Ivar Otto Bendixson (1 August 1861 – 29 November 1935) was a
Swedish
Swedish or ' may refer to:
Anything from or related to Sweden, a country in Northern Europe. Or, specifically:
* Swedish language, a North Germanic language spoken primarily in Sweden and Finland
** Swedish alphabet, the official alphabet used by ...
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, structure, space, models, and change.
History
On ...
.
Biography
Bendixson was born on 1 August 1861 at Villa Bergshyddan,
Djurgården
Djurgården ( or ) or, more officially, ''Kungliga Djurgården'' (), 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 ...
,
Oscar Parish
Oscar Parish ( sv, Oscars församling) 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.
His ...
,
Stockholm, 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 or post-primary education covers two phases on the International Standard Classification of Education scale. Level 2 or lower secondary education (less commonly junior secondary education) is considered the second and final pha ...
in
Stockholm, he obtained his school certificate on 25 May 1878.
On 13 September 1878 he enrolled to the
Royal Institute of Technology in Stockholm. In 1879 Bendixson went to
Uppsala University
Uppsala University ( sv, Uppsala universitet) is a public research university in Uppsala, Sweden. Founded in 1477, it is the oldest university in Sweden and the Nordic countries still in operation.
The university rose to significance during ...
and graduated with the equivalent of a
Master's degree
A master's degree (from Latin ) is an 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 practice. on 27 January 1881. Graduating from Uppsala, he went on to study at the newly opened
Stockholm University College
Stockholm University ( sv, Stockholms universitet) is a 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, soci ...
after which he was awarded a
doctorate
A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism ''li ...
by Uppsala University on 29 May 1890.
On 10 June 1890 Bendixson was appointed as a
docent
The title of docent is conferred by some European universities to denote a specific academic appointment within a set structure of academic ranks at or below the full professor rank, similar to a British readership, a French " ''maître de con ...
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 (m ...
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, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
and
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
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 at the Royal Institute of Technology and then he was promoted to professor there on 26 January 1900. On 16 June 1905 he became professor of higher mathematical analysis at Stockholm University College and from 1911 until 1927 he was its
rector
Rector (Latin for the member of a vessel's crew who steers) may refer to:
Style or title
*Rector (ecclesiastical), a cleric who functions as an administrative leader in some Christian denominations
*Rector (academia), a senior official in an edu ...
.
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 political ideologies that support and seek to achieve social equality and egalitarianism, often in opposition to social hierarchy. Left-wing politics typically involve a concern for those in soci ...
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 which investigated a
proportional representation
Proportional representation (PR) refers to a type of electoral system under which subgroups of an electorate are reflected proportionately in the elected body. The concept applies mainly to geographical (e.g. states, regions) and political divis ...
voting system
An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections m ...
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 mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
. His first research work was on
set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...
and the
foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
, following the ideas which
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
had introduced. He contributed important results in
point set topology
In mathematics, general 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 differential topology, geomet ...
. As a young student Bendixson made his name by proving that every
uncountable
In mathematics, an uncountable set (or uncountably infinite set) 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 num ...
closed set 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 point, limit points of S, also known as the derived set ...
(the
Bendixson derivative of the original set) and a
countable set. He also gave another important contribution when he gave an example of a perfect set which 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 an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
equation by
radicals 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 which intrigued Bendixson more than all others was the investigation of integral curves to first order
differential equations, in particular he was intrigued by the complicated behaviour of the integral curves in the neighbourhood 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 which does not end in a singular point has a limit cycle, was first proved by
Henri Poincaré but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901.
In 1902, he derived
Bendixson's inequality In mathematics, Bendixson's inequality is a quantitative result in the field of Matrix_(mathematics), matrices derived by Ivar Bendixson in 1902. The inequality puts limits on the imaginary and real parts of Eigenvalues_and_eigenvectors, characteris ...
which puts bounds on the characteristic roots of 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