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Leon Mirsky, 1918-l 983

H. Burkill and Hazel Perfect Department of Pure University of Shefield Shefield, England

Submitted by Richard Brualdi

This profile was originally conceived by the editors of and Its Applications as a tribute to Leon Mirsky on the occasion of his retirement in September 1983 from a chair of pure mathematics in the , England. However, sadly, it has to serve as an obituary, for Leon died suddenly on 1 December 1983, shortly before his 65th birthday. Leon was born in Russia on 19 December 1918; his father was a doctor, his mother a dentist. His parents were unable to leave the country, but, when he was 8, they sent their son to live with an uncle and aunt in Germany. After Hitler’s accession to power the family moved to England and Leon had a brief spell at an English boarding school. In 1936 he proceeded to King’s College, London to read mathematics and a year later he won a university scholarship. His contemporaries there recall his undergraduate enthusiasm for the theory of numbers and his astonishing erudition in the subject even then. Other intellectual passions, so characteristic of him in his later life, were also evident at that time. In 1940 he obtained a First Class Honours B.Sc. degree and two years later he was awarded the degree of M.Sc. with distinction for a thesis on the theory of numbers. He was then appointed to a temporary assistant lectureship at the University of Sheffield, where he was to remain, with one brief interlude, for the rest of his life. The academic year 1944-45 was spent in Manchester, again as a temporary assistant lecturer, but he then returned to Sheffield, this time as a full-fledged assistant lecturer. Thereafter, successive promotions took him through the grades of lecturer, senior lecturer, and reader finally, in 1971, to a personal chair in pure mathematics. When Leon came to Sheffield, Ph.D. degrees were relatively rare and it was not until 1949 that Leon obtained his as a staff candidate and therefore without the benefit of supervision. In 1953 Leon married Aileen Guilding, who was then a lecturer in Biblical History and Literature and who subsequently became professor and head of her department. Leon’s love of mathematics and the delight he took in communicating and expounding it were manifest in every aspect of his work-his research, his writing, and his teaching.

LINEAR ALGEBRA AND ITS APPLICATIONS 61:1-10 (1984) 1

0 Elsevier Science Publishing Co., Inc., 1984 52 Vanderbilt Ave., New York, NY 10017 0024.3795/84/$3,00 2 LEON MIRSKY

In spite of his impressive output, as evidenced by the list of his publica- tions, Leon was modest to a fault about his achievements. He was also singularly devoid of ambition. His motive for pursuing research-and he did so relentlessly-was sheer curiosity. The problems welled up in a mind steeped in mathematics and he simply wished to know the answers. He was eager to share his thoughts with anyone who showed an interest, and if others solved his problems he was genuinely delighted. He could never understand the dog-in-the-manager behavior which he observed in some mathematicians; to him research was always a cooperative rather than a competitive activity. Leon was, indeed, the ideal collaborator. Those of us who had the privilege of working with him not only experienced the stimulus of his ideas, but also had the encouragement of more than generous acknowledgment of their own contributions. Many of Leon’s research papers were the fruit of his remarkable mathe- matical insight which enabled him to perceive links between previously unrelated ideas. This talent also led to a number of survey papers, for he relished the task of creating order out of chaos. All Leon’s writings are recognizable by their distinguished and distinctive style, and elegance was invariably combined with clarity. Leon was the great communicator in the lecture room as well as in print. Although he was the last person to lose sight of the primary purpose of lecturing, he always strove to blend instruction with entertainment. In some ways he treated the lecture room as a theatre. The meticulously presented mathematical argument was spiced with witty asides, but the student had to be quick to catch the joke. Of course the actor needed a costume and Leon always wore a gown (somewhat threadbare towards the end of his career), though his colleagues had abandoned theirs many years before. The student audiences were known to show their appreciation with applause, but he himself never referred to this unusual honor. Leon’s public lectures were particularly polished performances. With his inaugural lecture on “The Elements of Mathematics” he accomplished the near-impossible feat of hold- ing spellbound a mixed audience of mathematicians and nonmathematicians. The memory of this delightful occasion will long be cherished by his listeners. Leon was for a number of years an editor of this journal and of the louma of Mathematical Analysis and Applications. He was also one of the original editors of the magazine Mathematical Spectrum which is designed for a school and university student readership; and the articles he wrote for this magazine [64,79,82] are imbued with his infectious zest for mathematics. Although Leon was primarily devoted to mathematics, his mind had room for many other enthusiasms. Literature, particularly poetry, in English, German, and Russian, was his constant joy; he read very widely in history and H. BURKILL AND H. PERFECT 3 philosophy. But he was no intellectual snob: for instance he took much pleasure in the works of Gilbert and Sullivan and he was a devoted admirer of P. G. Wodehouse. A phenomenal memory enabled him to quote long passages from his favorite authors. All this made him an ever-fascinating companion. But 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. Leon’s mathematical interests fell largely within three areas: the theory of numbers, linear algebra, and . He achieved distinction in each. Broadly, these studies occupied him over three disjoint periods of time, and they are discussed under separate headings below. Professor C. Hooley is the author of the section on Leon’s research in the theory of numbers; we are most grateful for his contribution.

