Ivar Otto Bendixson
Encyclopedia
Biography
Bendixson was born August 1, 1861 in Djurgårdsbrunn, StockholmStockholm
Stockholm is the capital and the largest city of Sweden and constitutes the most populated urban area in Scandinavia. Stockholm is the most populous city in Sweden, with a population of 851,155 in the municipality , 1.37 million in the urban area , and around 2.1 million in the metropolitan area...
Sweden
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
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
Secondary education is the stage of education following primary education. Secondary education includes the final stage of compulsory education and in many countries it is entirely compulsory. The next stage of education is usually college or university...
in Stockholm
Stockholm
Stockholm is the capital and the largest city of Sweden and constitutes the most populated urban area in Scandinavia. Stockholm is the most populous city in Sweden, with a population of 851,155 in the municipality , 1.37 million in the urban area , and around 2.1 million in the metropolitan area...
, he obtained his school certificate on May 25, 1878.
On September 13, 1878 he enrolled to the Royal Institute of Technology
Royal Institute of Technology
The Royal Institute of Technology is a university in Stockholm, Sweden. KTH was founded in 1827 as Sweden's first polytechnic and is one of Scandinavia's largest institutions of higher education in technology. KTH accounts for one-third of Sweden’s technical research and engineering education...
in Stockholm. In 1879 Bendixson went to Uppsala University
Uppsala University
Uppsala University is a research university in Uppsala, Sweden, and is the oldest university in Scandinavia, founded in 1477. It consistently ranks among the best universities in Northern Europe in international rankings and is generally considered one of the most prestigious institutions of...
and graduated with the equivalent of a Master's degree
Master's degree
A master's is an academic degree granted to individuals who have undergone study demonstrating a mastery or high-order overview of a specific field of study or area of professional practice...
on January 27, 1881. Graduating from Uppsala, he went on to study at the newly opened Stockholm University College after which he was awarded a doctorate
Doctorate
A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder to teach in a specific field, A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder...
by Uppsala University on May 29, 1890.
On June 10, 1890 Bendixson was appointed as a docent
Docent
Docent is a title at some European universities to denote a specific academic appointment within a set structure of academic ranks below professor . Docent is also used at some universities generically for a person who has the right to teach...
at Stockholm University College. He then worked as an assistant to the professor of mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
from March 5, 1891 until May 31, 1892. From 1892 until 1899 he taught at the Royal Institute of Technology and he also taught calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
and algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
at Stockholm University College. During this period he married Anna Helena Lind on December 19, 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
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
at the Royal Institute of Technology and then he was promoted to professor there on January 26, 1900. On June 16, 1905 he became professor of higher mathematical analysis at Stockholm University College and from 1911 until 1927 he was its rector
Rector
The word rector has a number of different meanings; it is widely used to refer to an academic, religious or political administrator...
.
For his outstanding contributions, Bendixson received many honours including an honorary doctorate on May 24, 1907.
Bendixson became more involved in politics as his career progressed. He was well known for his mild left-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 which investigated a proportional representation
Proportional representation
Proportional representation is a concept in voting systems used to elect an assembly or council. PR means that the number of seats won by a party or group of candidates is proportionate to the number of votes received. For example, under a PR voting system if 30% of voters support a particular...
voting system
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
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 mathematicsApplied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
. His first research work was on set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
and the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
, following the ideas which Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
had introduced. He contributed important results in point set topology. As a young student Bendixson made his name by proving that every uncountable closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
can be partitioned into a perfect set (the Bendixson derivative of the original set) and a countable set
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
. He also gave another important contribution when he gave an example of a perfect set which is totally disconnected.
Concerning solution of a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
equation by radicals
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
Bendixson returned to Niels Henrik Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
'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
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
, in particular he was intrigued by the complicated behaviour of the integral curves in the neighbourhood of singular points. The Poincaré-Bendixson theorem, which says an integral curve which does not end in a singular point has a limit cycle, was first proved by Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
but a more rigorous proof with weaker hypotheses was given by Bendixson in 1901.