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Anti-commute
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped arguments. The notion '' inverse'' refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of gives for example, Another prominent example of an anticommutative operation is the Lie bracket. In mathematical physics, where symmetry is of central importance, or even just in multilinear algebra these operations are mostly (multilinear with respect to some vector structures and then) called antisymmetric operations, and when they are not already of arity greater than two, extended in an associative setting to cover more than two arguments. Definition If A, B are two abelian groups, a bilinear map f\colon A^2 \to B is anticommuta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. A bilinear map can also be defined for modules. For that, see the article pairing. Definition Vector spaces Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function B : V \times W \to X such that for all w \in W, the map B_w v \mapsto B(v, w) is a linear map from V to X, and for all v \in V, the map B_v w \mapsto B(v, w) is a linear map from W to X. In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. Such a map B satisfies the following properties. * For any \lambda \in F, B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w). * The map B is additive in both components: if v_1, v_2 \in V an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry In Mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. Given a structured object ''X'' of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This can occur in many ways; for example, if ''X'' is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. If the object ''X'' is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (i.e., an isometry). In general, every kind of structure in mathematics will have its own kind of symmetry, many of which are listed in the given points mentioned above. Symmetry in geometry The types of symmetry considered in basic geometry include ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded-commutative Ring
In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy :xy = (-1)^ yx, where , ''x'', and , ''y'', denote the degrees of ''x'' and ''y''. A commutative (non-graded) ring, with trivial grading, is a basic example. For a nontrivial example, an exterior algebra is generally not a commutative ring but is a ''graded''-commutative ring. A cup product on cohomology satisfies the skew-commutative relation; hence, a cohomology ring is graded-commutative. In fact, many examples of graded-commutative rings come from algebraic topology and homological algebra. References * David Eisenbud, ''Commutative Algebra. With a view toward algebraic geometry'', Graduate Texts in Mathematics, vol 150, Springer-Verlag, New York, 1995. * See also * DG algebra * graded-symmetric algebra *alternating algebra In mathematics, an alternating algebra is a -graded algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade (geometry), blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude (mathematics), magnitude of a bivector, -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the Parallelepiped#Parallelotope, parallelotope defined by the ... [...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 (that is, 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 group theorists define the commutator as : . Using the first definition, this can be expressed as . Identities (group theory) Commutator identities are an important tool in group th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when new algebraic structures started to be studied. Definition A binary operation * on a set ''S'' is ''commutative'' if x * y = y * x for all x,y \in S. An operation that is not commutative is said to be ''noncommutative''. One says ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Ring
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-dimen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Multilinear Map
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of its arguments is equal. This generalizes directly to a module over a commutative ring. The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map of which all arguments belong to the same space. Definition Let R be a commutative ring and , W be modules over R. A multilinear map of the form f: V^n \to W is said to be alternating if it satisfies the following equivalent conditions: # whenever there exists 1 \leq i \leq n-1 such that x_i = x_ then . # whenever there exists 1 \leq i \neq j \leq n such that x_i = x_j then . Vector spaces Let V, W be vector spaces over the same field. Then a multilinear map of the form f: V^n \to W is alternating if it satisfies the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transposition (mathematics)
In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has ''k'' elements, it may be called a ''k''-cycle. Some authors widen this definition to include permutations with fixed points in addition to at most one non-trivial cycle. In Permutation#Cycle notation, cycle notation, cyclic permutations are denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted. For example, the permutation (1 3 2 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a 4-cycle, and the permutation (1 3 2)(4) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 is considered a 3-cycle by some authors. On the other hand, the permutation (1 3)(2 4) that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs and . For the wider definition of a cyclic permutation, allowi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion (algebra)
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements (in cases when this is indeed a submodule, such as when the ring is commutative). A torsion module is a module consisting entirely of torsion elements. A module is torsion-free if its only torsion element is the zero element. This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements. This terminology applies to abelian groups (with "module" and "submodule" replaced by "group" and "subgroup"). This is just a special case of the more general situation, because abelian groups are modules over the ring of integers. (In fact, this is the origin of the terminology, which was introduced for abelian groups before being generalized to modules.) In the case of gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