THE THEORY OF NUMBERS

One of Leon’s main interests in the theory of numbers was the subject of r-free numbers. An r-free number is an integer (usually positive) that is not divisible by the rth power of any integer other than 1, or, what is equivalent, by the rth power of any prime. Thus r-free numbers are like primes in that they can be isolated by an exclusion process analogous to the sieve of Eratosthenes. But the restriction of being r-free is of a less demanding nature than that of primality so that problems about r-free numbers are usually more tractable than their counterparts involving primes. Leon, in a series of papers, obtained theorems concerning r-free numbers that are parallel to such results as Vinogradov’s theorem for primes or to such hypotheses as the prime twins conjecture. As well as deriving new results, he substantially improved upon the work of earlier writers. Perhaps his most impressive contribution in this area is his theorem [ll] on the occurrence of groups of 1 integers n,n+a,,..., n + al_ i in which all members are r-free. This work, which has frequently been cited in the later literature, is a fine example of the techniques that Leon introduced into and that have had an important influence on other writers. Among the other topics in number theory that occupied Leon, we have only space to single out the subject of the d(n) and its distribution. Defined as the number of divisors of the positive integer n, the function d(n) can be shown to be O(nE) and therefore the number D(x) of values taken by it in the range n < x must be small compared with X. The 4 LEON MIRSKY more detailed behavior of D(x) is of considerable interest, but its full study is beset with difficulties. The striking asymptotic formula

2m(2y2 [(log X)] 1’2 (r logD(x) - -4 (3)“” log log x proved by Leon and Paul Erdos in 1952 [20] was therefore a significant development in the theory of D(x) and related sums. In 1977 Leon published an article [80] to commemorate the centenary of the birth of Edmund Landau. He wrote it, he said, partly as a tribute to Landau and partly “to discharge an almost personal obligation.” Modestly he called it “an extended footnote” to the obituaries by Hardy and Heilbronn and by Knopp, but it is a considerable piece of work on which Leon lavished much time and effort. After a biographical sketch it traces the later develop- ment of a number of topics on which Landau had worked, and in such a way that the nonexpert is kept wholly absorbed.

LINEAR ALGEBRA

In 1950 Leon was asked to give a course of lectures on linear algebra. Characteristically he threw himself wholeheartedly into the project and very soon the subject replaced number theory as his principal love. The first fruit of his devotion was his text book An Zntroduction to Linear Algebra, published in 1955, which went through a second edition and was recently reissued by Dover Publications. A mathematician educated during the last 20 years is accustomed to a host of books on linear algebra and may therefore find it difficult to appreciate the extent to which Leon’s book was a pioneering venture. But, unlike many such undertakings, it has stood the test of time: its wealth of material gives it the status of a work of reference, while its clarity and elegant style make it a delight to read. As the list of publications shows, the book was soon followed by a stream of papers on a variety of topics in linear algebra. However, there is one theme that recurs over and over again: the relation, first investigated by Hardy, Littlewood, and Polya, between doubly-stochastic matrices, the majorization of vectors, and convex functions. Thus, for instance, in one paper [38] Leon shows that parallel results hold for convex functions of a vector variable; in another [44] he derives a surprising consequence from the seemingly minor constraint that the convex functions should be positive. An example of a paper in quite a different area is [48]. Here s(f), the maximum modulus of H. BURKILL AND H. PERFECT 5 partial sums of roots of a manic polynomial f(x)= X” + alXnpl + . . . + a, is compared with m(f)=rnax(n,l~,1~,...,lu~1~). By means of an intricate argument various inequalities are obtained which can roughly be summarized in the statement s(f) = 0( m( f)). Th e survey papers [50,52] provide beguil- ing introductions to two aspects of theory, and are full of ideas and problems to stimulate the reader.

