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Trigintaduonion
In abstract algebra, the trigintaduonions, also known as the , , form a commutative property, noncommutative and associative property, nonassociative algebra over a field, algebra over the real numbers. Names The word ''trigintaduonion'' is derived from Latin ' 'thirty' + ' 'two' + the suffix -''nion'', which is used for hypercomplex number systems. Other names include , , , and . Definition Every trigintaduonion is a linear combination of the unit trigintaduonions e_0, e_1, e_2, e_3, ..., e_, which form a Basis (linear algebra), basis of the vector space of trigintaduonions. Every trigintaduonion can be represented in the form :x = x_0 e_0 + x_1 e_1 + x_2 e_2 + \cdots + x_ e_ + x_ e_ with real coefficients . The trigintaduonions can be obtained by applying the Cayley–Dickson construction to the sedenions. Applying the Cayley–Dickson construction to the trigintaduonions yields a 64-dimensional algebra called the ''64-ions'', ''64-nions'', ''sexagintaquatronions'', or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Dickson Construction
In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, each with twice the dimension of a vector space, dimension of the previous one. It is named after Arthur Cayley and Leonard Eugene Dickson. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics. The Cayley–Dickson construction defines a new algebra as a Cartesian product of an algebra with itself, with multiplication defined in a specific way (different from the componentwise operation, componentwise multiplication) and an involution (mathematics), involution known as ''conjugation''. The product of an element and its complex conjugate, conjugate (or sometimes the square ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercomplex Number
In mathematics, hypercomplex number is a traditional term for an element (mathematics), element of a finite-dimensional Algebra over a field#Unital algebra, unital algebra over a field, algebra over the field (mathematics), field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory. History In the nineteenth century, number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, extending the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them. The cataloguing project began in 1872 when Benjamin Peirce first published his ''Linear Associative Algebra'', and was carried forward by his son Charles Sanders Peirce. Most significantly, they identified the nilpotent and the idempotent element (ring theory), idempotent elements as useful ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathematics), product''. Multiplication is often denoted by the cross symbol, , by the mid-line dot operator, , by juxtaposition, or, in programming languages, by an asterisk, . The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''; both numbers can be referred to as ''factors''. This is to be distinguished from term (arithmetic), ''terms'', which are added. :a\times b = \underbrace_ . Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression 3 \times 4 , can be phrased as "3 ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercomplex Number
In mathematics, hypercomplex number is a traditional term for an element (mathematics), element of a finite-dimensional Algebra over a field#Unital algebra, unital algebra over a field, algebra over the field (mathematics), field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory. History In the nineteenth century, number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, extending the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them. The cataloguing project began in 1872 when Benjamin Peirce first published his ''Linear Associative Algebra'', and was carried forward by his son Charles Sanders Peirce. Most significantly, they identified the nilpotent and the idempotent element (ring theory), idempotent elements as useful ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternative Algebra
In abstract algebra, an alternative algebra is an algebra over a field, algebra in which multiplication need not be associative, only alternativity, alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The associator Alternative algebras are so named because they are the algebras for which the associator is alternating form, alternating. The associator is a trilinear map given by :[x,y,z] = (xy)z - x(yz). By definition, a multilinear map is alternating if it Vanish_(mathematics), vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to :[x,x,y] = 0 :[y,x,x] = 0 Both of these identities together imply that: :[x,y,x]=[x,x,x]+[x,y,x]+ :-[x,x+y,x+y] = := [x,x+y,-y] = := [x,x,-y] - [x,y,y] = 0 for all x and y. This is equivalent to the ''f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flexible Algebra
In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: : a \bullet \left(b \bullet a\right) = \left(a \bullet b\right) \bullet a for any two elements ''a'' and ''b'' of the set. A magma (that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a nonassociative algebra is flexible if its multiplication operator is flexible. Every commutative or associative operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the multiplication of sedenions, which are not even alternative. In 1954, Richard D. Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed that they satisfy the flexible identity.Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”, American Journal of Mathematics 76: 43 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Associativity
In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity. Definition An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element is associative. Concretely, this means that if an element x is performed an operation * by itself several times, it doesn't matter in which order the operations are carried out, so for instance x*(x*(x*x)) = (x*(x*x))*x = (x*x)*(x*x). Examples and properties Every associative algebra is power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even non-alternative flexible algebras like the sedenions, trigintaduonions, and Okubo algebras. Any algebra whose elements are idempotent is also power-associative. Exponentiation to the power of any positive integer can be defined consistently whenever multiplication is power-associative. For example, there is no ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sedenion
In abstract algebra, the sedenions form a 16-dimension of a vector space, dimensional commutative property, noncommutative and associative property, nonassociative algebra over a field, algebra over the real numbers, usually represented by the capital letter S, boldface or blackboard bold \mathbb S. The sedenions are obtained by applying the Cayley–Dickson construction to the octonions, which can be mathematically expressed as \mathbb=\mathcal(\mathbb,1). As such, the octonions are isomorphism, isomorphic to a Subalgebra#Subalgebras for algebras over a ring or field, subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, called the trigintaduonions or sometimes the 32-nions. The term ''sedenion'' is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions, or the algebra of 4 × 4 matri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PG(3,2)
In finite geometry, PG(3, 2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane, ''PG(2, 2)''. Elements It has 15 points, 35 lines, and 15 planes. Each point is contained in 7 lines and 7 planes. Each line is contained in 3 planes and contains 3 points. Each plane contains 7 points and 7 lines. These can be summarized in a rank 3 configuration matrix counting points, lines, and planes on the diagonal. The incidences are expressed off diagonal. The structure is self dual, swapping points and planes, expressed by rotating the configuration matrix 180 degrees. :\left begin15&7&7\\3&35&3\\7&7&15\end\right /math> It has the following properties: * Each plane is isomorphic to the Fano plane. * Every pair of distinct planes intersects in a line. * A line and a plane not containing the line intersect in exactly one point. has 20160 automorphisms. The number of automorphisms is given by finding the number of ways of selecting 4 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Plane
In finite geometry, the Fano plane (named after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces, is . Here, stands for "projective geometry", the first parameter is the geometric dimension (it is a plane, of dimension 2) and the second parameter is the order (the number of points per line, minus one). The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study. In a separate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
