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Ternary Commutator
In mathematical physics, the ternary commutator is an additional ternary operation on a triple system defined by : ,b,c= abc-acb-bac+bca+cab-cba. \, Also called the ternutator or alternating ternary sum, it is a special case of the ''n''-commutator for ''n'' = 3, whereas the 2-commutator is the ordinary commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a .... Properties *When one or more of ''a'', ''b'', ''c'' is equal to 0, 'a'', ''b'', ''c''is also 0. This statement makes 0 the absorbing element of the ternary commutator. **The same happens when ''a'' = ''b'' = ''c''. Further reading * * * * * * Algebras Ternary operations {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Operation
In mathematics, a ternary operation is an ''n''- ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''. In computer science, a ternary operator is an operator that takes three arguments as input and returns one output. Examples The function T(a, b, c) = ab + c is an example of a ternary operation on the integers (or on any structure where + and \times are both defined). Properties of this ternary operation have been used to define planar ternary rings in the foundations of projective geometry. In the Euclidean plane with points ''a'', ''b'', ''c'' referred to an origin, the ternary operation , b, c= a - b + c has been used to define free vectors. Since (''abc'') = ''d'' implies ''a'' – ''b'' = ''c'' – ''d'', these directed segments are equipollent and are associated with the same free vector. Any three points in the plane ''a, b, c'' thus determine a parallelogram with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triple System
In algebra, a triple system (or ternar) is a vector space ''V'' over a field F together with a F-trilinear map : (\cdot,\cdot,\cdot) \colon V\times V \times V\to V. The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators ''u'', ''v'' ''w''] and triple Commutator, anticommutators . In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations (symmetric R-spaces and their noncompact duals). Lie triple systems A triple system is said to be a ''Lie triple system'' if the trilinear map, denoted cdot,\cdot,\cdot, satisfies the following identities: : ,v,w= - ,u,w : ,v,w+ ,u,v+ ,w,u= 0 : ,x,y.html"_;"title=",v,[w,x,y">,v,[w,x,y_=_u,v,wx,y.html" ;"title=",x,y">,v,[ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, and , of a group , is the element : . This element is equal to the group's identity if and only if and commute (from the definition , being equal to the identity if and only if ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of ''G'' generated by all commutators is closed and is called the ''derived group'' or the ''commutator subgroup'' of ''G''. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as :. Identities (group theory) Commutator identities are an important tool in group theory. The expr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absorbing Element
In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero elementM. Kilp, U. Knauer, A.V. Mikhalev pp. 14–15 because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation ''zero'' may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous. Definition Formally, let be a set ''S'' with a closed binary operation • on it (known as a magma). A zero element is an element ''z'' such that for all ''s'' in ''S'', . This notion can be refined to the notions of left zero, where one requires only that , and right zero, where . Absorbing elements are particularly interesting for semigroups, especially the mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity inste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