COMBINATORICS

The change in direction of Leon’s research from algebra to combinatorics came about quite naturally. His abiding interest in doubly-stochastic matrices led him to investigate some of their fundamental properties, such as the “pattern” of their elements, by means of simple applications of Philip Hall’s theorem on distinct representatives (or transversals). Leon’s subsequent con- tributions to transversal theory are numerous and penetrating. With particular insight he recognized the “self-refining” nature of Hall’s deceptively simple theorem and, by means of elementary constructions such as adjunction, extension, replication, etc., he was able to demonstrate that many results of great generality are simply corollaries of this one basic theorem. A theorem of his on common transversals [61] (which was later generalized by other writers in the context of linkages) provides a particularly good example of the power of his methods. These have, furthermore, proved especially effective in the solution of existence problems for integral matrices [62]. Leon was also one of the first to recognize the significance for transversal theory of the notion of an independence structure () [59]. A short paper [70], outside transversal theory, provides another example of Leon’s perspicacity. He observes that, if the roles of chains and antichains in Dilworth’s decomposition theorem are interchanged, the resulting “dual” statement is much more easily proved than the original but will nevertheless often yield the same corollaries. In the late 196Os, transversal theory was emerging as a subject in its own right, and Leon was just the right person to codify it and to chart the ground. He had found a project ideally suited to his taste and to his remarkable talent for exposition. The work culminated in the book Transversal Theory, pub- lished in 1971 by Academic Press. For well over a decade it has remained the standard reference book for combinatorialists in this field. While writing this book, Leon was simultaneously editing the volume Studies in Pure Mathe- matics which consists of 27 individual papers (about half of them in combina- torics, including one by himself [69]) presented to on his 65th birthday. Rado’s influence on Leon was as strong as Landau’s and even more 6 LEON MIRSKY pervasive, and this work fittingly celebrates a friendship that began in Sheffield in the early 1940s. Leon had a great interest in Ramsey theory but published little on the subject. Two contributions may, however, be noted: the mathematical intro- duction [83] which he wrote to a book on F. P. Ramsey that appeared in 1978, and a discussion paper [76] on an aspect of the work of Issai Schur which has close affinities with Ramsey’s theorem.

While writing this notice we have been in correspondence with many of Leon’s friends, particularly Professor and Mrs. Richard Rado, and we are grateful for the reminiscences of Leon which they have communicated to us.

LIST OF PUBLICATIONS

Books

An Zntroduction to Linear Algebra, Oxford University Press, 1955; reprinted 1961, 1972. (Reissued by Dover Publications Inc. New York, 1982.) Transversal Theory: An Account of some Aspects of Combinatorial Mathe- matics, Academic Press, New York, 1971. (Editor) Studies in Pure Mathematics: Papers on Combinatorial Theory, Analysis, Geometry, Algebra and the Theory of Numbers, presented to Richard Rado on the occasion of his sixty-fifth birthday, Academic Press, London, 1971.

Papers

1. (With T. D. H. Baber) Note on certain integrals involving Hermite’s polynomials, Phil. Magazine 25:532-537 (1944). 2. Note on an asymptotic formula connected with r-free integers, Quart. 1. Math. (Oxford) 18:178-182 (1947). 3. On the number of representations of an integer as the sum of three r-free integers, Proc. Cambridge Phil. Sot. 43:433-441 (1947). 4. On coprime values taken by given polynomials, Amer. Math. Monthly 55:88-89 (1948). 5. The additive properties of integers of a certain class, Duke Math. .Z. 15:515-533 (1948). 6. On a theorem in the additive theory of numbers due to Evelyn and Linfoot, Proc. Cambridge Phil Sot. 44:305-312 (1948). 7. Note on a theorem of Carlitz, Duke Math. .Z. 15:803-815 (1948). H. BURKILL AND H. PERFECT 7

