# Leon Mirsky

### Quick Info

Born
19 December 1918
Russia
Died
1 December 1983
Sheffield, England

Summary
Leon Mirsky worked in Number Theory, Linear Algebra and Combinatorics.

### Biography

Leon Mirsky was usually known as Leon but his full name was Leonid. His father was a medical doctor while his mother was a dentist. Conditions in Russia were difficult at the time Leon was born for the Russian Revolution had begun in 1917. The Revolution marked the start of a Civil War which took place over the years 1918-1921. After 1921 the New Economic Policy chose not to enforce total socialism, which the leaders wanted, but moved strongly towards state control of major industries. Mirsky's parents wanted to leave Russia but were unable to do so. However they were able to send Leon to Germany when he was 8 years old to live with an uncle and aunt. This move, Leon's parents hoped, would give Leon a chance of a better life than Soviet Russia presented. Leon's uncle was in the wool trade and the family were comfortably off. Leon spent five years in Germany where his education was of high quality since his family were passionately education minded. However, the Germany to which young Leon had entered in 1926 was suddenly dramatically changed in 1933 when Hitler and the Nazi party came to power. Leon's uncle and aunt decided quickly to move to England where they set up home in Bradford.

Herne Bay College, at Eddington south east of Herne Bay in Kent, was run by Captain Eustace Turner and it specialised in engineering, having one of the best equipped engineering workshops in England during the 1930s. The school had a high reputation for successes in engineering examinations. The school, from which Mirsky graduated in 1936, was taken over by the military for the war effort in 1939 and never reopened as a school. Leaving Herne Bay College, at age seventeen, Mirsky entered King's College, London, where he had to study four subjects in his first year for the Intermediate Science Examination. Three of these subjects were Pure Mathematics, Applied Mathematics, and Physics. Mirsky had already discovered at Herne Bay College that laboratory work was not his strong point so he was quite proud to cope with the laboratory work in the Physics course. His performance in the Intermediate Science Examination was outstanding and, as a consequence, he was awarded a scholarship to study the Course for the B.Sc. Special Degree in Mathematics. This degree was designed as a three-year course.

One of the friends that Mirsky made at King's College, London, was Geoffrey Kneebone who was one year ahead of him. Mirsky and Kneebone shared interests in philosophy, literature and history as well, of course, as mathematics. They shared their interest in exploring these topics throughout the rest of Mirsky's life. Another friend he made at King's College was Ron Clark who introduced Mirsky to rock climbing and the two shared this interest in the vacations when they went youth hostelling. Another fellow student with Mirsky was Peter Gant who later taught mathematics at Felsted School. Gant described Mirsky in his student days as having [3]:-
... formal dress, complete with rolled umbrella, which went well with his remarkable erudition but contrasted delightfully with his lively sense of fun.
During this time he took a passionate interest in the theory of numbers and became a great admirer of Edmund Landau. He read widely on his own and he kept a record of the parts of the theory that pleased him by filling up a whole series of notebooks.

On 1 September 1939 German armies invaded Poland and, following the British Prime Minister Chamberlain's ultimatum to Hitler, which was ignored, Britain declared war on Germany on 3 September. In August 1939, Russia and Germany had signed a secret pact, the so-called Ribbentrop-Molotov pact, to divide Poland between them. Soviet troops invaded Poland on 17 September and by 29 September Poland was partitioned between Russia and Germany. Mirsky was now in a very unhappy situation of being educated in England having been born in Russia and brought up for five years in Germany.

He graduated from King's College, London, in 1940 with a First Class Honours B.Sc. Special Degree in Mathematics. However the war meant that students from King's College were sent to Bristol University and it was there that he began to study for a Master's Degree. It may appear surprising that a student with such an outstanding undergraduate record did not immediately register for a Ph.D. but, feeling awkward as an alien in war-time Britain, he wanted to complete his studies and start a career as quickly as he could. He was lucky that with the move to Bristol, he now came in contact with Hubert Linfoot. Linfoot had spend the academic year 1928-29 at Göttingen in Germany attending lectures by Edmund Landau on Waring's problem and on Schlicht functions. Linfoot had taken notes of high quality of all courses he attended. He already spoke German fluently before his Göttingen visit, and after spending the year in Germany he was often mistaken for being German. In fact the courses Linfoot attended in Göttingen were delivered in German but written up by him in English. By the time Mirsky arrived in Bristol, Linfoot had begun to undertake research in optics which he did for patriotic reasons when war started. However he was still an excellent advisor for Mirsky who was keen to write his Master's thesis on number theory. He completed his Master's degree in a year and was awarded the degree with distinction.

