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List Of Things Named After Arthur Cayley
{{Short description, none Arthur Cayley (1821 – 1895) is the eponym of all the things listed below. * Cayley absolute * Octonion, Cayley algebra * Cayley computer algebra system * Cayley diagrams – used for finding cognate linkages in mechanical engineering * Cayley graph * Cayley numbers * Cayley plane * Cayley table * Cayley transform * Cayleyan * Cayley–Bacharach theorem * Cayley–Dickson construction * Cayley–Hamilton theorem in linear algebra * Cayley–Klein metric * Klein model, Cayley–Klein model of hyperbolic geometry * Cayley–Menger determinant * Cayley–Purser algorithm * Cayley's formula * Hyperdeterminant, Cayley's hyperdeterminant * Cayley's mousetrap — a card game * Cayley's nodal cubic surface * Normal_p-complement#Cayley_normal_2-complement_theorem, Cayley normal 2-complement theorem * Cayley's ruled cubic surface * Cayley's sextic * Cayley's theorem * Cayley's Ω process * Chasles–Cayley–Brill formula * Grassmann–Cayley algebra * The crater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. He worked as a lawyer for 14 years. He postulated 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 a group in the modern way—as a set with a binary operation satisfying certain laws. Formerly, when mathematicians spoke of "groups", they had meant permutation groups. Cayley tables and Cayley graphs as well as Cayley's theorem are named in honour of Cayley. Early years Arthur Cayley was born in Richmond, London, England, on 16 August 1821. His father, Hen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Klein Metric
In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance"Cayley (1859), p 82, §§209 to 229 where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and subsequent books and papers. The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics. Foundations The algebra of throws by Karl von Staudt (1847) is an approach to geometry that is independent of metric. The idea was to use the relation of projective harmonic conjugates and cross-ratios as fundamental to the measure on a line. Another important insight was the Laguerr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Ω Process
In mathematics, Cayley's Ω process, introduced by , is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action. As a partial differential operator acting on functions of ''n''2 variables ''x''''ij'', the omega operator is given by the determinant : \Omega = \begin \frac & \cdots &\frac \\ \vdots& \ddots & \vdots\\ \frac & \cdots &\frac \end. For binary forms ''f'' in ''x''1, ''y''1 and ''g'' in ''x''2, ''y''2 the Ω operator is \frac - \frac. The ''r''-fold Ω process Ω''r''(''f'', ''g'') on two forms ''f'' and ''g'' in the variables ''x'' and ''y'' is then # Convert ''f'' to a form in ''x''1, ''y''1 and ''g'' to a form in ''x''2, ''y''2 # Apply the Ω operator ''r'' times to the function ''fg'', that is, ''f'' times ''g'' in these four variables # Substitute ''x'' for ''x''1 and ''x''2, ''y'' for ''y''1 and ''y''2 in the result The result of the ''r''-fold Ω process Ω''r''(''f'', ''g'') on the two ... [...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's Sextic
In geometry, Cayley's sextic (sextic of Cayley, Cayley's sextet) is a plane curve, a member of the sinusoidal spiral family, first discussed by Colin Maclaurin in 1718. Arthur Cayley was the first to study the curve in detail and it was named after him in 1900 by Raymond Clare Archibald. The curve is symmetric about the ''x''-axis (''y'' = 0) and self-intersects at ''y'' = 0, ''x'' = −''a''/8. Other intercepts are at the origin, at (''a'', 0) and with the ''y''-axis at ± ''a'' The curve is the pedal curve (or ''roulette'') of a cardioid with respect to its cusp. Equations of the curve The equation of the curve in polar coordinates is :''r'' = ''4a'' cos3(''θ''/3) In Cartesian coordinates the equation is :4(''x''2 + ''y''2 − (''a''/4)''x'')3 = 27(''a''/4)2(''x''2 + ''y''2)2 . Cayley's sextic may be parametrised (as a periodic function A periodic function is a functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Ruled Cubic Surface
In differential geometry, Cayley's ruled cubic surface is the ruled cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather tha ... :x^3 + (4 x\, z + y) x =0.\ It contains a nodal line of self-intersection and two cuspital points at infinity. In projective coordinates it is x^3 + (4 x\, z + y\, w) x =0.\ . References External linksCubical ruled surface* {{DEFAULTSORT:Cayley's Ruled Cubic Surface Algebraic surfaces Differential geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal P-complement
