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Generalized N-gon
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized ''n''-gons encompass as special cases projective planes (generalized triangles, ''n'' = 3) and generalized quadrangles (''n'' = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the '' Moufang property'' have been completely classified by Tits and Weiss. Every generalized ''n''-gon with ''n'' even is also a near polygon. Definition A generalized ''2''-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line. For ''n \geq 3'' a generalized ''n''-gon is an incidence structure (P,L,I), where P is the set of points, L is the set of lines and I\subseteq P\times L is the incidence relation, such that: * It is a partial linear space. * It has no ordinary ''m''-gons as subge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Split Cayley Hexagon
Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, entertainment, and media Films * ''Split'' (1989 film), a science fiction film * ''Split'' (2016 American film), a psychological horror thriller film * ''Split'' (2016 Canadian film), also known as ''Écartée'', a Canadian drama film directed by Lawrence Côté-Collins * ''Split'' (2016 South Korean film), a sports drama film * '' Split: A Divided America'', a 2008 documentary on American politics * ''The Split'' (1959 film) or ''The Manster'', a U.S.-Japanese horror film * ''The Split'' (film), a 1968 heist film Games * Split (poker), the division of winnings in the card game * Split (blackjack), a possible player decision in the card game Music Albums * ''Split'' (The Groundhogs album), 1971 * ''Split'' (Lush album), 1994 * ''Split' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
(B, N) Pair
In mathematics, a (''B'', ''N'') pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were introduced by the mathematician Jacques Tits, and are also sometimes known as Tits systems. Definition A (''B'', ''N'') pair is a pair of subgroups ''B'' and ''N'' of a group ''G'' such that the following axioms hold: * ''G'' is generated by ''B'' and ''N''. * The intersection, ''T'', of ''B'' and ''N'' is a normal subgroup of ''N''. *The group ''W'' = ''N/T'' is generated by a set ''S'' of elements of order 2 such that **If ''s'' is an element of ''S'' and ''w'' is an element of ''W'' then ''sBw'' is contained in the union of ''BswB'' and ''BwB''. **No element of ''S'' normalizes ''B''. The set ''S'' is uniquely determined by ''B'' and ''N'' and the pair (''W'',''S'') is a Coxeter sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Building (mathematics)
In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type. The more specialized theory of Bruhat–Tits buildings (named also after François Bruhat) plays a role in the study of -adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups. Overview The notion of a building was invented by Jacques Tits as a means of describing simple algebraic groups over an arbitrary field. Tits demonstrated how to every such group one can associate a simplicial complex with an action of , called the spherical building of . The group imposes very strong combinatorial regularity conditions on the complexes that can arise in this fashion. By tre ... [...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 mathematics 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?" More specifically, Ron Graham described Ramsey theory as a "branch of combinatorics". 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expander Graph
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes. Definitions Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not "too large" has a "large" boundary. Different formalisations of these notions give rise to different notions of expanders: ''edge expanders'', ''vertex expanders'', and ''spectral expanders'', as defined below. A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected graph is an expander; however, different connected graphs have different expansion parameters. The complete graph has the best expansion property, but it has largest possible degree. Informal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cage (graph Theory)
In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an is defined to be a graph in which each vertex has exactly neighbors, and in which the shortest cycle has length exactly . An is an with the smallest possible number of vertices, among all . A is often called a . It is known that an exists for any combination of and . It follows that all exist. If a Moore graph exists with degree and girth , it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth must have at least :1+r\sum_^(r-1)^i vertices, and any cage with even girth must have at least :2\sum_^(r-1)^i vertices. Any with exactly this many vertices is by definition a Moore graph and therefore automatically a cage. There may exist multiple cages for a given combination of and . For instance there are three nonisomorphic , each with 70 vertices: the Balaban 10-cage, the Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andries Brouwer
Andries Evert Brouwer (born 1951) is a Dutch mathematician and computer programmer, Professor Emeritus at Eindhoven University of Technology (TU/e). He is known as the creator of the greatly expanded 1984 to 1985 versions of the roguelike computer game ''Hack'' that formed the basis for ''NetHack''. He is also a Linux kernel hacker. He is sometimes referred to by the handle ''aeb''. Biography Born in Amsterdam, Brouwer attended the gymnasium, and obtained his MSc in mathematics at the University of Amsterdam in 1971. In 1976 he received his Ph.D. in mathematics from Vrije Universiteit with a thesis entitled "Treelike Spaces and Related Topological Spaces", under the supervision of Maarten Maurice and Pieter Baayen, both of whom were in turn students of Johannes de Groot. In 2004 he received an honorary doctorate from Aalborg University. After graduation Brouwer started his academic career at the Mathematisch Centrum, later Centrum Wiskunde & Informatica. From 1986 to 2012 he w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Cameron (mathematician)
Peter Jephson Cameron FRSE (born 23 January 1947) is an Australian mathematician who works in group theory, combinatorics, coding theory, and model theory. He is currently half-time Professor of Mathematics at the University of St Andrews, and Emeritus Professor at Queen Mary University of London. Cameron received a B.Sc. from the University of Queensland and a D.Phil. in 1971 from the University of Oxford as a Rhodes Scholar, with Peter M. Neumann as his supervisor. Subsequently, he was a Junior Research Fellow and later a Tutorial Fellow at Merton College, Oxford, and also lecturer at Bedford College, London. Work Cameron specialises in algebra and combinatorics; he has written books about combinatorics, algebra, permutation groups In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graham Higman
Graham Higman FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory. Biography Higman was born in Louth, Lincolnshire, and attended Sutton High School, Plymouth, winning a scholarship to Balliol College, Oxford. In 1939 he co-founded The Invariant Society, the student mathematics society, and earned his DPhil from the University of Oxford in 1941. His thesis, ''The units of group-rings'', was written under the direction of J. H. C. Whitehead. From 1960 to 1984 he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. Higman was awarded the Senior Berwick Prize in 1962 and the De Morgan Medal of the London Mathematical Society in 1974. He was the founder of the Journal of Algebra and its editor from 1964 to 1984. Higman had 51 D.Phil. students, including Jonathan Lazare Alperin, Rosemary A. Bailey, Marston Conder, John Mackintosh Howie, and Peter M. Neumann. He was also a local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