In 1942 he was appointed as a temporary assistant lecturer at Sheffield University where Percy John Daniell was Town Trust Professor of Mathematics and the Head of Department. This was a difficult period with professors involved in war work and having relatively few students to teach due to the war. In fact Daniell, despite deteriorating health, worked for the Ministry of Supply. One important consequence of his period at Sheffield was the friendship that Mirsky formed with Richard Rado [3]:-
[Rado] and his wife Luise took Leon under their wing: they provided him with the affectionate home atmosphere that he craved as a newcomer to a strange city, and they enormously influenced his intellectual development as well. Leon greatly admired Richard's power as a mathematician and often paid glowing tribute to the benefit he derived from the association ... Luise Rado encouraged Leon's literary interests, particularly by reading English and German poetry with him. Jointly with Richard she opened the world of music to him and he became especially fond of the 19th century German songs which Luise used to sing to him.
After completing the two year temporary position, Mirsky went to the University of Manchester where he had again a temporary assistant lectureship. After one year in Manchester, he was offered an assistant lectureship (this time not a temporary one) back at Sheffield and he took up this position in 1945. This meant that he was again a colleague of Rado but this only lasted two years since, in 1947, Rado moved to King's College, London. In the same year of 1947 Mirsky was promoted to lecturer at Sheffield. We noted before that Mirsky had not undertaken research for a Ph.D. but once on the permanent staff at Sheffield he set his sights on earning a doctorate. Being a staff member, this was not a standard route to a Ph.D. and Mirsky worked on his own without the benefit of a thesis advisor. He began publishing many short papers on number theory: On the number of representations of an integer as the sum of three r-free integers (1947); Note on an asymptotic formula connected with r-free integers (1947); On coprime values taken by given polynomials (1948); The additive properties of integers of a certain class (1948); On a theorem in the additive theory of numbers due to Evelyn and Linfoot (1948); On a problem in the theory of numbers (1948); Note on a theorem of Carlitz (1948); A remark on D H Lehmer's solution of the Tarry-Escott problem (1948); Arithmetical pattern problems relating to divisibility by rth powers (1949); The number of representations of an integer as the sum of a prime and a k-free integer (1949); A property of square-free integers (1949); Generalizations of a problem of Pillai (1949); Summation formulae involving arithmetic functions (1949); A theorem on representations of integers in the scale of r (1949); On the distribution of integers having a prescribed number of divisors (1949), and On the frequency of pairs of square-free numbers with a given difference (1949). We see that he had at least 15 papers published in the three years 1947-49. He was awarded a Ph.D. in 1949 by Sheffield University.

In 1953 Mirsky married Aileen Guilding who was, at that time, a lecturer in Biblical History and Literature at Sheffield but later became a professor and Head of Department. Aileen and Leon Mirsky shared a love of music and literature. They also enjoyed long walks together in the Derbyshire countryside. Except for the year 1951-52 spent at Bristol (working with Heilbronn), he was to spend the rest of his career at Sheffield. He was promoted to Senior Lecturer in 1958, Reader in 1961 and Professor in 1971.

Mirsky had three main areas of research.
1. The theory of numbers, where he studied $r$-free numbers, i.e. numbers not divisible by the $r$th power of any integer. He obtained analogues of Vinogradov's result on the representation of an odd integer as the sum of three primes, the Goldbach conjecture on the representation of an even integer as the sum of two primes, and the twin primes conjecture.

2. Linear algebra, where he wrote his famous text An introduction to linear algebra (1955) and went on to publish 35 papers on the topic. In particular he proved results on the existence of matrices with given eigenvalues and given diagonal elements.
For extracts from reviews of his linear algebra book see THIS LINK.

3. Combinatorics, where he also wrote an important book Transversal Theory and he developed ideas coming from Hall's theorem:-
A finite family of sets has a system of distinct representatives if and only if the union of every $k$ sets of the family contains at least $k$ elements.
For extracts from reviews of his combinatorics book see THIS LINK.
Referring to the success of his two books, Linear Algebra and Transversal Theory the author of [1] writes that it:-
... bears witness to Mirsky's gift for exposition; his hallmark was elegance combined with clarity.

