douglas samuel jones mbe. 10 january 1922 - royal...

26 November 2013−−Douglas Samuel Jones MBE. 10 January 1922

B. D. Sleeman and I. D. Abrahams

originally published online June 24, 2015, 203-224, published 24 June 2015612015 Biogr. Mems Fell. R. Soc.

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DOUGLAS SAMUEL JONES MBE10 January 1922 — 26 November 2013

Biogr. Mems Fell. R. Soc. 61, 203–224 (2015)

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DOUGLAS SAMUEL JONES MBE

10 January 1922 — 26 November 2013

Elected FRS 1968

By B. D. Sleeman1 and I. D. Abrahams2

1School of Mathematics, University of Leeds, Leeds LS2 9JT, UKand

Division of Mathematics, University of Dundee, Dundee DD1 4HN, UK2School of Mathematics, University of Manchester, Manchester M13 9PL, UK

Douglas Jones was an extremely creative and influential mathematician. His contributions to the theory of electromagnetic and acoustic waves and his development of original and exceptionally powerful mathematical techniques with which to study them has led to the solution of problems of both practical and social importance. His work is fundamental to the design and performance of radar antennae wherein it is necessary to optimize their transmitting and receiving characteristics. Jones also investigated the manner in which electromagnetic waves interact with objects having sharp edges. These studies are basic to the construction of ‘stealth’ aircraft, in which the geometrical shape is designed to minimize the aircraft’s signature. When supersonic airliner capability was realized in the development of Concorde there was considerable public concern regarding the excessive noise created during take-off and landing and the impact of ‘sonic boom’ on built-up areas. This prompted investigations into the noise levels experienced on the ground due to a moving acoustic source. This inspired Douglas to develop a mathematical theory of noise shielding. To address these difficult problems he developed powerful techniques of analysis: these included the asymptotic expansion of multidimensional integrals and the generalization of the method of stationary phase; the solution of integral equations arising in diffraction and obstacle scattering theory; the development of multidimensional generalized functions; uniform asymptotics and Stokes’ phenomenon; the Wiener–Hopf technique; and powerful numerical techniques to solve integral equations arising in electromagnetic wave theory.

Douglas Jones was a very private man, not given to small talk, but once engaged was stimulating and amusing company and always happy to engage in the exchange of ideas. He

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206 Biographical Memoirs

was an important guiding light to young staff and research students, many of whom have gone on to distinguished careers.

He and his wife Ivy, who predeceased him, were a devoted and mutually supportive team. They were both very active in the work of Tenovus Scotland and the World Wildlife Fund for Nature. Douglas Jones was a fine man, a friend and mentor and is greatly missed. He was survived by his sisters Dot and Joyce (sadly Joyce died in early 2014) and two children Helen and Philip.

Early life and education

Douglas Samuel Jones was born on 10 January 1922 in Corby, Northamptonshire, the son of Jesse Dewis Jones (1887–1932) and Bessie Jones (1900–92; née Streather). At that time Corby was a small mainly farming village of two streets. At an early age Douglas was introduced to the things that a farmer had to know: scything and drying grass, milking cows and building haycocks. Although iron ore had been known to exist in Corby since Roman times, this was not exploited until the 1930s with the result that the farming community was destroyed and Corby was converted to an industrial town, the process being accelerated by the steel tube manufacturer Stewarts & Lloyds moving the inhabitants of a whole Scottish town down to Corby. The last time Douglas visited there, many years ago, a main road had been driven through the fields and lanes where he used to herd cattle and hitch up the pony and cart.

Douglas was the eldest of four children; he had one brother, Jimy (Gerald), and two sisters, Dot (Doris) and Joyce. Douglas’s father Jesse became a general manager for Tarmac (which merged with Lafarge to become Lafarge Tarmac in 2013) when Douglas was born. This position, however, brought with it the demand to move with the work. So, after a short move to Redcar, the family finally settled in Bilston, where Douglas spent his childhood.

Douglas’s father was a great sportsman. He captained the village cricket team and played soccer for Nottingham Forest. He was also a keen gambler, playing cards (at which he invariably won) and attending greyhound race meetings (where he frequently lost). He was also fond of horse racing, which he managed to attend more often than might be imagined because of his habit of working every day of the week including Sunday; this allowed him some flexibility in arranging time off. Although severely wounded in World War I and suffering therefrom to the end of his life, he maintained his interest in sport and was manager of the works sports teams in addition to his other duties. Whenever possible the Jones family was expected to accompany him on all these activities. Douglas’s mother was a keen sportswoman and she became the club tennis champion. It is fair to say that the Jones children probably saw more greyhounds, horses, football matches, etc., and, on Sundays, slag heaps (with the occasional side trip to the seaside) than many adults of that time saw in their whole lives. It was inevitable that such a hectic and busy life would take its toll on Jesse. Although he survived one bout of pneumonia in 1930, two years later he died of double pneumonia just two days after Douglas’s tenth birthday.

Jesse left behind the memory of a warm, loving, vigorous and kindly man, a house covered by an insurance policy, but no money; it had all gone on gambling but none on drink, for he was a strict teetotaller. Although he had little formal education, Jesse was of high intellectual capacity and taught himself shorthand at the age of 12 years. He had a great belief in education and left one injunction on Bessie, namely to see that the children had the best education possible.



Douglas Samuel Jones 207

With no money coming into the Jones household, Bessie was in dire circumstances because there was no widow’s pension. Jesse’s brothers and sisters offered to help by each taking one of the four children into their homes and bringing them up, leaving Bessie to look after herself. However, she had the courage and love to reject this offer; Douglas’s life might otherwise have been very different.

Douglas’s education began in 1927 at Ettingshall Primary School, known locally as the ‘The Tin’ because it was constructed from corrugated iron. From there he attended Wolverhampton Grammar School (1931–40). Among its alumni were A. E. H. Love (1863–1940; FRS 1894), famous for A treatise on the mathematical theory of elasticity (Love 1920) and the development of surface waves, now known as ‘Love waves’; A. Goldie (Professor of Pure Mathematics, University of Leeds 1963–86), known for ‘Goldie’s theorem’ in ring theory; and Mervyn King (now Baron King of Lothbury), Governor of the Bank of England and chairman of its Monetary Policy Committee from 2003 to 2013.

