John Hilton Grace

John Hilton Grace
Born	(1873-05-21)21 May 1873 Halewood, Lancashire
Died	4 March 1958(1958-03-04) (aged 84)
Nationality	British
Fields	Mathematics
Known for	Grace–Walsh–Szegő theorem
Notable awards	Fellow of the Royal Society^[1]

John Hilton Grace FRS^[1] (21 May 1873 – 4 March 1958) was a British mathematician. The Grace–Walsh–Szegő theorem is named in part after him.^[2]

Early life

He was born in Halewood, near Liverpool, the eldest of the six children of farmer William Grace and Elizabeth Hilton. He was educated at the village school and the Liverpool Institute. From there in 1892 he went up to Peterhouse College, Cambridge to study mathematics.^[1] His nephew, his younger sister's son, was the animal geneticist, Alan Robertson FRS.

Career

He was made a Fellow of Peterhouse in 1897 and became a Lecturer of Mathematics at Peterhouse and Pembroke colleges. An example of his work was his 1902 paper on The Zeros of a Polynomial. In 1903 he collaborated with Alfred Young on their book Algebra of Invariants.^[1]

He was elected a Fellow of the Royal Society in 1908.^[1]

He spent 1916–1917 as Visiting Professor in Lahore and deputised for Professor MacDonald at Aberdeen University during the latter part of the war.^[3]

In 1922 a breakdown in health forced his retirement from academic life and he spent the next part of his life in Norfolk.^[1]

He died in Huntingdon in 1958 and was buried in the family grave at St. Nicholas Church, Halewood.

Theorem on zeros of a polynomial

a(z)=a_0+\tbinom{n}{1}a_1 z+\tbinom{n}{2}a_2 z^2+\dots+a_n z^n

b(z)=b_0+\tbinom{n}{1}b_1 z+\tbinom{n}{2}b_2 z^2+\dots+b_n z^n

are two polynomials that satisfy the apolarity condition, i.e. $a_0 b_n - \tbinom{n}{1}a_1 b_{n-1} + \tbinom{n}{2}a_2 b_{n-2} - \cdots +(-1)^n a_n b_0 = 0$ , then every neighborhood that includes all zeros of one polynomial also includes at least one zero of the other.^[4]^[5]

Corollary

Let $a(z)$ and $b(z)$ be defined as in the above theorem. If the zeros of both polynomials lie in the unit disk, then the zeros of the "composition" of the two, $c(z)=a_0 b_0 + \tbinom{n}{1}a_1 b_1 z + \tbinom{n}{2}a_2 b_2 z^2 + \cdots + a_n b_n z^n$ , also lie in the unit disk.^[4]