Let us look briefly at how he came to work on these three topics. His early work was on number theory which was his first love but, after Geoffrey Walker was appointed as his head of department at Sheffield in 1947 and asked Mirsky to teach a course on linear algebra, he became fascinated in that topic in addition to his number theory interest. However, when he was a morning speaker at the British Mathematical Colloquium at St Andrews in 1956 his topic was Additive prime number theory. He worked on doubly stochastic matrices in the first half of the 1960s, publishing papers such as Results and problems in the theory of doubly-stochastic matrices (1963) and two papers with Hazel Perfect, Spectral properties of doubly-stochastic matrices (1965) and The distribution of positive elements in doubly-stochastic matrices (1965). This led to him applying Hall's theorem and so his interests turned towards combinatorics. For example he published (with Hazel Perfect) Systems of representatives (1966), Systems of representatives with repetition (1967), (with Hazel Perfect) Applications of the notion of independence to problems of combinatorial analysis (1967) and A theorem on common transversals (1968).

Mirsky's talent for teaching was clearly related to his gifts as a writer and is described in [3] as follows:-
Leon was a born teacher. He welcomed the challenge of presenting a whole theory or just one proof in a logical, efficient, clear and elegant manner. ... His lectures ... were highly individual performances. There was never any hint of familiarity with his audience and Leon always wore a gown to emphasize the formality of the occasion. On the other hand, the alert student could spot a succession of jokes all made with an entirely straight face and no change of tone.
He believed that mathematics research should not be a solitary occupation, but firmly believed in collaborating with others [1]:-
He was particularly eager to share ideas at research level and firmly believed that research should always be a cooperative rather than competitive venture.
Outside mathematics Mirsky had a wide variety of interests [1]:-
His knowledge of literature, history and philosophy was wide and, in some areas, almost professional. Many of his former colleagues will miss the stimulus of conversations with him on these and a host of other subjects.
The authors of [3] go into more detail:-
Amongst scientists the awe-inspiring spread and depth of Leon's learning in the humanities must have been almost unique. In philosophical, historical and literary discussions he could hold his own with professionals. A phenomenal memory and an exceptional reading speed made this possible, but enthusiasm was the driving force. Literature, particularly poetry, gave him most pleasure. He was fluent in German and Russian, read comfortably in French and taught himself enough Italian to appreciate Dante in the original. But Leon's tastes reflected the lighter side of his nature as well. He was captivated by P G Wodehouse and he knew the Gilbert and Sullivan operas by heart. At one time he also read great quantities of detective novels - at great speed, of course. A somewhat unexpected interest of his was cricket. Friends were apt to regard this as an amusing pose, but they were wrong, for he found the spectacle of the game aesthetically pleasing and the tactics appealed to his intellect.
Mirsky retired from his chair in Sheffield in September 1983 and the paper [4] was planned to honour his achievements at Sheffield. However, he died suddenly in December 1983 and the paper [4] became an obituary. We end our biography with the following quote from [4]:-
... no account of Leon, however brief, is complete without mention of his kindness and the steadfast friendship he displayed throughout his life. In any crisis he was the person his friends would first turn to, and they were never disappointed.

### References (show)

1. Obituary in The Times
2. R A Brualdi, Review: Transversal Theory: An account of some aspects of combinatorial mathematics, by Leon Mirsky, SIAM Review 15 (2) (1973), 393-395.
3. H Burkill, W Ledermann, C Hooley and H Perfect, Obituary: Leon Mirsky, Bull. London Math. Soc. 18 (2) (1986), 195-206.
4. H Burkill and H Perfect, Leon Mirsky, 1918-1983, Linear Algebra Appl. 61 (1984), 1-10.
5. G M Merriman, Review: An Introduction to Linear Algebra, by Leon Mirsky, Amer. Math. Monthly 63 (10) (1956), 735-736.
6. W Ledermann, Review: An Introduction to Linear Algebra, by Leon Mirsky, The Mathematical Gazette 41 (337) (1957), 239.
7. D W Sharpe, Professor Leon Mirsky, Math. Spectrum 16 (2) (1983/84), 55.