Douglas had been at Wolverhampton Grammar School as a fee-paying pupil for only a year when his father died and so the only hope of remaining there was to win a scholarship. This he managed to do; moreover, the scholarship included free lunches, free travel and a contribution towards the cost of clothing. His sisters and brother secured places at the Royal Wolverhampton School as boarders as soon as they were old enough to do so, and the expenses of keeping them during term time were therefore minimal. The school also had some notable alumni, including Eric Idle of ‘Monty Python’ fame, and Gilbert Harding, the journalist and radio and television personality of the 1950s. The company that had employed Douglas’s father contributed a small pension. To supplement the family income Douglas’s mother learned shorthand and became a shorthand teacher in the evenings, teaching local office girls at 6d. (2½p) an hour. So the family was able to survive financially, although when funds were at a low ebb it was necessary to take in lodgers. In these circumstances Douglas had to do his homework in the kitchen and in so doing acquired the ability to concentrate, which was so valuable in later life. Although the Joneses led a frugal life it was a harmonious family, thanks to the unfailing cheerfulness of Bessie. Douglas remembered Christmas as a glorious time—mixing the cake and pudding from carefully hoarded ingredients, making mince pies and preparing banana cream. Bessie’s sisters, not much better off than the Joneses, all managed to send the family Christmas presents and, on occasion, one of Jesse’s sisters would provide a Melton Mowbray pork pie together with a string of sausages.

At this time Douglas was in the classics stream at school, concentrating on Latin and Greek on the advice of his form master. In this stream he took the School Certificate, achieving second place among all entrants from the school in that year. Having obtained his School Certificate, Douglas thought about getting a job. However, it was decided that he should go into the sixth form and aim for the Executive Class of the Civil Service. The family doctor advised mortgaging the house to permit Douglas to train for a medical career, but that avenue was quite out of the question with three other children to think about. Furthermore, Latin and Greek offered few prospects for employment should Douglas miss out in the fierce competition for the Executive Class. So he decided to switch to mathematics in the hope that it would give him more opportunities in the employment scene. Douglas confessed that at this stage he never really liked the subject but was greatly influenced by his mother, who while at Kettering High School had developed a love for geometry. (Always mentally active, Bessie at the age of 70 years decided it was high time she learnt German.) For Douglas, school was six days a week with a large chunk of homework set for Sunday. Two half-days were devoted




to sport, of which Douglas took full advantage, eventually representing the school at chess (captain), football, cricket (captain) and fives (Eton and Rugby). Another school activity was the Officers’ Training Corps (OTC), in which he rose to the rank of company quartermaster sergeant. In such a system there was sufficient flexibility for a switch of subjects to be feasible, if rare, provided that one ‘put one’s back into it’. He was also Senior Prefect, Victor Ludorum and House Captain.

Douglas never took the Civil Service examinations because recruitment was stopped by World War II. Bessie now thought that a job was both possible and essential for her. She became a book-keeping clerk, a job that gave her much pleasure because she always enjoyed arithmetic. In addition there was more money coming into the household, so she contemplated the possibility of Douglas’s going to university. After her discussions with the headmaster, Douglas found himself taking the examinations for the Mathematical Scholarship at Corpus Christi College, Oxford. Douglas was awarded the second of these scholarships, the first going to R. H. Tuck (latterly Professor of Agricultural Economics at the University of Reading, 1965–82). However, F. B. Pidduck, Fellow of Corpus Christi College, at the time remarked that Douglas’s knowledge of mechanics was abysmal.

There was then the problem of finding enough money to supplement the scholarship, which was inadequate on its own. Fortunately, this was resolved by Douglas’s winning a Staffordshire Major Scholarship and by the allocation of a Kitchener Scholarship. Together these were sufficient to relieve his mother of the cost of supporting him. In fact, he was able to pay her a small weekly sum during the vacations. So armed with a Higher School Certificate in mathematics, physics and English together with a Mathematical Scholarship at Oxford, Douglas was able to give up gas-mask making and digging air-raid shelters, which he had been doing at home while his mother became a fire watcher and air-raid warden.

World War II service, 1942–45

Before entering active service, Douglas intended to train as an electrical engineer under a special scheme just begun at the Clarendon Laboratory (Oxford). Shortly afterwards he joined the Royal Air Force Volunteer Reserve (figure 1) and was appointed to the Signals Branch (Radio Detection Finding), later renamed the Signals Radar Branch. Although Douglas’s training had equipped him with some basic principles he had little knowledge of the practical realities of engineering, so the latter had to be absorbed rapidly under rather pressing wartime conditions.

The elder of Douglas’s sisters, Dot, joined the Women’s Auxiliary Air Force as soon as she was eligible; later, his brother, Jimy, joined the Tank Corps but his young life was cut short by Hodgkin’s disease. His younger sister, Joyce, was at school for the duration of the war.

The first military unit that Douglas served with was 85 Squadron, which was equipped with a modified version of the Douglas Havoc. The front of the Havoc was sliced off and replaced by a searchlight (Turbinlite; figures 2 and 3) powered by banks of accumulators distributed throughout the aircraft. The idea was that the Havoc would use its A.I. (Airborne Interception radar—an early form of radar) to find a German aircraft, then catch it in the spotlight and let a normal day fighter (such as a Hawker Hurricane; figure 4) attack it. The difficulties of coordinating such an attack at night can well be imagined, and the whole project was finally abandoned in early 1943, although Douglas had left the squadron by then.




After a short spell with a conventional night fighter squadron Douglas was posted to a squadron equipped with Boulton Paul Defiants (figure 5), which, because of their cumbersome turret and vulnerability, had been assigned the job of being the first airborne countermeasures group. These aircraft were installed with the exotically named ‘Moonshine’ and ‘Mandrel’ spoofer/jammer devices used to defeat German radar. This technology was developed at the Telecommunications Research Establishment, where Douglas came into contact with Martin (later Sir Martin) Ryle (1918–84 FRS 1952), the radio astronomer and later Nobel laureate, and William Cochrane. Douglas recalls flying many nights in the gun turret of the Defiant, making crucial measurements of German radar frequencies for jamming purposes. Although the turret allowed the Defiant to defend itself from more angles of attack than a conventional fighter, it also inhibited manoeuvrability. As a result the Defiant was a ‘sitting duck’ when attacked from below.

