翻訳と辞書 Words near each other ・ Octomeria rohrii ・ Octomeria scirpoidea ・ Octomeria tricolor ・ Octomeria truncicola ・ Octomino ・ Octomitus ・ Octomom Home Alone ・ Octomylodon ・ Octon ・ Octon, East Riding of Yorkshire ・ Octon, Hérault ・ Octonaire ・ Octonary ・ Octonauts Rollercoaster Adventure ・ Octonia Stone ・ Octonion ・ Octonion algebra ・ Octonium ・ Octonodula ・ Octonodula binotella ・ Octonodula inumbrata ・ Octopamine ・ Octopamine dehydratase ・ Octopart ・ Octophone ・ Octophonic sound ・ Octopiler ・ Octopine ・ Octopine dehydrogenase family ・ Octopod External Fixator Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Octonion ： ウィキペディア英語版 Octonion In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold $\backslash mathbb\; O$. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H. The octonions are the largest such algebra, with eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, namely they are alternative. Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Despite this, they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic. The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Hamilton's discovery of quaternions. Graves called his discovery octaves, and mentioned them in a letter to Hamilton dated 16 December 1843, but his first publication of his result in was slightly later than Cayley's article on them. The octonions were discovered independently by Arthur Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra. described the early history of Graves' discovery. ==Definition== The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions: :$\backslash ,\backslash ,$ where ''e''0 is the scalar or real element; it may be identified with the real number 1. That is, every octonion ''x'' can be written in the form :$x\; =\; x\_0e\_0\; +\; x\_1e\_1\; +\; x\_2e\_2\; +\; x\_3e\_3\; +\; x\_4e\_4\; +\; x\_5e\_5\; +\; x\_6e\_6\; +\; x\_7e\_7,\backslash ,$ with real coefficients . Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the product of all the terms, again like quaternions. The product of each term can be given by multiplication of the coefficients and a multiplication table of the unit octonions, like this one:〔 This table is due to Arthur Cayley (1845) and John T. Graves (1843). See 〕 Most off-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for which is an operand. The table can be summarized by the relations:〔 〕 :$e\_i\; e\_j\; =\; -\; \backslash delta\_e\_0\; +\; \backslash varepsilon\; \_\; e\_k,\backslash ,$ where $\backslash varepsilon\; \_$ is a completely antisymmetric tensor with value +1 when ''ijk'' = 123, 145, 176, 246, 257, 347, 365, and: :$e\_ie\_0\; =\; e\_0e\_i\; =\; e\_i;\backslash ,\backslash ,\backslash ,\backslash ,e\_0e\_0\; =\; e\_0,\backslash ,$ with ''e''0 the scalar element, and ''i'', ''j'', ''k'' = 1 ... 7. The above definition though is not unique, but is only one of 480 possible definitions for octonion multiplication with . The others can be obtained by permuting and changing the signs of the non-scalar basis elements. The 480 different algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used. Each of these 480 definitions is invariant up to signs under some 7-cycle of the points (1234567), and for each 7-cycle there are four definitions, differing by signs and reversal of order. A common choice is to use the definition invariant under the 7-cycle (1234567) with as it is particularly easy to remember the multiplication. A variation of this sometimes used is to label the elements of the basis by the elements ∞, 0, 1, 2, ..., 6, of the projective line over the finite field of order 7. The multiplication is then given by and , and all expressions obtained from this by adding a constant (mod 7) to all subscripts: in other words using the 7 triples (124) (235) (346) (450) (561) (602) (013). These are the nonzero codewords of the quadratic residue code of length 7 over the field of 2 elements. There is a symmetry of order 7 given by adding a constant mod 7 to all subscripts, and also a symmetry of order 3 given by multiplying all subscripts by one of the quadratic residues 1, 2, 4 mod 7.〔 〕〔 Available as (ArXive preprint ) Figure 1 is located (here ).〕 The multiplication table for a Geometric algebra of signature (----) can be given in terms of the following 7 quaternionic triples (omitting the identity element): (I,j,k), (i,J,k), (i,j,K), (I,J,K), ( *I,i,m), ( *J,j,m), ( *K,k,m) in which the lowercase items are vectors (mathematics and physics) and the uppercase ones are bivectors and *=mijk (which is in fact the Hodge dual operator). If the * is forced to be equal to the identity then the multiplicaton ceases to be associative, but the * may be removed from the multiplication table resulting in an octonion multiplication table. Note that in keeping *=mijk associative and thus not reducing the 4 dimensional geometric algebra to an octonion one, the whole multiplication table can be derived from the equation for *. Consider the gamma matrices. The formula definining the fifth gamma matrix shows that it is the * of a four dimensional geometric algebra of the gamma matrices. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Octonion」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.