8. A remark on D. H. Lehmer’s solution of the Tarry-Escott problem, Scripta Math. 14: 126-127 (1948). 9. On a problem in the theory of numbers, Simon Stevin 26:25-27 (1948-9). 10. On the distribution of integers having a prescribed number of divisors, Simon Stevin 26: 168- 175 (1948-9). 11. An arithmetical pattern problem relating to divisibility by r-th powers, Proc. London Math. Sec. 50:497-508 (1949). 12. The number of representations of an integer as the sum of a prime and a k-free integer, Amer. Math. Monthly 56:17-19 (1949). 13. A theorem on representations of integers in the scale of r, Scriptu Math. 15:11-12 (1949). 14. Summation formulae involving arithmetic functions, Duke Math. J. 16:261-272 (1949). 15. A property of square-free integers, .Z. Indian Math. Sot. 13:1-3 (1949). 16. On the frequency of pairs of square-free numbers with a given difference, Bull. Amer. Math. Sot. 55:636-639 (1949). 17. Generalizations of a problem of Pi& Proc. Boy. Sot. Edinburgh (A) 62:460-469 (1947-9). 18. A theorem on sets of coprime integers, Amer. Math. Monthly 57:8-14 (1950). 19. Generalizations of some results of Evelyn-Linfoot and Page, Nieuw Arch. v. Wisk. 27:111-116 (1950). 20. (With P. Erdos) The distribution of values of the divisor function d(n), PTOC.London Math. Sot. 2:257-271 (1952). 21. An inequality for positive definite matrices, Amer. Math. Monthly 62:428-430 (1955). 22. (With H. K. Farahat) A condition for diagonabihty of matrices, Amer. Math. Monthly 63:410-412 (1956). (Reprinted in Selected Papers on Algebra, Math Assoc. of Am., 1977.) 23. The spread of a matrix, Muthematicu 3:126-130 (1956). 24. The norms of adjugate and inverse matrices, Archiv der Math. 7:276-277 (1956). 25. A note on normal matrices, Amer. Math. Monthly 63:479 (1956). (Reprinted in Selected Papers Algebra, Math. Assoc. of Am., 1977.) 26. (With R. Rado) A note on matrix polynomials, Quart. 1. Math. (Oxford) 8: 128- 132 (1957). 27. On a generalization of Hadamard’s determinantal inequality due to Szasz, Archiv der Math. 8:174-175 (1957). 28. Inequalities for normal and hermitian matrices, Duke Math. I. 24:591-599 (1957). 29. Additive theory, Math. Gazette 42:7-10 (1958). 30. Matrices with prescribed characteristic roots and diagonal elements, J. London Math. Sot. 33:14-21 (1958). 8 LEON MIRSKY

31. On the minimization of matrix norms, Amer. Math. Monthly 65:106-107 (1958). (Reprinted in Selected Papers on Algebra, Math. Assoc. of Am., 1977.) 32. Proofs of two theorems on doubly-stochastic matrices, Proc. Amer. Math. Sot. 9:371-374 (1958). 33. (With H. K. Farahat) Group membership in rings of various types, Math. Zeitschrifi 70:231-244 (1958). 34. Maximum principles in matrix theory, Proc. Glasgow Math. Assoc. 4:34-37 (1958). 35. Diagonal elements of orthogonal matrices, Amer. Math. Monthly 66: 19-22 (1959). 36. Remarks on an existence theorem in matrix theory due to A. Horn, Monatsh. fir Math. 63:241-243 (1959). 37. On a convex set of matrices, Archiv der Math. 10:88-92 (1959). 38. Inequalities for certain classes of convex functions, Proc. Edinburgh Math. Sot. 11:231-235 (1959). 39. On the trace of matrix products, Math. Nachfichten 20:171-174 (1959). 40. An algorithm relating to symmetric matrices, Monatsh. fir Math. 64: 35-38 (1960). 41. Symmetric gauge functions and unitarily invariant norms, Quart J. Math. (Oxford) 11:50-59 (1960). 42. Problems of arithmetical geometry, Math. Gazette 44:182-191 (1960). 43. (With H. K. Farahat) Permutation endomorphisms and refinement of a theorem of Birkhoff, Proc. Cambridge Phil. Sot. 56:322-328 (1960). 44. Majorization of vectors and inequalities for convex functions, Monatsh. fiir Math. 65: 159- 169 (1961). 45. An existence theorem for infinite matrices, Amer. Math. Monthly 68:465-469 (1961). 46. Even doubly-stochastic matrices, Math. Ann&en 144:418-421 (1961). 47. Estimates of zeros of a polynomial, Proc. Cambridge Phil. Sot. 58:229-234 (1962). 48. Partial sums of zeros of a polynomial, Quart. _I. Math. (Oxford) 13:151-155 (1962). 49. A note on cyclotomic polynomials, Amer. Math. Monthly 69:772-775 (1962). 50. Results and problems in the theory of doubly-stochastic matrices, Z. Wahrscheinlichkeitstheorie 1:319-334 (1962-3). 51. Some applications of a minimum principle in linear algebra, Monatsh. fir Math. 67:104-112 (1963). 52. Inequalities and existence theorems in the theory of matrices, 1. Math. Analysis Appl. 9:99-118 (1964). 53. (With Hazel Perfect) Extreme points of certain convex polytopes, H. BURKILL AND H. PERFECT 9