From then on, Douglas was always involved with countermeasures, arriving eventually with the formation of the 90 Group. Douglas remained a serving officer with an operational squadron with the rank of flight lieutenant. His war service was recognized by the award of an MBE and by being Mentioned in Dispatches. Douglas recalled playing in 1945 for a services cricket XI against an overseas services XI. In the latter was an Australian (very) fast

Figure 1. Douglas on joining the Royal Air Force Volunteer Reserve, December 1941. (Source: Bennett Clark Ltd, Wolverhampton.)




Figure 2. The Turbinlite. (Source: photo-graph MH5711, from the collections of the Imperial War Museums.)

Figure 3. Douglas Havoc Turbinlite. (Source: photograph MH5710 from the collections of the Imperial War Museums.)

Figure 4. Hawker Hurricane. (Online version in colour.)

Figure 5. Boulton Paul Defiant. (Copyright © RAF Museum, Hendon; reproduced with permission.)




soon-to-be-test-match bowler Keith Miller. Douglas said that as far as he was concerned every ball from Miller was to be a quick single! At the end of the war Douglas had just been posted to the Far East when word came that he was to be demobilized and should return to Oxford immediately.

Oxford; Massachusetts Institute of Technology, 1945–48

At Oxford, Douglas turned to the matter of becoming a mathematician, with an actuarial career vaguely in mind. His tutor at Corpus Christi College was F. B. Pidduck, one of the kindest men as far as his students were concerned but somewhat at loggerheads with the other Fellows, so that he never graced the Senior Common Room with his presence. Jones recalled that Pidduck was not a particularly good lecturer but was not the worst. That accolade went to J. H. C. Whitehead (1904–60; FRS 1944), who totally changed his notation and started the course afresh at every lecture. For Douglas the best lecturer was U. S. Haslam-Jones. During his time at Oxford, Douglas took part actively in field sports and also in squash and chess. In his final year (1947) Pidduck suggested that Jones widen his experience by going to the USA by trying for a Commonwealth Fund Scholarship. He was fortunate in his application. The other successful candidate that year was Freeman Dyson (FRS 1952), an already well-established theoretical physicist. It was as an undergraduate that Douglas wrote his first paper (1)*. This little note pointed out pitfalls in the use of complex conjugate functions as a means of determining equipotentials of an electrified curve as a result of the possible presence of singularities. After graduating with a 1st in Moderations and Finals (specializing in applied mathematics), Douglas went to Massachusetts Institute of Technology (MIT) with the intention of doing research under the supervision of the electrical engineer J. A. Stratton, noted for his classic text Electromagnetic theory (Stratton 1941). However, it was very clear soon after Douglas’s arrival that Stratton was fully occupied in his pursuit of the presidency of MIT (1959–66) and therefore had little intention of supervising any research students. All callers were discouraged by having to get past three secretaries in sequence before setting foot in his office. It was a technique that ensured that research students really undertook independent research. Undaunted, Douglas took the opportunity to learn physics from Victor Weisskopf (1908–2002), H. Feshbach (1917–2000) and R. D. Evans (1907–95). Douglas was particularly inspired by Evans’s lectures on radiation and its medical applications. Indeed, it was through the good offices of Evans that Douglas was able to participate in the first experiments on the absorption of calcium by bone, using tracer methods. Apart from helping with blood samples and ultracentrifuges, his main contribution to the research was in devising a formula governing the absorption. It also brought into sharp contrast the differences between research in the biomedical sciences and that in the physical sciences and engineering. At the end of this year Douglas went on a grand tour of the USA, visiting about 40 states in the company of John Westcott (1920–2014; FRS 1983). It was a wonderful experience and he would have been tempted to seek a position had it not been for his desire to return to see his mother and the rest of the family. In addition, Douglas had secured an appointment as assistant lecturer at Manchester University.

* Numbers in this form refer to the bibliography at the end of the text.




Manchester, 1948–57

The Mathematics Department of Manchester University at the time of Douglas’s appointment was led by Sydney Goldstein (1903–89; FRS 1937), Beyer Professor of Applied Mathematics (1945–50), Max Newman (1897–1984; FRS 1939), Fielden Professor of Pure Mathematics, and James (later Sir James) Lighthill (1924–98; FRS 1953), who had recently been appointed to a senior lectureship (1946) and who succeeded Goldstein in 1950 to the Beyer Chair. Alan Turing (1912–54; FRS 1951) was appointed to a readership in 1948 before becoming Deputy Director of the Computing Laboratory and working on the software for the Manchester Ferranti Mark 1 computer. With his future secure, Douglas married Ivy Styles on 23 September 1950 in Bilston, Wolverhampton. It was a most happy, loving and companionable marriage.

Douglas’s first researches concerned acoustic and electromagnetic diffraction by bodies with sharp edges. With F. B. Pidduck (2) he developed a perturbation method, valid for high conductivity, to find an approximate solution to the problem of diffraction by a highly conductive wedge. Between 1950 and 1952 (3–5) Douglas set down a sound theoretical basis for obtaining rigorous solutions to diffraction problems involving edges. In particular he established sufficient conditions at edges and corners to ensure uniqueness of solutions. At that time no such conditions had been rigorously established, although their need had been noted by C. J. Bouwkamp in 1946. Jones also demonstrated (5) that some of the solutions then available did not satisfy these conditions. He also demonstrated (6) that the usual method of reducing an electromagnetic scattering problem, when the scattering object is of infinite extent, to an integral equation produced a field that failed to satisfy the radiation condition. His alternative formulation avoided this difficulty.

Having established the theoretical basis for the correct formulation of scattering problems, Douglas developed powerful new methods to handle a variety of scattering problems involving parallel planes, and cylinders of semi-infinite or finite length. In particular, Douglas modified and extended the Wiener–Hopf technique so that it was directly applicable to such problems; for bodies of finite length his is the established approach taken to this day. In the 1950s the standard procedure to employing the Wiener–Hopf technique was first to use Green’s theorem, or a similar identity, to construct an integral equation (or a system of such equations) defined on a half-line in physical space. This could then be transformed into a functional equation in a suitable complex-variable plane. Such an approach was used by J. Schwinger in the USA and by E. T. Copson in the UK; however, it was cumbersome and often required the construction of complicated Green functions. Douglas (7) showed that the process could be very significantly simplified by directly applying the two-sided Laplace transform (or equivalently a Fourier transform) to the governing equation and boundary conditions and thence relatively easily obtaining the associated Wiener–Hopf functional equation holding in a strip in the Laplace transform complex plane. This is now the standard approach for formulating diffraction problems via the Wiener–Hopf technique. Douglas’s implementation of the Wiener–Hopf technique is referred to as ‘Jones’s method’.

