Ivar Otto Bendixson | |
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| Born | (1861-08-01)1 August 1861 Stockholm, Sweden |
| Died | 29 November 1935(1935-11-29) (aged 74) Stockholm, Sweden |
| Occupation | Mathematician |
| Known for | Bendixson's inequality Bendixson–Dulac theorem Cantor–Bendixson rank Cantor–Bendixson theorem Poincaré–Bendixson theorem |
Ivar Otto Bendixson[pronunciation?] (1 August 1861 – 29 November 1935) was aSwedishmathematician.
Bendixson was born on 1 August 1861 at Villa Bergshyddan,Djurgården,Oscar Parish,Stockholm, Sweden,[1] to a middle-class family. His father Vilhelm Emanuel Bendixson was a merchant, and his mother was Tony Amelia Warburg. On completingsecondary education inStockholm, he obtained his school certificate on 25 May 1878.
On 13 September 1878 he enrolled to theRoyal Institute of Technology in Stockholm. In 1879 Bendixson went toUppsala University and graduated with the equivalent of amaster's degree on 27 January 1881. Graduating from Uppsala, he went on to study at the newly openedStockholm University College after which he was awarded adoctorate by Uppsala University on 29 May 1890.
On 10 June 1890 Bendixson was appointed as adocent at Stockholm University College. He then worked as an assistant to the professor ofmathematical analysis from 5 March 1891 until 31 May 1892. From 1892 until 1899 he taught at the Royal Institute of Technology and he also taughtcalculus andalgebra at Stockholm University College. During this period he married Anna Helena Lind on 19 December 1887. Anna, who was about eighteen months older than Bendixson, was the daughter of the banker Johan Lind.
In 1899 Bendixson substituted for the Professor ofPure Mathematics at the Royal Institute of Technology and then he was promoted to professor there on 26 January 1900. On June 16, 1905, he assumed the position of professor of higher mathematical analysis at Stockholm University College, and he served as itsrector from 1911 until 1927.
For his outstanding contributions, Bendixson received many honours including an honorary doctorate on 24 May 1907.
Bendixson became more involved in politics as his career progressed. He was well known for his mildleft-wing views and he put his beliefs into practice being head of a committee to help poor students. He served on many other committees and he was an advisor to a committee that investigated aproportional representationvoting system in Sweden in 1912–13. In this capacity he was able to make use of his mathematical skills in advising the committee.
Bendixson started out very much as a pure mathematician but later in his career he turned to also consider problems fromapplied mathematics. His first research work was onset theory and thefoundations of mathematics, following the ideas thatGeorg Cantor had introduced. He contributed important results inpoint-set topology. As a young student, Bendixson made his name by proving that everyuncountableclosed set ofreals can be partitioned into aperfect set (theCantor–Bendixson derivative of the original set) and acountable set. He also gave another important contribution when he gave an example of a perfect set that istotally disconnected.
Concerning solution of apolynomial equation byradicals, Bendixson returned toNiels Henrik Abel's original contribution and showed that Abel's methods could be extended to describe precisely which equations could be solved by radicals.
The analysis problem that intrigued Bendixson more than all others was the investigation ofintegral curves to first-orderdifferential equations, in particular he was intrigued by the complicated behaviour of the integral curves in theneighbourhood ofsingular points. ThePoincaré–Bendixson theorem, which says an integral curve that does not end in a singular point has a limit cycle, was first proved byHenri Poincaré, but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901.
In 1902, he derivedBendixson's inequality, which puts bounds on theeigenvalues of real matrices.
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