Ivar Otto Bendixson (1 August 1861 – 29 November 1935) was a Swedish

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Biography

Bendixson was born on 1 August 1861 at Villa Bergshyddan,

Djurgården Djurgården ( or ) or, more officially, , is an island in central Stockholm, Sweden. Djurgården is home to historical buildings and monuments, museums, galleries, the amusement park Gröna Lund, the open-air museum Skansen, the small resident ...

Oscar Parish Oscar Parish () is a parish in Östermalm's church district (''kontrakt'') in the Diocese of Stockholm, Sweden. The parish is located in Stockholm Municipality in Stockholm County. The parish forms its own pastorship. History Oscar Parish was fo ...

Stockholm Stockholm (; ) is the Capital city, capital and List of urban areas in Sweden by population, most populous city of Sweden, as well as the List of urban areas in the Nordic countries, largest urban area in the Nordic countries. Approximately ...

, Sweden, to a middle-class family. His father Vilhelm Emanuel Bendixson was a merchant, and his mother was Tony Amelia Warburg. On completing

secondary education Secondary education is the education level following primary education and preceding tertiary education. Level 2 or ''lower secondary education'' (less commonly ''junior secondary education'') is considered the second and final phase of basic e ...

, he obtained his school certificate on 25 May 1878. On 13 September 1878 he enrolled to the

Royal Institute of Technology KTH Royal Institute of Technology (), abbreviated KTH, is a public research university in Stockholm, Sweden. KTH conducts research and education in engineering and technology and is Sweden's largest technical university. Since 2018, KTH consist ...

in Stockholm. In 1879 Bendixson went to

Uppsala University Uppsala University (UU) () is a public university, public research university in Uppsala, Sweden. Founded in 1477, it is the List of universities in Sweden, oldest university in Sweden and the Nordic countries still in operation. Initially fou ...

and graduated with the equivalent of a

master's degree A master's degree (from Latin ) is a postgraduate academic degree awarded by universities or colleges upon completion of a course of study demonstrating mastery or a high-order overview of a specific field of study or area of professional prac ...

on 27 January 1881. Graduating from Uppsala, he went on to study at the newly opened

Stockholm University College Stockholm University (SU) () is a public university, public research university in Stockholm, Sweden, founded as a college in 1878, with university status since 1960. With over 33,000 students at four different faculties: law, humanities, social ...

after which he was awarded a

doctorate A doctorate (from Latin ''doctor'', meaning "teacher") or doctoral degree is a postgraduate academic degree awarded by universities and some other educational institutions, derived from the ancient formalism '' licentia docendi'' ("licence to teach ...

by Uppsala University on 29 May 1890. On 10 June 1890 Bendixson was appointed as a

docent The term "docent" is derived from the Latin word , which is the third-person plural present active indicative of ('to teach, to lecture'). Becoming a docent is often referred to as habilitation or doctor of science and is an academic qualifi ...

at Stockholm University College. He then worked as an assistant to the professor of

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

from 5 March 1891 until 31 May 1892. From 1892 until 1899 he taught at the Royal Institute of Technology and he also taught

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

and

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

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 of

Pure Mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...

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 its rector 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 mild

left-wing Left-wing politics describes the range of Ideology#Political ideologies, political ideologies that support and seek to achieve social equality and egalitarianism, often in opposition to social hierarchy either as a whole or of certain social ...

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 a

proportional representation Proportional representation (PR) refers to any electoral system under which subgroups of an electorate are reflected proportionately in the elected body. The concept applies mainly to political divisions (Political party, political parties) amon ...

voting system An electoral or voting system is a set of rules used to determine the results of an election. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, nonprofit organizations and inf ...

in Sweden in 1912–13. In this capacity he was able to make use of his mathematical skills in advising the committee.

Scientific achievements

Bendixson started out very much as a pure mathematician but later in his career he turned to also consider problems from

applied mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...

. His first research work was on

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

and the

foundations of mathematics Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...

, following the ideas that

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...

had introduced. He contributed important results in

point-set topology In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...

. As a young student, Bendixson made his name by proving that every

uncountable In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...

closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...

of reals can be partitioned into a

perfect set In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all limit points of S, also known as the derived set of S. (Some ...

(the Cantor–Bendixson derivative of the original set) and a

countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...

. He also gave another important contribution when he gave an example of a perfect set that is

totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...

. Concerning solution of a

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

equation by

radicals Radical (from Latin: ', root) may refer to: Politics and ideology Politics *Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century *Radical politics ...

, Bendixson returned to

Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...

'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 of

integral curve In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Name Integral curves are known by various other names, depending on the nature and interpre ...

s to first-order differential equations, in particular he was intrigued by the complicated behaviour of the integral curves in the

neighbourhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...

of singular points. The

Poincaré–Bendixson theorem In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. Theorem Given a differentiable real dynamical system defined on an op ...

, which says an integral curve that does not end in a singular point has a limit cycle, was first proved by

Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...

, but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901. In 1902, he derived Bendixson's inequality, which puts bounds on the

eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

of real matrices.

References

Notes

Print

*Bendixson, Ivar Otto, ''Svenskt Biografiskt Lexikon 3'' (Stockholm, 1922), 146–150. *L. Garding, ''Mathematics and Mathematicians : Mathematics in Sweden before 1950'' (Providence, R.I., 1998), 109–112.

External links

* {{DEFAULTSORT:Bendixson, Ivar Otto 1861 births 1935 deaths Academic staff of Stockholm University Rectors of Stockholm University Academic staff of the KTH Royal Institute of Technology KTH Royal Institute of Technology alumni Stockholm University alumni Uppsala University alumni 19th-century Swedish mathematicians 20th-century Swedish mathematicians Linear algebraists