As an aside, Ben Noble’s book (Noble 1958) entitled Methods based on the Wiener–Hopf technique, first published in 1958, remains to this day the definitive text on the subject. It was Ian Sneddon (1919–2000; FRS 1983), in his capacity as editor of the International Series of Monographs in Pure and Applied Mathematics, who suggested to Noble (1922–2006) that he write a book on the subject. It is believed that Douglas supported Ben Noble in the writing of the book, which was mostly based on Douglas’s own notes, articles and bibliography list.




Douglas Jones turned his attention in the mid 1950s to the solution of scattering of acoustic and electromagnetic waves by a circular disc. The scattering problem for an acoustically soft disc can be reduced to finding the solution of a Fredholm integral equation of the first kind with kernel

2 22ð i 2 cos

2 20

e d .2 cos

k x y xy

x y xy

θ

θθ

+ −

+ −∫

He then made the extraordinary observation (8) that by writing the Fredholm equation as a pair of integral equations of Volterra type it was fairly easy to invert the integral equation with kernel

2ð 2 2

2 20

cos 2 cos d ,2 cos

k x y xyx y xy

θθ

θ

+ −

+ −∫

which leads to a more effective way of solving the problem. Indeed, the final result of a series of complicated manipulations leads to the solution of the original problem in terms of a Fredholm integral equation of the second kind with kernel

sin ( ) .k x yx y

−−

This equation is especially well suited to numerical evaluation in the low-frequency limit.In 1956 Douglas held a visiting professorship at the Courant Institute of Mathematical

Sciences in New York, where he collaborated with Morris Kline (1908–92) on the asymptotic behaviour of integrals. This led to their seminal paper (11) on the asymptotics of two-dimensional integrals of the form

i ( , )( , )e d d ,kf x yg x y x y∫∫which occur frequently in physical applications and are of intrinsic mathematical interest. Although there had been earlier work on the topic, this paper pioneered techniques and results that are now standard textbook material. Furthermore, it gave powerful impetus to the use of generalized functions (or distributions) in addressing the problem of the asymptotic behaviour of integrals with oscillatory integrands. Before the final draft of the paper was completed, Jones and Kline entered into a flurry of correspondence across the Atlantic. In a letter to Douglas dated 19 April 1956, Morris wrote:

I attended a meeting on electromagnetic problems about ten days ago and Prof. Erdélyi [1908–77; FRS 1975] was there. I discussed the idea of this paper with him and in fact gave him a verifaxed copy whose content is exactly the same as the enclosed except that the figures were not ready at the time. As you know from previous correspondence I have been a little concerned as to the justification of this paper in view of Focke’s results [Focke 1954]. I gave some arguments in the text which I hope you will comment on if you feel more or less should be said. I discussed this point with Erdélyi and he believes that the method of this paper is simpler and more direct. He feels that Focke’s method is rather clumsy.

In response Douglas wrote back on 5 July:

I agree with Erdélyi that the simpler the method the better, but I do not see how we can get a simpler approach than we have already. The basic ideas are in a sense almost trivial, that of transforming a double integral to a single Fourier integral. The actual labour of evaluating the

2π

2π




expression can scarcely be avoided in view of complicated form that it takes. Having made that remark, I shall, of course discover a paper in which everything is put quite trivially.

On returning to Manchester, Douglas was promoted to senior lecturer but was soon to move to the University of Keele as head of department. During his time at Manchester, Douglas supervised several research students, some of whom went on to significant mathematical careers. Most notably these included W. E. Williams (Professor of Mathematics, University of Surrey), T. B. A. Senior (Radiation Laboratory, University of Michigan) and A. Sharples (New Mexico Institute of Mining and Technology, 1968–2000).

Keele, 1957–64

During his visit to New York in 1956 Douglas was thinking about his future and had applied to the University of Glasgow for the position of head of a new Department of Applied Mathematics and Simson Professor of Mathematics. He travelled to Glasgow to hear about their plans for the new department, having been one of two candidates on the shortlist. When he got there he found that the department was intended to teach engineering students, would have no undergraduates of its own and would not be able to take on any mathematical research students. Douglas was unsure how he could form a department along these lines. The matter was resolved by the appointment of the other candidate, Ian Sneddon. Sneddon had been the first holder of the Chair of Mathematics at the University of Keele. So as Ian Sneddon headed north to Glasgow, Douglas and Ivy set off for Keele. It was during the early years in Keele that their daughter, Helen Elizabeth, was born on 28 May 1958, followed by their son, Philip Andrew, on 9 February 1960.

High-frequency scattering was the subject that Douglas Jones researched on arrival in Keele. His work on scattering by convex bodies already contained the ingredients that were so brilliantly brought to fruition in Joe Keller’s theory of geometrical diffraction (Keller 1962). To give the flavour of Jones’s arguments (9, 10), consider the scattering of a plane wave by a perfectly conducting convex obstacle. It is assumed that the radii of curvature at all points of the body are large compared with the wavelength so that the high-frequency approximation can be used. The points of glancing incidence will form a curve D (the shadow boundary) on the obstacle. At points on the illuminated side, not near D, the field is given by the geometrical acoustics approximation and here the coefficient σ, defined as the total energy flux outwards from the obstacle in the scattered wave divided by the energy flux in the beam of the incident wave that falls on the obstacle will be 2. In the neighbourhood of a point P of D, D will be well approximated by a cylinder whose axis is parallel to the tangent to D at P and whose radius is the radius of curvature R of the obstacle in a plane through P perpendicular to the tangent. Here Douglas ignored a small correction due to the variation of the radius of curvature in the penumbra. Then, if the tangent at P makes an angle π/2 − β with the direction of propagation of the incident wave, Douglas showed that the energy scattered per unit length of D, for an incident wave carrying unit energy per unit area, is

2/3 1/3 1/30 cos ,b k R β−

where k is the incident wavenumber and b0 = 0.9962 in the acoustically sound hard case and −0.8640 in the sound soft case. The scattering coefficient of the obstacle can then be shown to be given by




1/3 1/302/3

0

2 cos dD

b R sk S

σ β= + ∫ ,

where S0 is the projected area of the body on a plane normal to the direction of propagation of the incident wave and s is the arc length of D. Douglas then went on to determine explicit values of σ for the sphere and the spheroid and extended the ideas to high-frequency scattering by electromagnetic waves.