In mathematical group theory, a normal p-complement of a finite group for a prime ''p'' is a normal subgroup of order coprime to ''p'' and index a power of ''p''. In other words the group is a semidirect product of the normal ''p''-complement and any Sylow ''p''-subgroup. A group is called p-nilpotent if it has a normal ''p''-complement. Cayley normal 2-complement theorem Cayley showed that if the Sylow 2-subgroup of a group ''G'' is cyclic then the group has a normal 2-complement, which shows that the Sylow 2-subgroup of a simple group of even order cannot be cyclic. Burnside normal p-complement theorem showed that if a Sylow ''p''-subgroup of a group ''G'' is in the center of its normalizer then ''G'' has a normal ''p''-complement. This implies that if ''p'' is the smallest prime dividing the order of a group ''G'' and the Sylow ''p''-subgroup is cyclic, then ''G'' has a normal ''p''-complement. Frobenius normal p-complement theorem The Frobenius normal ''p''-complement the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Nodal Cubic Surface
In algebraic geometry, the Cayley surface, named after Arthur Cayley, is a cubic nodal surface in 3-dimensional projective space with four conical points. It can be given by the equation : wxy+ xyz+ yzw+zwx =0\ when the four singular points are those with three vanishing coordinates. Changing variables gives several other simple equations defining the Cayley surface. As a del Pezzo surface of degree 3, the Cayley surface is given by the linear system of cubics in the projective plane passing through the 6 vertices of the complete quadrilateral. This contracts the 4 sides of the complete quadrilateral to the 4 nodes of the Cayley surface, while blowing up its 6 vertices to the lines through two of them. The surface is a section through the Segre cubic. The surface contains nine lines, 11 tritangents and no double-sixes. A number of affine forms of the surface have been presented. Hunt uses (1-3 x-3y-3z)(xy+xz+yz)+6xyz = 0 by transforming coordinates (u_0, u_1, u_2, u_3) to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley's Mousetrap
Mousetrap is the name of a game introduced by the English mathematician Arthur Cayley. In the game, cards numbered 1 through n ("say thirteen" in Cayley's original article) are shuffled to place them in some random permutation and are arranged in a circle with their faces up. Then, starting with the first card, the player begins counting 1, 2, 3, ... and moving to the next card as the count is incremented. If at any point the player's current count matches the number on the card currently being pointed to, that card is removed from the circle and the player starts all over at 1 on the next card. If the player ever removes all of the cards from the permutation in this manner, then the player wins. If the player reaches the count n+1 and cards still remain, then the game is lost. In order for at least one card to be removed, the initial permutation of the cards must not be a derangement In combinatorial mathematics, a derangement is a permutation of the elements of a set, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperdeterminant
In algebra, the hyperdeterminant is a generalization of the determinant. Whereas a determinant is a scalar valued function defined on an ''n'' × ''n'' square matrix, a hyperdeterminant is defined on a multidimensional array of numbers or tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens .... Like a determinant, the hyperdeterminant is a homogeneous polynomial with integer coefficients in the components of the tensor. Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of volumes. There are at least three definitions of hyperdeterminant. The first was discovered by Arthur Cayley in 1843 presented to the Cambridge Philosophical Society. A. Cayley, ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Purser Algorithm
The Cayley–Purser algorithm was a public-key cryptography algorithm published in early 1999 by 16-year-old Irishwoman Sarah Flannery, based on an unpublished work by Michael Purser, founder of Baltimore Technologies, a Dublin data security company. Flannery named it for mathematician Arthur Cayley. It has since been found to be flawed as a public-key algorithm, but was the subject of considerable media attention. History During a work-experience placement with Baltimore Technologies, Flannery was shown an unpublished paper by Michael Purser which outlined a new public-key cryptographic scheme using non-commutative multiplication. She was asked to write an implementation of this scheme in Mathematica. Before this placement, Flannery had attended the 1998 ESAT Young Scientist and Technology Exhibition with a project describing already existing cryptographic techniques from the Caesar cipher to RSA. This had won her the Intel Student Award which included the opportunity to com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
