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Graham–Rothschild Theorem
In mathematics, the Graham–Rothschild theorem is a theorem that applies Ramsey theory to combinatorics on words and combinatorial cubes. It is named after Ronald Graham and Bruce Lee Rothschild, who published its proof in 1971. Through the work of Graham, Rothschild, and in 1972, it became part of the foundations of structural Ramsey theory. A special case of the Graham–Rothschild theorem motivates the definition of Graham's number, a number that was popularized by Martin Gardner in ''Scientific American'' and listed in the ''Guinness Book of World Records'' as the largest number ever appearing in a mathematical proof. Background The theorem involves sets of strings, all having the same length n, over a finite alphabet, together with a group acting on the alphabet. A combinatorial cube is a subset of strings determined by constraining some positions of the string to contain a fixed letter of the alphabet, and by constraining other pairs of positions to be equal to each othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey Theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?" Examples A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces. How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property? This idea can be defined as partition regularity. For example, consider a complete graph of order ''n''; that is, there are ''n'' vertices and each vertex is connected to every other vertex by an edge. A complete graph of order 3 is called a triangle. Now colour each edge either red or blue. How large must ''n'' be in order to ensure that there is either a blue triangle or a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called '' coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the '' origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synthese
''Synthese'' () is a monthly peer-reviewed academic journal covering the epistemology, methodology, and philosophy of science, and related issues. The name ''Synthese'' (from the Dutch for '' synthesis'') finds its origin in the intentions of its founding editors: making explicit the supposed internal coherence between the different, highly specialised scientific disciplines. Jaakko Hintikka was editor-in-chief from 1965 to 2002. The current editors-in-chief are Otávio Bueno (University of Miami), Wiebe van der Hoek (University of Liverpool), and Kristie Miller (University of Sydney). Editorial decision controversies In 2011, the journal became involved in a controversy over intelligent design. The printed version of the special issue ''Evolution and Its Rivals'', which appeared two years after the online version, was supplied with a disclaimer from the then editors of the journal that "appeared to undermine he authorsand the guest editors". The journal engendered controversy a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