It was at Keele that Douglas first showed his remarkable ability to balance outstanding leadership and commitment to administrative duties with mathematical research and scholarship. For a while as the head of department busily engaged in building on the work done by Ian Sneddon to secure a lively and active department, and as Dean of Science from 1959 to 1962, he continued his researches into high-frequency scattering and wrote his monumental 807-page book The theory of electromagnetism (12). This book alone would have cemented Douglas’s reputation as world leader. It was an ambitious work with the aim, as Douglas wrote in the preface, ‘to provide a text which will take the student from a first acquaintance with Maxwell’s equations to within striking distance of modern research.’

In 1962–63 Douglas made a return visit to the Courant Institute of Mathematical Sciences, New York, and embarked on his remarkable contributions (13, 14) to the problem of high-frequency diffraction by a circular disc. Here he showed that the governing integral equation for scattering by a sound soft circular disc could be reduced to a Fredholm equation with kernel

i ( ) i ( )e e .k x y k x y

x y x y

− +

+− +

He then recast this equation into an equation of the second kind that could be solved very effectively at high frequency.

At about the time of his return from New York, Queen’s College in the University of St Andrews was wishing to found a Department of Applied Mathematics, and Douglas and Ivy turned their thoughts to a future career in Dundee.

Dundee, 1965–92

At the time of Douglas Jones’s appointment as Ivory Professor of Applied Mathematics, the Department of Mathematics was small and headed by W. N. (Norrie) Everitt (1924–2011), who had been appointed to the Baxter Chair of Mathematics in 1963. Douglas’s appointment came amid the result of the Robbins Report on higher education, which recommended that at least one and perhaps two new universities be founded in Scotland. As a consequence, Queen’s College became the University of Dundee in 1967. Douglas wasted no time in setting about building Applied Mathematics with several appointments. One of us (B.D.S.) was indeed fortunate to have been one of his earliest appointees. In addition to recruiting young applied mathematicians, Douglas also had the foresight to see that numerical analysis had to be a vital component in the development of applied mathematics. So it was that a chair in numerical analysis was founded in 1967. The first holder of this chair was the inspirational A. R. (Ron) Mitchell (1921–2007). Under their leadership the department grew rapidly in both number and prestige, and its influence was felt not only in Scotland but also throughout the UK and internationally. The late 1960s and early 1970s saw the initiation of the biennial conferences




devoted to numerical analysis held in the odd years and to the theory of differential equations held in the even years. The Numerical Analysis Conference continues to this day and is now hosted by the University of Strathclyde. In addition, the Department of Mathematics initiated two Master of Science courses: one in numerical analysis run by Ron Mitchell, and one in functional analysis and differential equations, run by Norrie Everitt. The research environment in the department also benefited from a constant flow of visitors, research students and postdoctoral fellows, many of whom have gone on to distinguished careers. Among the more notable applied mathematicians are W. G. C. Boyd (Bristol), P. Davies OBE (Strathclyde), R. Fletcher (Dundee; FRS 2003), P. Grindrod CBE (Oxford), the late D. Morgan, A. D. Rawlins (Brunel), B. P. Rynne (Heriot-Watt), N. H. Scott (East Anglia), P. D. Smith (Macquarie, Australia), D. Wall (Canterbury, New Zealand) and G. E. Tupholme (Bradford).

In his early years in Dundee, Douglas embarked on a programme of developing existing and new mathematical techniques with which to rigorously address many problems arising in acoustic and electromagnetic wave theory. Perhaps as a result of his work on the asymptotic behaviour of integrals (11) he worked on multidimensional theory of generalized functions, a subject dealt with earlier in the one-dimensional case by Lighthill and G. Temple. In this regard, it is fitting to note the following remark of Lighthill relating to the theory of generalized functions made at a conference in 1992 at Dundee University to mark Douglas’s 70th birthday; it concerned Douglas’s book The theory of generalised functions (21):

I have moreover been overjoyed that my tiny 80-page Introduction to Fourier analysis and generalised functions [Lighthill 1958], which concentrates on functions of just one variable, has proved to be a suitable appetite-whetting ‘starter’, as it were, leading up to Douglas’s superbly concocted ‘main dish’ in 540 pages which extends all the results in a comprehensive fashion and includes the corresponding properties of functions of many variables.

The method of geometrical diffraction is an important technique for studying the propagation of sound at high frequencies. It is based on an assumption of locally plane wave behaviour, except in certain regions where properties change abruptly. In these latter regions, coefficients are determined from certain canonical problems that enable the calculation of the transition from one kind of local behaviour to another. Douglas brought a new insight into diffraction by considering the problem of reflection and transmission at high frequencies of an acoustic wave by a curved interface between two media of differing refractive indices. Contrary to what would be predicted by geometrical acoustics, total reflection does not completely annihilate the transmission of energy. Rather, propagating waves emerge from a region of evanescence and are responsible for an energy flow. The rays that give rise to this flow were called ‘tunnelling rays’ by Douglas (19). There is a connection here with the concept of ‘electromagnetic wormholes’ that arise in the study of ‘cloaking devices’ (Greenleaf et al. 2009). Although the energy transmitted by tunnelling rays on a single reflection may not be large, there is the possibility, in situations where multiple reflections are important, of accumulation with a significant flux in directions where it may not be expected. Tunnelling rays do not occur with plane interfaces but are a feature of the effects of curvature.

Another topic in which Douglas’s insight has led to significant advances is in the solution of the problem of scattering of waves by an obstacle. For simplicity we consider the two-dimensional acoustic sound-hard scattering problem. If D is the exterior of a simple closed curve C and the function G is a solution of Helmholtz’s equation that has the property of behaving like a source on the curve C and satisfies the Sommerfeld radiation condition, then




the solution ϕ of the scattering problem when restricted to the curve C must be a solution of the integral equation

1 ( ) ( ) ( , )d ( ) ( ),2 qC

p q G p q s q f pn

φ φ ∂− + =

∂∫where

( ) ( ),q

q f qn

φ∂=

∂

for all points q on the bounding curve C.If, as is frequently done, we choose G to be the free-space Green function, then it is

well known that the uniqueness of the solution fails at those values of the wavenumber that coincide with an eigenvalue of the interior sound soft problem for C. This difficulty can be overcome by a variety of devices but at the price of introducing other complications, which lead to increased computer time and storage. Fritz Ursell (1923–2012; FRS 1972) suggested (Ursell 1973) a beautiful modification that was relatively simple to apply and was potentially useful in computations. It did have the practical drawback of requiring the computation of an infinite series. To address this point, Douglas (16) proposed a further modification in which an arbitrary limit is imposed on the size of the wavenumber k. His analysis then led to the replacement of the above integral equation by

1 ( ) ( ) ( , ) ( , ) d ( ) ( ),2 pC

p q G p q p q s q f pn

µ µ χ ∂

− + + = ∂ ∫

to be solved for the unknown function μ, where G is the free space Green function and χ represents a suitable finite series of radiating surface harmonics. The solution ϕ to the scattering problem is then given by

[ ]( ) ( ) ( , ) ( , ) d ( ).C

P q G P q P q s qφ µ χ= + ∫The solution obtained in this manner overcomes the non-uniqueness inherent in the classical approach.

