**Joseph Miller Thomas** (16 January 1898 – 1979) was an American mathematician, known for the Thomas decomposition of algebraic and differential systems.^{[1]}

Thomas received his Ph.D., supervised by Frederick Wahn Beal, from the University of Pennsylvania with thesis *Congruences of Circles, Studied with reference to the Surface of Centers*.^{[2]} He was a mathematics professor at Duke University for many years. His graduate students include Mabel Griffin (later married to L. B. Reavis) and Ruth W. Stokes.^{[3]} In 1935 he was one of the founders of the *Duke Mathematical Journal*. For the academic year 1936–1937 was a visiting scholar at the Institute for Advanced Study.^{[4]}

Based upon earlier work by Charles Riquier and Maurice Janet, Thomas's research was important for the introduction of involutive bases.^{[5]}^{[6]}

## Selected publications

### Articles

- with Oswald Veblen: Veblen, O.; Thomas, J. M. (1925). "Projective Normal Coördinates for the Geometry of Paths".
*Proceedings of the National Academy of Sciences*.**11**(4): 204–7. doi:10.1073/pnas.11.4.204. PMC 1085921. PMID 16576871. - Note on the projective geometry of paths. Proceedings of the National Academy of Sciences 11, no. 4 (1925): 207–209.
- The number of even and odd absolute permutations of
*n*letters. Bull. Amer. Math. Soc. 31 (1925) 303–304. MR1561049 - Conformal correspondence of Riemann spaces. Proceedings of the National Academy of Sciences 11, no. 5 (1925): 257–259.
- Conformal invariants. Proceedings of the National Academy of Sciences 12, no. 6 (1926): 389–393.
- Asymmetric displacement of a vector. Trans. Amer. Math. Soc. 28 (1926) 658–670. MR1501370
- with Oswald Veblen: Projective invariants of affine geometry of paths. Annals of Mathematics 27 (1926): 279–296. doi:10.2307/1967848
- Riquier's existence theorems. Annals of Mathematics 30 (1928): 285–310. doi:10.2307/1968282
- Matrices of integers ordering derivatives. Trans. Amer. Math. Soc. 33 (1931) 389–410. MR1501594
- The condition for an orthonomic differential system. Trans. Amer. Math. Soc. 34 (1932) 332–338. MR1501640
- Pfaffian systems of species one. Trans. Amer. Math. Soc. 35 (1933) 356–371. MR1501689
- Riquier's existence theorems. Annals of Mathematics 35 (1934): 306–311. doi:10.2307/1968434 (addendum to 1928 publication in
*Annals of Mathematics*) - An existence theorem for generalized pfaffian systems. Bull. Amer. Math. Soc. 40 (1934) 309–315. MR1562842
- The condition for a pfaffian system in involution. Bull. Amer. Math. Soc. 40 (1934) 316–320. MR1562843
- Sturm's theorem for multiple roots. National Mathematics Magazine 15, no. 8 (1941): 391–394. JSTOR 3028551
- Equations equivalent to a linear differential equation. Proc. Amer. Math. Soc. 3 (1952) 899–903. MR0052001

### Books

*Differential systems*. 1937.^{[7]}*Theory of equations*. McGraw-Hill. 1938; 211 pages*Elementary mathematics in artillery fire, by Joseph Miller Thomas with tables prepared by Vincent H. Haag*. McGraw-Hill. 1942; 256 pages*Systems and roots*. William Byrd Press. 1962; 123 pages*A primer on roots*. William Byrd Press. 1974; 106 pages

## References

**^***Thomas Decomposition of Algebraic and Differential Systems*by Thomas Bächler, Vladimir Gerdt, Markus Lange-Hegermann, Daniel Robertz, 2010**^**Joseph Miller Thomas at the Mathematics Genealogy Project**^**Green, Judy; LaDuke, Jeanne (2009).*Pioneering Women in American Mathematics: The Pre-1940 PhD's*. American Mathematical Society. ISBN 9780821843765. (Griffin) Reavis and Stokes biographies on p.513-515 and p.580-582 of the Supplementary Material at AMS, respectively.**^**Joseph Miller Thomas | Institute for Advanced Study**^**Kondratieva, M. V. (1998).*Differential and Difference Dimension Polynomials*. Springer Science & Business Media. p. ix (preface). ISBN 978-0-7923-5484-0.**^**Astrelin, A. V.; Golubitsky, O. D.; Pankratiev, E. V. (2000). "Involutive bases of ideals in the ring of polynomials".*Programming and Computer Software*.**26**(1): 31–35. doi:10.1007/bf02759177.**^**Bochner, Salomon (1938). "Review:*Differential systems*by J. M. Thomas" (PDF).*Bull. Amer. Math. Soc*.**44**(5): 314–315. doi:10.1090/s0002-9904-1938-06724-9.