Number theory

Mirsky's early research concerned number theory. He was particularly interested in the r-free numbers, a generalization of the square-free integers consisting of the numbers not divisible by any rth power. These numbers are a superset of the prime numbers, and Mirsky proved theorems for them analogous to Vinogradov's theorem, Goldbach's conjecture, and the twin prime conjecture for prime numbers.^[2][3]

With Paul Erdős in 1952, Mirsky proved strong asymptotic bounds on the number of distinct values taken by the divisor function d(n) counting the number of divisors of the number n. If D(n) denotes the number of distinct values of d(m) for m ≤ n, then^[2][3]

${\displaystyle D(n)={\bigl (}1+o(1){\bigr )}\exp \left({\frac {\pi {\sqrt {8\log n}}}{{\sqrt {3}}\log \log n}}\right).}$

The Mirsky–Newman theorem concerns partitions of the integers into arithmetic progressions, and states that any such partition must have two progressions with the same difference. That is, there cannot be a covering system that covers every integer exactly once and has distinct differences. This result is a special case of the Herzog–Schönheim conjecture in group theory; it was conjectured in 1950 by Paul Erdős and proved soon thereafter by Mirsky and Donald J. Newman. However, Mirsky and Newman never published their proof. The same proof was also found independently by Harold Davenport and Richard Rado.^[6]

Linear algebra

In 1947, Mirsky was asked to teach a course in linear algebra. He soon after wrote a textbook on the subject, An introduction to linear algebra (Oxford University Press, 1955), as well as writing a number of research papers on the subject.^[2][3]

In his research, Mirsky provided necessary and sufficient conditions for the existence of matrices of various types (real symmetric matrices, orthogonal matrices, Hermitian matrices, etc.) with specified diagonal elements and specified eigenvalues.^[2]

He obtained a tightening of the Birkhoff–von Neumann theorem with H. K. Farahat stating that every doubly stochastic matrix can be obtained as a convex combination of permutation matrices. In Mirsky's version of this theorem, he showed that at most ${\displaystyle n^{2}-2n+2}$ permutation matrices are needed to represent every ${\displaystyle n\times n}$ doubly stochastic matrix, and that some doubly stochastic matrices need this many permutation matrices. In modern polyhedral combinatorics, this result can be seen as a special case of Carathéodory's theorem applied to the Birkhoff polytope. He also worked with Hazel Perfect on the spectra of doubly stochastic matrices.^[2]

Combinatorics

In the mid 1960s, Mirsky's research focus shifted again, to combinatorics, after using Hall's marriage theorem in connection with his work on doubly stochastic matrices. In this area, he wrote the textbook Transversal Theory (Academic Press, 1971), at the same time editing a festschrift for Richard Rado.^[3] He derived conditions for pairs of set families to have simultaneous transversals, closely related to later work on network flow problems.^[2] He also was one of the first to recognize the importance of transversal matroids,^[2][3] and he showed that transversal matroids can be represented using linear algebra over transcendental extensions of the rational numbers.^[2]

Mirsky's theorem, a dual version of Dilworth's theorem published by Mirsky in 1971, states that in any finite partially ordered set the size of the longest chain equals the smallest number of antichains into which the set may be partitioned. Although much easier to prove than Dilworth's theorem, it has many of the same consequences.^[2][3]