Kazimierz Zarankiewicz
Encyclopedia
Kazimierz Zarankiewicz was a Polish
mathematician. He was born in Częstochowa
and died in London
, England
.
His main interest was topology
. He studied at the University of Warsaw
, together with Zygmunt Janiszewski
, Stefan Mazurkiewicz
, Wacław Sierpiński, Kazimierz Kuratowski
, and Stanisław Saks. He wrote works on cut-point
s in connected space
s, on conformal mappings, on complex functions
and number theory
, and triangular number
s.
During World War II
, Zarankiewicz took part in illegal teaching, forbidden by the German
authorities, and eventually was sent to a concentration camp. He successfully survived and became a teacher at Warsaw University of Technology
.
He visited universities in Tomsk
, Harvard, London
, Vienna
. He served as president of the Warsaw
section of the Polish Mathematical Society
and the International Astronautical Federation
.
The Zarankiewicz problem
is named after Zarankiewicz; this problem asks, for a given size of (0,1)-matrix, how many matrix entries must be set equal to 1 in order to ensure that some a × b submatrix is made up only of 1's.
Also the Zarankiewicz Crossing Number Conjecture in the mathematical field of graph theory
is named after Zarankiewicz. The conjecture
states that the crossing number
of a complete bipartite graph
equals
Zarankiewicz proved that this formula is an upper bound for the actual crossing number. The problem of determining the number
was suggested by Paul Turán
and became known as Turán's Brick Factory Problem.
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
mathematician. He was born in Częstochowa
Czestochowa
Częstochowa is a city in south Poland on the Warta River with 240,027 inhabitants . It has been situated in the Silesian Voivodeship since 1999, and was previously the capital of Częstochowa Voivodeship...
and died in London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
, England
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
.
His main interest was topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. He studied at the University of Warsaw
University of Warsaw
The University of Warsaw is the largest university in Poland and one of the most prestigious, ranked as best Polish university in 2010 and 2011...
, together with Zygmunt Janiszewski
Zygmunt Janiszewski
Zygmunt Janiszewski was a Polish mathematician.-Life:His mother was Julia Szulc-Chojnicka. His father, Czeslaw Janiszewski, was a graduate of the University of Warsaw and was an important person in finance, being the director of the Société du Crédit Municipal in Warsaw.Janiszewski taught at the...
, Stefan Mazurkiewicz
Stefan Mazurkiewicz
Stefan Mazurkiewicz was a Polish mathematician who worked in mathematical analysis, topology, and probability. He was a student of Wacław Sierpiński and a member of the Polish Academy of Learning...
, Wacław Sierpiński, Kazimierz Kuratowski
Kazimierz Kuratowski
Kazimierz Kuratowski was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.-Biography and studies:...
, and Stanisław Saks. He wrote works on cut-point
Cut-point
In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. For example every point of a line is a cut-point, while no point of a circle is a cut-point...
s in connected space
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
s, on conformal mappings, on complex functions
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, and triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
s.
During World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, Zarankiewicz took part in illegal teaching, forbidden by the German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
authorities, and eventually was sent to a concentration camp. He successfully survived and became a teacher at Warsaw University of Technology
Warsaw University of Technology
The Warsaw University of Technology is one of the leading institutes of technology in Poland, and one of the largest in Central Europe. It employs 2,453 teaching faculty, with 357 professors . The student body numbers 36,156 , mostly full-time. There are 17 faculties covering almost all fields of...
.
He visited universities in Tomsk
Tomsk
Tomsk is a city and the administrative center of Tomsk Oblast, Russia, located on the Tom River. One of the oldest towns in Siberia, Tomsk celebrated its 400th anniversary in 2004...
, Harvard, London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
, Vienna
Vienna
Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...
. He served as president of the Warsaw
Warsaw
Warsaw is the capital and largest city of Poland. It is located on the Vistula River, roughly from the Baltic Sea and from the Carpathian Mountains. Its population in 2010 was estimated at 1,716,855 residents with a greater metropolitan area of 2,631,902 residents, making Warsaw the 10th most...
section of the Polish Mathematical Society
Polish Mathematical Society
The Polish Mathematical Society began in Kraków, Poland in 1917. It was originally simply called the Mathematical Society. It was officially constituted on April 2, 1919.Hugo Steinhaus, Stefan Banach and Otto Nikodym were among the founders....
and the International Astronautical Federation
International Astronautical Federation
International Astronautical Federation , the world's foremost space advocacy organisation, is based in Paris. It was founded in 1951 as a non-governmental organization. It has 206 members from 58 countries across the world. They are drawn from space agencies, industry, professional associations,...
.
The Zarankiewicz problem
Zarankiewicz problem
In the mathematical field of extremal graph theory, the Zarankiewicz problem asks how many edges can be added to a bipartite graph while avoiding a specific bipartite subgraph...
is named after Zarankiewicz; this problem asks, for a given size of (0,1)-matrix, how many matrix entries must be set equal to 1 in order to ensure that some a × b submatrix is made up only of 1's.
Also the Zarankiewicz Crossing Number Conjecture in the mathematical field of graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
is named after Zarankiewicz. The conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
states that the crossing number
Crossing number (graph theory)
In graph theory, the crossing number cr of a graph G is the lowest number of edge crossings of a planar drawing of the graph G. For instance, a graph is planar if and only if its crossing number is zero.The concept originated in...
of a complete bipartite graph
Complete bipartite graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.- Definition :...
equalsZarankiewicz proved that this formula is an upper bound for the actual crossing number. The problem of determining the number
was suggested by Paul TuránPál Turán
Paul Turán was a Hungarian mathematician who worked primarily in number theory. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers.- Life and education :...
and became known as Turán's Brick Factory Problem.