With the growth of air travel in the 1970s, accompanied by the introduction of large-capacity jet airlines, the noise generated on the ground by such aircraft became a real concern. The noise produced by the supersonic airliner Concorde at take off and landing and also flying at speeds in excess of Mach 1 over land proved, in particular, a major issue. The problem of noise pollution from aircraft remains to this day, especially for people living in densely populated areas in close proximity to airports.

Armed with his considerable array of modelling and mathematical techniques, Douglas investigated a wide range of problems in which the occurrence of noise and the means to limit it was of major interest. In particular he was concerned with acoustic noise generated by sharp edges such as the leading and trailing edges of wings, and the noise created by gaps in wings when flaps were lowered in the process of landing. Another problem was to assess the noise created on the ground when jet engines were either situated above or below aircraft wings. Douglas (17) developed rigorous theories of such situations and at the same time built on asymptotic ray methods of Keller and Lighthill’s theory of aerodynamic noise (Lighthill 1952).

To illustrate Douglas’s contributions with a concrete example, he sought to understand the way in which sound waves are refracted or scattered by jets, which is relevant to




turbomachinery noise (including the humble hand-dryer) and also in aeroacoustics. The simplest model that can be contemplated is the situation of a plane vortex sheet, separating a still medium from one moving with uniform velocity and illuminated by acoustic radiation from a line source. This model was examined by Douglas and others in several articles (for example (15), where earlier references will be found). Perhaps surprisingly for such a simple problem, difficulties were found to arise when ensuring that there was no sound field before the source was switched on (that is, satisfying causality), being resolved ultimately by working with what Douglas described as ‘rather abstruse entities’ known as ultradistributions. The physical interpretation of these obscure generalized functions is not obvious and so doubt was cast on the adequacy of the model.

The trouble with the investigation of the vortex sheet stemmed from the presence of a certain non-real pole in complex wavenumber space. To attempt a deeper understanding of the issue, Douglas resolved to obtain an exact analytical solution of the sound scattered by a simple shear layer. In this model the flow increases linearly, from 0 below the layer to a constant speed U, say, above it. Suppose the layer is of height h and a monochromatic acoustic line source irradiates the layer with radian frequency ω so that a Strouhal number can be defined as ωh/U. This work was reported in a ‘tour-de-force’ article of 42 pages in Philosophical Transactions A (18). In it Douglas displayed masterful powers of analysis, extensive knowledge of special functions (Airy and Whittaker functions) and their properties, very delicate asymptotics including ray theory, and profound physical interpretation of the results. This allowed him to make a very careful examination of the singularity structure for arbitrary values of the Strouhal number, from which he discovered that there was an infinite number of poles, some of which lie on the real axis. The multiplicity of solutions for the shear layer is thereby far worse than that for the vortex sheet. However, it was found that as the Strouhal number was reduced, two of the real zeros came into coincidence and shifted off the real axis. The observation turned out to be crucial in resolving some of the non-uniqueness, especially because most of the poles ultimately gave an insignificant contribution to the field. The final step that disposed of all questions of uniqueness was causality, and this Douglas showed could be complied with by conventional functions; that is, there is no need to employ ultradistributions in the solution. Physically, Douglas deduced that for a Strouhal number below a critical value (when the poles have coalesced) the field contains a (standard) Helmholtz instability wave in the shear layer, whereas above this value there is no instability wave.

Perhaps stimulated by his war experiences, Douglas also made significant contributions to antenna theory. In (20) he wrote:

The concept of an antenna as a piece of wire or portion of dielectric which radiates electromagnetic energy is simple enough in principle, but the derivation of quantitative results of value for design purposes is fraught with difficulties. Even when the isolated antenna can be described as a straightforward boundary-value problem, it can rarely be solved with any ease. In fact the antenna, to be of any use as an element of a communication system, must be coupled with a transmission line or waveguide, and coupling forms an important but complicated part of any real system. For these reasons a substantial amount of analysis has been devoted to antennas, not always with success. The advent of large computers has made it possible to generate numerical answers to problems which had hitherto defied solution.

It must be confessed, however, that the mathematical detail has often obscured the physical principles involved leaving the engineer up in the air when both analysis and computer fail. For example, to keep computer requirements reasonable, some type of symmetry is often assumed but




the symmetry is usually lost as soon as a transmission line is connected. While it is our purpose to enumerate some of the analytical and numerical techniques that have been tried, it is hoped not to lose sight completely of physical principles which may be helpful.

These principles are clearly adhered to in Douglas’s contributions to antenna theory. On the one hand he provided a rigorous and complete analysis of problems, and on the other he gave clear formulations of the underlying principles and a simple exposition of formulae necessary for computational modelling.

Douglas returned to his studies of Wiener–Hopf problems in the early 1980s, motivated by new scattering problems involving multiple bodies or barriers. These led to coupled systems of such equations, for which no general constructive method of solution has yet been found. It remains to this day an important open problem in mathematics. However, Douglas obtained elegant solutions in some special cases of physical interest, and also offered the first class of ‘large’ coupled Wiener–Hopf systems that yields an exact solution (23), an extension of the Khrapkov–Daniele commutative (2 × 2) matrix form. Interested readers may wish to refer to Lawrie & Abrahams (2007) for details and an extensive bibliography.

During the 1970s and 1980s mathematicians began to direct their attention to the potential of exploiting mathematical ideas to address problems arising in the biological and medical sciences. This initiative arose, in part, from the development of the groundbreaking work done by Alan Turing on biological pattern formation and carried forward by J. D. Murray (FRS 1985) and others.

