Axel Johannes Malmquist

Axel Johannes Malmquist , better known as " Johannes Malmquist ", (born October 19, 1882 in Hammar , † February 24, 1952 in Solna ), was a Swedish mathematician .

Life

He was the last student of Magnus Gösta Mittag-Leffler and received his doctorate in 1909, the first year he received his doctorate there. The dissertation Sur les équations différentielles du premier ordre dont 1'intégrale générale admet un nombre fini de branches permutables autour des points critiques mobiles took up studies by Paul Painlevé . For a number of years he was also called in by Mittag-Leffler and Ivar Bendixson to support their lectures at Stockholm University, with Mittag-Leffler lecturing primarily on function theory and Bendixson on differential equations. From 1903 to 1909 he was a tutor (Amanuens) at Stockholm University for Mittag-Leffler and Bendixson and, after completing his dissertation, lecturer from 1910. In 1906/07 he taught mathematics at the War College and from 1908 to 1913 he worked in an office for statistics. From 1911 he taught as an assistant professor at the Royal Technical University of Stockholm , where he was professor from 1913 to 1948. He worked in the editorial team of Acta Mathematica and was editorial secretary for many years.

He was married to Elsa Sofia Melander, with whom he had a son and a daughter.

plant

He mainly dealt with differential equations in the complex. In 1905 he solved a function-theoretical problem by Mittag-Leffler, the construction of entire functions that go towards zero in all directions except one.

He made his most important mathematical contribution in 1913 when he concluded that certain nonlinear algebraic differential equations were based on the existence of a transcendent meromorphic solution and that it was a Riccatic differential equation . Kōsaku Yosida gave a new proof of this theorem in 1932 with the help of the Nevanlinnaschen value distribution theory . This created the basis for the systematic work of Hans Wittich in the 1950s. Other authors later gave various generalizations of Malmquist's theorem, for example Norbert Steinmetz (1978). More precisely, the sentence of Malmquist and Yosida reads:

Differential equations (binomial differential equations) with an in and rational function that do not vanish identically are considered. If the differential equation has a transcendent meromorphic solution, then R has the form with rational functions . ${\displaystyle {({\frac {dw}{dz}})}^{n}=R(z,w)}$ ${\displaystyle z}$ ${\displaystyle w}$ ${\displaystyle R(z,w)}$ ${\displaystyle R(z,w)=\sum _{j=0}^{2n}a_{j}(z)w^{j}}$ ${\displaystyle a_{i}(z)}$

For the special case one has the Riccatic differential equation ( ). ${\displaystyle n=1}$ ${\displaystyle R(z,w)=a(z)+b(z)w+c(z)w^{2}}$

In the case treated by Malmquist, the theorem can also be formulated in such a way that the Riccatian differential equations are distinguished from the differential equations with rational functions by the fact that they are unambiguous on a large scale and have non-rational solutions. ${\displaystyle n=1}$ ${\displaystyle {\frac {dw}{dz}}=R(z,w)}$ ${\displaystyle R(z,w)}$

Malmquist expanded the theory in various directions, including systems of differential equations. With it he was able to confirm a theory of the physicist and mathematician Carl Størmer about the orbits of charged particles in a magnetic field in 1944 .

Fonts

• Om singulära ställen till differentialekvationer av första ordningen , Stockholm, 1918 (German: About singular places of first-order differential equations)
• Föreläsningar i matematik , Stockholm 1923 (German: lectures on mathematics)
• with V. Stenström, Sture Danielson: Matematisk analys , 3 volumes, Stockholm 1951 to 1953 (German: Mathematical Analysis).
• Étude d'une fonction entière , Acta Mathematica, Volume 29, 1905, pp. 203-215
• Sur les fonctions a un nombre fini de branches définies par les equations différentielles du premier ordre , Acta Mathematica, Volume 36, 1913, pp. 297-343
• Sur les fonctions à un nombre fini de branches satisfaisant à une equation différentielle du premier ordre , Acta Mathematica, Volume 42, 1920, pp. 317-325
• Sur l'étude analytique des solutions d'un système d'équations différentielles dans le voisinage d'un point singulier d'indétermination , Acta Mathematica, part 1, volume 73, 1941, pp. 87–129, part 2, volume 74, 1941, pp. 1-64, Part 3, pp. 109-128

literature

• Åke Pleijel: Johannes Malmquist in memoriam . In: Acta Mathematica , 88, 1, 1952, pp. Ix-xii
• Einar Hille : On some generalizations of the Malmquist theorem . In: Math. Scand. , 39, 1976, pp. 59-79
• Axel Johannes Malmquist . In: Theodor Westrin, Ruben Gustafsson Berg, Eugen Fahlstedt (eds.): Nordisk familjebok konversationslexikon och realencyklopedi . 2nd Edition. tape 37 : Supplement: L – Riksdag . Nordisk familjeboks förlag, Stockholm 1925, Sp. 416 (Swedish, runeberg.org ).

Web links

• A Johannes Malmquist . In: Svenskt biografiskt lexikon

Individual evidence

1. ^ Lars Gårding: Mathematics and Mathematicians: Mathematics in Sweden Before 1950 . 1998, p. 136
2. ^ Domar, Mathematical research during the first decades of the University of Stockholm, pdf
3. ^ Arild Stubhaug: Gösta Mittag-Leffler: A Man of Conviction . 2010, p. 602
4. ^ J. Malmquist, Sur les fonctions à un nombre fini de branches définies par les équations differentielles du premier ordre, Acta Mathematica, Volume 36, 1913, pp. 297-343, Project Euclid
5. ↑ K. Yosida, A generalization of Malmquist's theorem, Japan J. Math., Vol. 9, 1932, pp. 253-256. The proof according to the Nevanlinna theory is presented in Ludwig Bieberbach, Theory of Ordinary Differential Equations based on Function Theory, Springer 1953, pp. 88ff. The sentence is only formulated there for , the extension to general binomial differential equations comes from Yosida. ${\displaystyle n=1}$
6. ↑ Guido Walz (Ed.), Lexikon der Mathematik, Spektrum Verlag, article Nevanlinna theory