Publications

Grace, J. H. (January 1936). "The actual irreducibility of some finite systems of invariant forms". Journal of the London Mathematical Society. s1-11 (1): 20–21. doi:10.1112/jlms/s1-11.1.20. JFM 62.0075.02. Zbl 0013.14601.
Grace, J. H. (April 1930). "Two contract theorems". Journal of the London Mathematical Society s1–5 (2): 121–124. doi:10.1112/jlms/s1-5.2.121.
Grace, J. H. (January 1930). "The algebraic expression of projective properties". Journal of the London Mathematical Society s1–5 (1): 62–67. doi:10.1112/jlms/s1-5.1.62.
Grace, J. H. (1928). "Binary and ternary forms with prescribed polar systems". Proceedings of the London Mathematical Society. s2-28 (1): 421–430. doi:10.1112/plms/s2-28.1.421. JFM 54.0133.03.
Grace, J. H. (January 1928). "Surfaces related to a rational normal curve". Journal of the London Mathematical Society s1–3 (1): 34–38. doi:10.1112/jlms/s1-3.1.34.
Grace, J. H. (July 1927). "Note on ternary forms". Journal of the London Mathematical Society s1–2 (3): 182–185. doi:10.1112/jlms/s1-2.3.182.
Grace, J. H. (January 1927). "The four square theorem". Journal of the London Mathematical Society s1–2 (1): 3–8. doi:10.1112/jlms/s1-2.1.3.
Grace, J. H. (July 1926). "A point in enumerative geometry". Journal of the London Mathematical Society s1–1 (3): 167–170. doi:10.1112/jlms/s1-1.3.167.
Grace, J. H. (1918). "Note on a Diophantine approximation". Proceedings of the London Mathematical Society. s2-17 (1): 316–319. doi:10.1112/plms/s2-17.1.316.
Grace, J. H. (1918). "Tetrahedra in relation to spheres and quadrics". Proceedings of the London Mathematical Society. s2-17 (1): 259–271. doi:10.1112/plms/s2-17.1.259. JFM 47.0612.01.
Grace, J. H. (1918). "The classification of rational approximations". Proceedings of the London Mathematical Society. s2-17 (1): 247–258. doi:10.1112/plms/s2-17.1.247. JFM 47.0166.01.
Grace, J. H. (1904). "Note on the foregoing paper". Proceedings of the London Mathematical Society s2–1 (1): 208–209. doi:10.1112/plms/s2-1.1.208.
Grace, J. H. (1904). "Extension of two theorems on covariants". Proceedings of the London Mathematical Society s2–1 (1): 151–153. doi:10.1112/plms/s2-1.1.151. JFM 34.0120.03.
Grace, J. H.; A. Young (1903). The Algebra of Invariants. Cambridge University Press.
Grace, J. H. "On the zeros of a polynomial". Proceedings of the Cambridge Philosophical Society 11: 352–357.
Grace, J. H. (May 1902). "On perpetuants". Proceedings of the London Mathematical Society. s1-35 (1): 319–331. doi:10.1112/plms/s1-35.1.319.
Grace, J. H. (May 1902). "Types of perpetuants". Proceedings of the London Mathematical Society. s1-35 (1): 107–114. doi:10.1112/plms/s1-35.1.107.
Grace, J. H. (March 4, 1901). "A theorem on curves in a linear complex". Proceedings of the Cambridge Philosophical Society 11: 132–133.
Grace, J. H. (March 1901). "Linear null systems of binary forms". Proceedings of the London Mathematical Society. s1-34 (1): 168–172. doi:10.1112/plms/s1-34.1.168. JFM 33.0126.02.
Grace, J. H. (November 1900). "On a class of plane curves". Proceedings of the London Mathematical Society. s1-33 (1): 193–196. doi:10.1112/plms/s1-33.1.193.
Grace, J. H. (1898). "Circles, spheres, and linear complexes". Transactions of the Cambridge Philosophical Society 16: 153–190.

References

1 2 3 4 5 6 Todd, J. A. (1958). "John Hilton Grace 1873-1958". Biographical Memoirs of Fellows of the Royal Society 4: 92–97. doi:10.1098/rsbm.1958.0008. JSTOR 769502.
↑ Hörmander, Lars (1954). "On a theorem of Grace". Mathematica Scandinavica 2: 55–64.
↑ Todd, J. A. (1959). "John Hilton Grace". Journal of the London Mathematical Society: 113–117. doi:10.1112/jlms/s1-34.1.113.
1 2 Szegő, Gábor (1922). "Bemerkungen zu einem Satz von J H Grace über die Wurzeln algebraischer Gleichungen". Mathematische Zeitschrift (in German) 13: 28–55. doi:10.1007/BF01485280.
↑ Rahman, Qazi I.; Gerhard Schmeisser (2002). "Grace's theorem and equivalent forms". Analytic Theory of Polynomials. Oxford University Press. p. 107. ISBN 0-19-853493-0.

External links

Works written by or about John Hilton Grace at Wikisource

Authority control	WorldCat VIAF: 49586237 LCCN: n86864707 ISNI: 0000 0001 2279 7477 SUDOC: 13993524X NLA: 36001772

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