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Arthur Cayley (d
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He postulated what is now known as the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of an abstract group, a set with a binary operation satisfying certain laws, as opposed to Évariste Galois' concept of permutation groups. In group theory, Cayley tables, Cayley graphs, and Cayley's theorem are named in his honour, as well as Cayley's formula in combinatorics. Early life Arthur Cayley was born in Richmond, London, England, on 16 August 1821. His father, Henry Cayley, was a distant cousin of George Cayley, the aeronautics engineer innovator, and descended from an ancient Yorkshire family. He settled in S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richmond, London
Richmond is a town in south-west London,The London Government Act 1963 (c.33) (as amended) categorises the London Borough of Richmond upon Thames as an Outer London borough. Although it is on both sides of the River Thames, the Boundary Commission for England defines it as being in South London or the South Thames sub-region, pairing it with Kingston upon Thames for the purposes of devising constituencies. However, for the purposes of the London Plan, Richmond now lies within the West London (sub region), West London region. west-southwest of Charing Cross. It is on a meander of the River Thames, with many Richmond upon Thames parks and open spaces, parks and open spaces, including Richmond Park, and many protected conservation areas, which include much of Richmond Hill, London, Richmond Hill. A specific Richmond, Petersham and Ham Open Spaces Act 1902, Act of Parliament protects the scenic view of the River Thames from Richmond. Richmond was founded following Henry VII of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
King's College School
King's College School, also known as Wimbledon, KCS, King's and KCS Wimbledon, is a public school in Wimbledon, southwest London, England. The school was founded in 1829 by King George IV, as the junior department of King's College London and had part of the school's premises in Strand, London, Strand, prior to relocating to Wimbledon in 1897. KCS is a member of the Eton Group of schools. It is predominantly a boys' school but accepts girls into the Sixth Form. In the Sixth Form pupils can choose between the International Baccalaureate and A-Level qualifications. History A royal charter by King George IV of the United Kingdom, George IV founded the school in 1829 as the junior department of the newly established King's College, London. The school occupied the basement of the college in Strand, London, The Strand. Most of its original eighty-five pupils lived in the city within walking distance of the school. During the early Victorian Era, the school grew in numbers and r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aeronautics
Aeronautics is the science or art involved with the study, design, and manufacturing of air flight–capable machines, and the techniques of operating aircraft and rockets within the atmosphere. The British Royal Aeronautical Society identifies the aspects of "aeronautical Art, Science and Engineering" and "The profession of Aeronautics (which expression includes Astronautics)." While the term originally referred solely to ''operating'' the aircraft, it has since been expanded to include technology, business, and other aspects related to aircraft. The term "aviation" is sometimes used interchangeably with aeronautics, although "aeronautics" includes lighter-than-air craft such as airships, and includes ballistic vehicles while "aviation" technically does not. A significant part of aeronautical science is a branch of dynamics called aerodynamics, which deals with the motion of air and the way that it interacts with objects in motion, such as an aircraft. History Early ideas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Cayley
Sir George Cayley, 6th Baronet (27 December 1773 – 15 December 1857) was an English engineer, inventor, and aviator. He is one of the most important people in the history of aeronautics. Many consider him to be the first true scientific aerial investigator and the first person to understand the underlying principles and forces of flight and the first man to create the wire wheel. * * * In 1799, he set forth the concept of the modern aeroplane as a fixed-wing flying machine with separate systems for lift, propulsion, and control. He was a pioneer of aeronautical engineering and is sometimes referred to as "the father of aviation." He identified the four forces which act on a heavier-than-air flying vehicle: weight, lift, drag and thrust. Modern aeroplane design is based on those discoveries and on the importance of cambered wings, also proposed by Cayley. He constructed the first flying model aeroplane and also diagrammed the elements of vertical flight. He also designed the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Formula
In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n, the number of trees on n labeled vertices is n^. The formula equivalently counts the number of spanning trees of a complete graph with labeled vertices . Proof Many proofs of Cayley's tree formula are known. One classical proof of the formula uses Kirchhoff's matrix tree theorem, a formula for the number of spanning trees in an arbitrary graph involving the determinant of a matrix. Prüfer sequences yield a bijective proof of Cayley's formula. Another bijective proof, by André Joyal, finds a one-to-one transformation between ''n''-node trees with two distinguished nodes and maximal directed pseudoforests. A proof by double counting due to Jim Pitman counts in two different ways the number of different sequences of directed edges that can be added to an empty graph on n vertices to form from it a rooted tree; see . History The formula was fir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Theorem
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group. More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose elements are the permutations of the underlying set of . Explicitly, * for each g \in G, the left-multiplication-by- map \ell_g \colon G \to G sending each element to is a permutation of , and * the map G \to \operatorname(G) sending each element to \ell_g is an injective homomorphism, so it defines an isomorphism from onto a subgroup of \operatorname(G). The homomorphism G \to \operatorname(G) can also be understood as arising from the left translation action of on the underlying set . When is finite, \operatorname(G) is finite too. The proof of Cayley's theorem in this case shows that if is a finite group of order , then is isomorphic to a subgroup of the standard symmetric group S_n. But might also be isomorphic to a subgroup o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs. Definition Let G be a group and S be a generating set of G. The Cayley graph \Gamma = \Gamma(G,S) is an edge-colored directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405. * Each element g of G is assigned a vertex: the vertex set of \Gamma is identified with G. * Each element s of S is assigned a color c_s. * For every g \in G and s \in S, there is a directed edge of color c_s from the vertex corresponding to g to the one corresponding to gs. Not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Table
Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a groupsuch as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's centercan be discovered from its Cayley table. A simple example of a Cayley table is the one for the group under ordinary multiplication: History Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation ''θ'' ''n'' = 1". In that paper they were referred to simply as tables, and were merely illustrativethey came to be known as Cayley tables later on, in honour of their creator. Structure and layout Because many Cayley tables describe groups that are not abelian, the product ''ab'' with respect to t ... [...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 Being a subgroup of a symmetric group, all that is necessary for a set of permutatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem that had been open for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra. He was a staunch republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830. As a result of his political activism, he was arrested repeatedly, serving one jail sentence of several months. For reasons that remain obscure, shortly after his release from prison he fought in a duel and died of the wounds he suffered. Life Early life Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante). His father was a Republican and was head of Bourg-la-Reine's liberal party. His father became may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Function
In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs. Precisely stated, a function f is binary if there exists sets X, Y, Z such that :\,f \colon X \times Y \rightarrow Z where X \times Y is the Cartesian product of X and Y. Alternative definitions Set-theoretically, a binary function can be represented as a subset of the Cartesian product X \times Y \times Z, where (x,y,z) belongs to the subset if and only if f(x,y) = z. Conversely, a subset R defines a binary function if and only if for any x \in X and y \in Y, there exists a unique z \in Z such that (x,y,z) belongs to R. f(x,y) is then defined to be this z. Alternatively, a binary function may be interpreted as simply a function from X \times Y to Z. Even when thought of this way, however, one generally writes f(x,y) instead of f((x,y)). (That is, the same pair of parentheses is used to indicate both function application and the formation of an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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