As a forward-thinking mathematician and scientist, Douglas realized that the new and rapidly evolving subject of ‘mathematical biology’ should be made accessible to undergraduate students. This led in 1983 to his co-authored book Differential equations and mathematical biology (26, 27). Mathematical biology is now recognized as a major field of applied mathematical research, and most universities in the UK and worldwide offer mathematical biology courses to students.

Notwithstanding all this inspiring and creative mathematical activity, Douglas was also an outstanding administrator and mentor. In addition to serving as head of department on several occasions he was also Dean of the Faculty of Science from 1976 to 1979. He was also highly regarded by the wider community for his wisdom and sound advice, serving on several influential boards and committees. For example, Douglas served on the Council of the Royal Society (1973–74), the Propagation Aerials and Waveguides Committee of the Electronics Research Council (1970–76), the Noise Research Committee of the Aeronautical Research Council (1971–79) and the Computer Board.

Douglas Jones was a tireless champion and campaigner for the promotion of mathematics and of the professional mathematician. He was appointed a member of the Mathematics Subcommittee of the University Grants Committee (UGC) in 1970, succeeding W. H. Cockcroft as its chairman in 1976 together with membership of the main committee. In 1981 he published the controversial report on behalf of the UGC entitled Whither mathematics?. The report highlighted the serious problems caused by the bulge in the group of academic staff 35–45 years of age that was reflected in the boom in recruitment in the 1960s as a consequence of the Robbins Report on university expansion. With a predicted fall by 36% in mathematically trained students, it was recommended that these staff in mid-career be compulsorily retired. As a result of both public and academic pressure, no government action was taken. As his fellow committee member B. G. (Brian) Gowenlock recalled (Gowenlock 1993):




Those were the days when smoking was still permitted in meetings and I was therefore treated to a sustained monthly dose of indirect smoking from Douglas. He was always content to say that the statistical chances of serious illness for himself were not sufficiently large to cause him to alter his habits and it was therefore left to me to assume that my own health was not at any serious risk.

Brian further recalled that Douglas was a delightful person to know and had the ‘gift of wisdom’.

As chairman of the UGC Mathematics Subcommittee, Douglas was also responsible for overseeing the development of computer science. In typical Douglas Jones style he began an address to the Inter-University Committee on Computing’s Colloquium in 1983 (22) as follows:

I must open with a disclaimer. Any views which I express must be treated as personal opinions and should not be attributed to the UGC Delphic oracle.

Secondly, when your kind invitation to address you was extended to me, the suggestion was made that I should attempt to assess where we were and peer into the murky bowl of the future to indicate the path we are about to tread. In the present foul political weather this is rather like batting when you can’t see the other wicket let alone the bowler and since my only claim to distinction in computing is that I am one of the few people in this room who learned it from Turing, I am none too sure whether it is the game of cricket or tennis which engages us.

Douglas then went on to try to outline his views on the course of the universities over the following decades.

Figure 6. Celebrating the 25th anniversary of the foundation of the Institute of Mathematics and its Applications, 1988. (Source: IMA; reproduced with permission.) (Online version in colour.)




Alan Turing must have had a significant effect on Douglas, so it should not come entirely as a surprise that he was also able to make a useful contribution to computing with his books on 80x86 assembly programming (24, 25). At the heart of nearly all major desktop computers today is an Intel or Intel-compatible central processing unit running in ‘virtual 80x86’ mode.

Douglas was a founding Fellow of the Institute of Mathematics and its Applications (IMA), served on its Council and was appointed President in 1988 (figure 6). It was during his presidency that he led the negotiations with the Privy Council that resulted in the IMA’s being incorporated by Royal Charter and subsequently being granted the right to award Chartered Mathematician status.

Personal thoughts (B.D.S.)

I was most fortunate to begin my academic career by being among Douglas’s early appointments at Queen’s College, Dundee. On my arrival I was immediately aware that teaching and scholarship in its widest form were of the highest priority. There was also no pressure on colleagues to write grant proposals but rather to pursue research for its own sake and to make original contributions. Douglas never directed the research of young staff but was always there to give encouragement and offer ideas. In my case, after a couple of years, Douglas told me that he thought it would be a good idea if I spent a year at the Courant Institute of Mathematical Sciences at New York University. In particular he arranged for me to work in Joe Keller’s group. So my wife, Julie, and I, together with two very young children, headed off to the ‘Big Apple’ and spent what for me was a momentous and exciting year, which had a fundamental influence on my career. There I met R. (Richard) Courant and enjoyed seminars by P. D. Lax, L. Nirenberg, J. Stoker and E. (Eugene) Isaacson as well as Joe Keller.

With regard to teaching back in Dundee, Douglas assigned lecturing duties that on the one hand one would enjoy and on the other he thought would be ‘good for the soul’. On my return to Dundee he assigned to me a new course on approximation theory, which was being offered to the first graduate students on the new Numerical Analysis and Programming Masters Course. I knew absolutely nothing about approximation theory and thought that Douglas had made a mistake with the assignment. So, plucking up courage I decided to go and discuss the matter with him. After knocking on his door and waiting for the red light to turn blue, indicating entry, I was ushered in. ‘Professor Jones,’ I said, ‘you have assigned the NAP course on approximation theory to me, but I know nothing about the subject.’ His response was firm and short, ‘Well you will do when you have given the course.’ Such was Douglas’s approach to many things, throwing out challenges and widening one’s horizons. In writing our book on Differential equations and mathematical biology, Douglas and I began by deciding what the book should contain and the audience for which we were writing, and we agreed on how the work should be shared in assigning chapters. From then on we worked independently and when one of us had finished writing a chapter it was passed to the other for comment. In this way we were both free to express our own ideas. Furthermore, the book has evolved over each edition, with new appraisals of content being assessed in each new writing.

Douglas was a great friend and mentor; he and Ivy (figure 7) were friends to our family. He will be greatly missed by all.




