This justifies the adjective “partial” that appears in the title. Therefore, it is perhaps better to declare from the start what one may expect from this survey. It is only natural to investigate which “classical”properties of normed spaces are preserved in the new setting. But it is probably more interesting to look for those properties that pertain to the new theory and which have no corresponding analogue in the classical theory. It must also be added that Probabilistic Normed spaces may be approached from different standpoints: they may be studied for their own sake, as a special subject in functional analysis or in topology that is worth investigating simply because it is there, or because it provides a tool to approach open problems or, again, to shed light on topics that one thought had been thoroughly investigated. In this, one thinks immediately of the possible applications in probability and statistics. This paper was written as a part of the project “Metodi stocastici in finanza matematica” of the Italian M.I.U.R..