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Cameron–Fon-Der-Flaass IBIS Theorem
In mathematics, Cameron–Fon-Der-Flaass IBIS theorem arises in the dynamical algebraic combinatorics. The theorem was discovered in 1995 by two mathematicians Peter Cameron and Dima Von-Der-Flaass. The theorem is considered to be a link between group theory and graph theory as it studies redundancy of a group. Statement Let P be a permutational group of \Omega, then the following are equivalent: * Irredundant bases of P are stored by re-ordering. * The bases of matroid In combinatorics, 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 Axiomatic system, axiomatically, the most significant being in terms ... are formed due to the irredundant bases of P. * Every irredundant base of P got the same size. References Further reading * https://www.theoremoftheday.org/GroupTheory/IBIS/TotDIBIS.pdf Algebraic combinatorics Theorems in combinatorics {{Combin-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. History The term "algebraic combinatorics" was introduced in the late 1970s. Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, ''Algebraic combinatorics'', of the AMS Mathematics Subject Classification, introduced in 1991. Scope Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and signific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Cameron (mathematician)
Peter Jephson Cameron Fellow of the Royal Society of Edinburgh, FRSE (born 23 January 1947) is an Australian mathematician who works in group theory, combinatorics, coding theory, and model theory. He is currently Emeritus Professor at the University of St Andrews and Queen Mary University of London. Education Cameron received a B.Sc. from the University of Queensland and a D.Phil. in 1971 from the University of Oxford as a Rhodes Scholarship, 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, Bedford College, London. Work Cameron specialises in algebra and combinatorics; he has written books about combinatorics, algebra, permutation groups, and logic, and has produced over 350 academic papers. In 1988, he posed the Cameron–Erdős conjecture with Paul Erdős. Honours and awards He was awarded the London Mathematical Soci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dima Von-Der-Flaass
D. G. Von Der Flaass (September 8, 1962 – June 10, 2010) was a Russian mathematician and educator, Candidate of Physical and Mathematical Sciences, senior researcher at the Sobolev Institute of Mathematics. He was a specialist in combinatorics, a popularizer of mathematics, and an author of International Mathematical Olympiad problems. He was also a jury member for numerous mathematical olympiads. Biography Early years D. G. von der Flaass was born in the city of Krasnokamsk, Perm Krai on September 8, 1962, into the family of Herman Sergeevich von der Flaass, a Doctor of Geological and Mineralogical Sciences and a professor. The von der Flaass family lineage traces back to an officer in Napoleon's army, a Dutchman by origin, who was taken prisoner and remained in Russia. In 1975, at the age of 13 (two years earlier than usual), Von Der Flaass was admitted to the Lavrentiev Physics and Mathematics School. He actively participated in school mathematics olympiads, cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Group
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'' to itself). The group of ''all'' permutations of a set ''M'' is the symmetric group of ''M'', often written as Sym(''M''). The term ''permutation group'' thus means a subgroup of the symmetric group. If then Sym(''M'') is usually denoted by S''n'', and may be called the ''symmetric group on n letters''. By Cayley's theorem, every group is isomorphic to some permutation group. The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry. Basic properties and terminology A ''permutation group'' is a subgroup of a symmetric group; that is, its elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
In combinatorics, 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 Axiomatic system, 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 simple matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terms used in 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 Cryptomorphism, equivalent ways to define a (finite) matroid. Independent sets In terms of independence, a finite matroid M is a pair (E, \mathcal), where E is a finite set (called the ''gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. History The term "algebraic combinatorics" was introduced in the late 1970s. Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, ''Algebraic combinatorics'', of the AMS Mathematics Subject Classification, introduced in 1991. Scope Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and signific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