Honours and awards

1943 Mentioned in Dispatches1945 MBE (Military)1964 Fellow of the Institute of Mathematics and its Applications1967 Fellow of the Royal Society of Edinburgh1968 Fellow of the Royal Society1971–73 Keith Prize of the Royal Society of Edinburgh1975 Honorary DSc of the University of Strathclyde Marconi Prize of the Institute of Electrical Engineers1980 Honorary Fellow of Corpus Christi College Oxford1981 The Balthasar van der Pol Gold Medal of the International Union of Radio

Science (figure 8)1986 Naylor Prize and Lectureship of the London Mathematical Society1989 Fellow of the Institution of Electrical Engineers2013 Life Member of the Institute of Electrical and Electronics Engineers

Acknowledgements

We are grateful for access to Douglas’s autobiographical notes deposited with the Royal Society. One of us (B.D.S.) is indebted to Dot and Joyce for allowing access to Douglas’s library and permitting us to consult some of his

Figure 7. Douglas and Ivy. (Online version in colour.)




personal correspondence, and to Peter Grosvenor and Jane Pyzniuk for helping sort through the archive of relevant papers. Special thanks go to Paul Martin for his help in preparing an up-to-date list of Douglas’s publications. We are also grateful to many people who have shared their reminiscences with us: Brian Gowenlock, Barbara and Greg Kriegsmann, Jack Lambert, Tony Rawlins, Norman Riley, Paul Smith, David Thomas and Alistair Watson. Thanks are also due to Sophie Abrahams, Mark Chaplain, Bill Horspool, David Youdan and Peter Grosvenor for help with photographs. Finally we thank David Sleeman for his technical assistance, and Peter Grosvenor, Paul Martin, Juliet Sleeman and David Colton for their careful reading of several previous drafts and for their invaluable support.

The frontispiece photograph was taken by Godfrey Argent and is reproduced with permission.

References to other authors

Focke, J. 1954 Asymptotische Entwicklungen mittels der Methode der stationären Phase. Ber. Verh. Sächs. Akad. Wiss. Leipzig 101(3), 1–48.

Gowenlock, B. G. 1993 University Grants Committee. Bull. Inst. Math. Applic. 29, 99.Greenleaf, A., Kurylev, Y., Lassas, M. & Uhlmann, G. 2009 Cloaking devices, electromagnetic wormholes and

transformation optics. SIAM Rev. 51, 3–33.Keller, J. B. 1962 Geometrical theory of diffraction. J. Opt. Soc. Am. 52, 116–130.Lawrie, J. B. & Abrahams, I. D. 2007 A brief historical perspective of the Wiener–Hopf technique. J. Engng Math.

59, 351–358.Lighthill, M. J. 1952 On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. A 211, 564–587.Lighthill, M. J. 1958 An introduction to Fourier analysis and generalised functions. Cambridge University Press.

Figure 8. Celebrating the award of The Balthasar van der Pol Gold Medal, of the International Union of Radio Science, to Douglas in 1981. Left to right: Professor Jack Lambert, Douglas, Professor Ron Mitchell, Professor Norrie Everitt, in the Mathematics Department, Dundee University.




Love, A. E. H. 1920 A treatise on the mathematical theory of elasticity. Cambridge University Press.Noble, B. 1958 Methods based on the Wiener–Hopf technique for the solution of partial differential equations.

London: Pergamon Press.Stratton, J. A. 1941 Electromagnetic theory. New York: McGraw-Hill.Ursell, F. 1973 On the exterior problems of acoustics. Proc. Camb. Phil. Soc. 74, 117–125.

Bibliography

The following publications are those referred to directly in the text. A full bibliography is available as electronic supplementary material at http://dx.doi.org/10.1098/rsbm.2015.0005 or via http://rsbm.royalsocietypublishing.org.(1) 1948 Note on an electrostatic problem. Math. Gaz. 32, 84–85.(2) 1950 (With F. B. Pidduck) Diffraction by a metal wedge at large angles. Q. J. Math. 1, 229–237.(3) Note on diffraction by an edge. Q. J. Mech. Appl. Math. 3, 420–434.(4) 1952 Diffraction by an edge and by a corner. Q. J. Mech. Appl. Math. 5, 363–378.(5) The behaviour of the intensity due to a surface distribution of charge near an edge. Proc. Lond. Math.

Soc. 2, 440–454.(6) The removal of an inconsistency in the theory of diffraction. Proc. Camb. Phil. Soc. 48, 733–741.(7) A simplifying technique in the solution of a class of diffraction problems. Q. J. Math. 3, 189–196.(8) 1956 A new method of calculating scattering with particular reference to the circular disc. Commun. Pure

Appl. Math. 9, 713–746.(9) 1957 Approximate methods in high frequency scattering. Proc. R. Soc. Lond. A 239, 338–348.(10) High-frequency scattering of electromagnetic waves. Proc. R. Soc. Lond. A 240, 206–213.(11) 1958 (With M. Kline) Asymptotic expansions of multiple integrals and the method of stationary phase. J.

Math. Phys. 37, 1–28.(12) 1964 The theory of electromagnetism. Oxford: Pergamon Press.(13) 1965 Diffraction at high frequencies by a circular disc. Proc. Camb. Phil. Soc. 61, 223–245.(14) Diffraction of a high-frequency plane electromagnetic wave by a perfectly conducting circular disc.

Proc. Camb. Phil. Soc. 61, 247–270.(15) 1972 (With J. D. Morgan) The instability of a vortex sheet on a subsonic stream under acoustic radiation.

Proc. Camb. Phil. Soc. 72, 465–488.(16) 1974 Integral equations for the exterior acoustic problem. Q. J. Mech. Appl. Math. 27, 129–142.(17) 1977 The mathematical theory of noise shielding. Prog. Aerospace Sci. 17, 149–229.(18) The scattering of sound by a simple shear layer. Phil. Trans. R. Soc. Lond. A 284, 287–328.(19) 1978 Acoustic tunnelling. Proc. R. Soc. Edinb. A 81, 1–21.(20) 1979 Methods in electromagnetic wave propagation. Oxford University Press.(21) 1982 The theory of generalised functions, 2nd edn. Cambridge University Press.(22) 1983 Future prospects. IUCC Bull. 5, 113–117.(23) 1984 Commutative Wiener–Hopf factorization of a matrix. Proc. R. Soc. Lond. A 393, 185–192.(24) 1988 Assembly programming and the 8086 microprocessor. Oxford University Press.(25) 1991 80x86 assembly programming. Oxford University Press.(26) 2003 (With B. D. Sleeman) Differential equations and mathematical biology, 2nd edn. Boca Raton:

Chapman & Hall/CRC.(27) 2010 (With M. J. Plank & B. D. Sleeman) Differential equations and mathematical biology, 3rd edn. Boca

Raton: Chapman & Hall/CRC.


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