Monatsh. j?ir Math. 68:143-149 (1964). 54. (With Hazel Perfect) Spectral properties of doubly-stochastic matrices, Monatsh. fir Math. 69:35-57 (1965). 55. (With Hazel Perfect) The distribution of positive elements in doubly- stochastic matrices, J. London Math. Sot. 40:689-698 (1965). 56. Transversals of subsets, Quurt. 1. Math. (Oxford) 17:58-60 (1966). 57. (With Hazel Perfect) Systems of representatives, J. Math. Analysis Appl. 15:520-568 (1966). 58. An inequality for characteristic roots and singular values of complex matrices, Monatsh. fir Math. 70:357-359 (1966). 59. (With Hazel Perfect) Applications of the notion of independence to problems of combinatorial analysis, J. Combinatorial Theory 2:327-357 (1967). 60. Systems of representatives with repetition, Proc. Cambridge Phil. Sot. 63:1135-1140 (1967). 61. A theorem on common transversals, Math. Annalen 177:49-53 (1968). 62. Combinatorial theorems and integral matrices, J. Combinatorial Theory 5:30-44 (1968). 63. (With Hazel Perfect) Comments on certain combinatorial theorems of Ford and Fulkerson, Archiv der Math. 19:413-416 (1968). 64. From rule of thumb to abstract structure, Math. Spectrum 1:21-24 (1968-9). 65. Transversal theory and the study of abstract independence, J. Math. Analysis Appl. 25:20!%217 (1969). 66. Pure and applied combinatorics, Bull. Inst. Math. Appl. 5:2-4 (1969). 67. (With R. A. Smith) Area1 spread of matrices, Linear Algebra Appl. 2: 127- 129 (1969). 68. HaII’s criterion as a self-refining result, Monatsh. ftir Math. 73:139-146 (1969). 69. A proof of Rado’s theorem on independent transversals, Studies in Pure Math. (Papers presented to Richard Rado on his sixty-fifth birthday, 1971) 151-156. 70. A dual of Dilworth’s decomposition theorem, Amer. Math. Monthly 78:876-877 (1971). 71. A footnote to a minimum problem of Mordeh, Math. Gazette 57:51-56 (1973). 72. (With H. Burkill) Monotonicity, J. Math. Analysis Appl. 41:391-410 (1973). 73. (With H. BurkiIl) Combinatorial problems on the existence of large submatrices I, Discrete Math. 6:15-28 (1973). 74. The rank formula of Nash-Williams as a source of covering and packing theorems, J. Math. Analysis Appl. 43:328-347 (1973). 10 LEON MIRSKY

75. Partitioned transversals, Glasgow Math. J. 15:14-16 (1974). 76. The combinatorics of arbitrary partitions, Bull. Inst. Math. Appl. 11:6-9 (1975). 77. A trace formula of John von Neumann, Mom&h.@ Math. 79:303-306 (1975). 78. (With H. BurkiIl) Comments on Chebycheff’s inequality, Periodica Math. Hung&x 6:3-16 (1975). 79. A case study in inequalities, Math. Spectrum 9:1-6 (1976-7). 80. In memory of Edmund Landau-Glimpses from the panorama of num- ber theory and analysis, Math. Scientist 2:1-26 (1977). 81. (With H. Burkill) Combinatorial problems on the existence of large submatrices II, Discrete Math. 20:103-108 (1977). 82. A medley of squares, Math. Spectrum 10:72-81 (1977-8). 83. An Introduction to the Mathematics of F. P. Ramsey contained in Foundations: Essays in Philosophy, Logic, Mathematics and Economics (D. H. Mellor, Ed.), Routledge, and Kegan Paul, 1978, pp. 10-13. 84. Review article on Combinatorics with Emphasis on the Theory of Graphs by J. E. Graver and M. E. Watkins (New York, Springer, 1977) Bull. Amer. Math. Sot. N.S. 1: 380-388 (1979). 85. The distribution of values of the partition function in residue classes, J. Math. Analysis AppZ. 93:593-598 (1983). 86. An inequality of the Markov-Bernstein type for polynomials, Siam 1. Math. Analysis 14: 10041008 (1983).

Received 10 April 1984