|
Richard Rado
Richard Rado FRS (28 April 1906 – 23 December 1989) was a German-born British mathematician whose research concerned combinatorics and graph theory. He was Jewish and left Germany to escape Nazi persecution. He earned two PhDs: in 1933 from the University of Berlin, and in 1935 from the University of Cambridge. He was interviewed in Berlin by Lord Cherwell for a scholarship given by the chemist Sir Robert Mond which provided financial support to study at Cambridge. After he was awarded the scholarship, Rado and his wife left for the UK in 1933. He was appointed Professor of Mathematics at the University of Reading in 1954 and remained there until he retired in 1971. Contributions Rado made contributions in combinatorics and graph theory including 18 papers with Paul Erdős. In graph theory, the Rado graph, a countably infinite graph containing all countably infinite graphs as induced subgraphs, is named after Rado. He rediscovered it in 1964 after previous works on the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reading, Berkshire
Reading ( ) is a town and borough in Berkshire, Southeast England, southeast England. Located in the Thames Valley at the confluence of the rivers River Thames, Thames and River Kennet, Kennet, the Great Western Main Line railway and the M4 motorway serve the town. Reading is east of Swindon, south of Oxford, west of London and north of Basingstoke. Reading is a major commercial centre, especially for information technology and insurance. It is also a regional retail centre, serving a large area of the Thames Valley with its shopping centre, the The Oracle, Reading, Oracle. It is home to the University of Reading. Every year it hosts the Reading and Leeds Festivals, Reading Festival, one of England's biggest music festivals. Reading has a professional association football team, Reading F.C., and participates in many other sports. Reading dates from the 8th century. It was an important trading and ecclesiastical centre in the Middle Ages, the site of Reading Abbey, one of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge
Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge became an important trading centre during the Roman and Viking ages, and there is archaeological evidence of settlement in the area as early as the Bronze Age. The first town charters were granted in the 12th century, although modern city status was not officially conferred until 1951. The city is most famous as the home of the University of Cambridge, which was founded in 1209 and consistently ranks among the best universities in the world. The buildings of the university include King's College Chapel, Cavendish Laboratory, and the Cambridge University Library, one of the largest legal deposit libraries in the world. The city's skyline is dominated by several college buildings, along with the spire of the Our Lady and the English Martyrs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. Definition There are many equivalent ( cryptomorphic) ways to define a (finite) matroid.A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh (1976). See Brylawsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chao Ko
Ke Zhao or Chao Ko (, April 12, 1910 – November 8, 2002) was a Chinese mathematician born in Wenling, Taizhou, Zhejiang. Biography Ke graduated from Tsinghua University in 1933 and obtained his doctorate from the University of Manchester under Louis Mordell in 1937. His main fields of study were algebra, number theory and combinatorics. Some of his major contributions included his work on quadratic forms, the Erdős–Ko–Rado theorem and his theorem on Catalan's conjecture. In 1955, he was one of the founding members of the Chinese Academy of Sciences. He was later a professor at Sichuan University , retrieved 2015-03-02. and became the president of the university and of the |
Hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) where X is a set of elements called ''nodes'' or ''vertices'', and E is a set of non-empty subsets of X called ''hyperedges'' or ''edges''. Therefore, E is a subset of \mathcal(X) \setminus\, where \mathcal(X) is the power set of X. The size of the vertex set is called the ''order of the hypergraph'', and the size of edges set is the ''size of the hypergraph''. A directed hypergraph differs in that its hyperedges are not sets, but ordered pairs of subsets of X, with each pair's first and second entries constituting the tail and head of the hyperedge respectively. While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same card ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rado's Theorem (Ramsey Theory)
Rado's theorem is a theorem from the branch of mathematics known as Ramsey theory. It is named for the German mathematician Richard Rado. It was proved in his thesis, ''Studien zur Kombinatorik''. Statement Let A \mathbf = \mathbf be a system of linear equations, where A is a matrix with integer entries. This system is said to be r''-regular'' if, for every r-coloring of the natural numbers 1, 2, 3, ..., the system has a monochromatic solution. A system is ''regular'' if it is ''r-regular'' for all ''r'' ≥ 1. Rado's theorem states that a system A \mathbf = \mathbf is regular if and only if the matrix ''A'' satisfies the ''columns condition''. Let ''ci'' denote the ''i''-th column of ''A''. The matrix ''A'' satisfies the columns condition provided that there exists a partition ''C''1, ''C''2, ..., ''C''''n'' of the column indices such that if s_i = \Sigma_c_j, then # ''s''1 = 0 # for all ''i'' ≥ 2, ''si'' can be written as a ration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey's Theorem
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on vertices or a red clique on vertices. (Here signifies an integer that depends on both and .) Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of ''monochromatic subsets'', that is, subsets of connected edges of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Set Theory
In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.Todd Eisworth, ''Successors of Singular Cardinals'' Chapter 15 in Handbook of Set Theory, edited by Matthew Foreman and Akihiro Kanamori, Springer, 2010 Ramsey theory for infinite sets Write κ, λ for ordinals, ''m'' for a cardinal number and ''n'' for a natural number. introduced the notation :\kappa\rightarrow(\lambda)^n_m as a shorthand way of saying that every partition of the set ºsup>''n'' of ''n''-element subsets of \kappa into ''m'' pieces has a homogeneous set of order type λ. A homogeneous set is in this case a subset of κ such that every ''n''-element subset is in the same element of the partition. When ''m'' is 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfréd Rényi
Alfréd Rényi (20 March 1921 – 1 February 1970) was a Hungarian mathematician known for his work in probability theory, though he also made contributions in combinatorics, graph theory, and number theory. Life Rényi was born in Budapest to Artúr Rényi and Borbála Alexander; his father was a mechanical engineer, while his mother was the daughter of philosopher and literary critic Bernhard Alexander; his uncle was Franz Alexander, a Hungarian-American psychoanalyst and physician. He was prevented from enrolling in university in 1939 due to the anti-Jewish laws then in force, but enrolled at the University of Budapest in 1940 and finished his studies in 1944. At this point, he was drafted to forced labour service, from which he escaped. He then completed his PhD in 1947 at the University of Szeged, under the advisement of Frigyes Riesz. He married Katalin Schulhof (who used Kató Rényi as her married name), herself a mathematician, in 1946; their daughter Zsuzsanna was bor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelm Ackermann
Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. Biography Ackermann was born in Herscheid, Germany, and was awarded a Ph.D. by the University of Göttingen in 1925 for his thesis ''Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit'', which was a consistency proof of arithmetic apparently without Peano induction (although it did use e.g. induction over the length of proofs). This was one of two major works in proof theory in the 1920s and the only one following Hilbert's school of thought. From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt, and then at Lüdenscheid until 1961. He was also a corresponding member of the Akademie der Wissenschaften (''Academy of Sciences'') in Göttingen, and was an honorary professor at the Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rado Graph
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of . The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other. Every finite or countably infinite graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
