Let and be any functions of a set of variables . Then the expression(1)is called a Poisson bracket (Poisson 1809; Whittaker 1944, p. 299). Plummer (1960, p. 136) uses the alternate notation .The Poisson brackets are anticommutative,(2)(Plummer 1960, p. 136).Let be independent functions of the variables . Then the Poisson bracket is connected with the Lagrange bracket by(3)where is the Kronecker delta. But this is precisely the condition that the determinants formed from them are reciprocal (Whittaker 1944, p. 300; Plummer 1960, p. 137).If and are physically measurable quantities (observables) such as position, momentum, angular momentum, or energy, then they are represented as non-commuting quantum mechanical operators in accordance with Heisenberg's formulation of quantum mechanics. In this case,(4)where is the commutator and is the Poisson bracket. Thus, for example, for a single particle..
A rational function can be rewritten using what is known as partial fraction decomposition. This procedure often allows integration to be performed on each term separately by inspection. For each factor of the form , introduce terms(1)For each factor of the form , introduce terms(2)Then write(3)and solve for the s and s.Partial fraction decomposition is implemented as Apart.
The number of nonassociative -products with elements preceding the rightmost left parameter is(1)(2)where is a binomial coefficient. The number of -products in a nonassociative algebra is(3)where is a Catalan number, 1, 1, 2, 5, 14, 42, 132, ... (OEIS A000108).
The nesting of two or more functions to form a single new function is known as composition. The composition of two functions and is denoted , where is a function whose domain includes the range of . The notation(1)is sometimes used to explicitly indicate the variable.Composition is associative, so that(2)If the functions is continuous at and is continuous at , then is also continuous at .A function which is the composition of two other functions, say and , is sometimes said to be a composite function.Faà di Bruno's formula gives an explicit formula for the th derivative of the composition .A combinatorial composition is defined as an ordered arrangement of nonnegative integers which sum to (Skiena 1990, p. 60). It is therefore a partition in which order is significant. For example, there are eight compositions of 4,(3)(4)(5)(6)(7)(8)(9)(10)A positive integer has compositions.The number of compositions of into parts (where..
Let , , ... be operators. Then the commutator of and is defined as(1)Let , , ... be constants, then identities include(2)(3)(4)(5)(6)(7)(8)Let and be tensors. Then(9)There is a related notion of commutator in the theory of groups. The commutator of two group elements and is , and two elements and are said to commute when their commutator is the identity element. When the group is a Lie group, the Lie bracket in its Lie algebra is an infinitesimal version of the group commutator. For instance, let and be square matrices, and let and be paths in the Lie group of nonsingular matrices which satisfy(10)(11)(12)then(13)
An operator defined on a set which takes two elements from as inputs and returns a single element of . Binary operators are called compositions by Rosenfeld (1968). Sets possessing a binary multiplication operation include the group, groupoid, monoid, quasigroup, and semigroup. Sets possessing both a binary multiplication and a binary addition operation include the division algebra, field, ring, ringoid, semiring, and unit ring.
A binary operation is an operation that applies to two quantities or expressions and .A binary operation on a nonempty set is a map such that 1. is defined for every pair of elements in , and 2. uniquely associates each pair of elements in to some element of . Examples of binary operation on from to include addition (), subtraction (), multiplication ) and division ().
Direct sums are defined for a number of different sorts of mathematical objects, including subspaces, matrices, modules, and groups.The matrix direct sum is defined by(1)(2)(Ayres 1962, pp. 13-14).The direct sum of two subspaces and is the sum of subspaces in which and have only the zero vector in common (Rosen 2000, p. 357).The significant property of the direct sum is that it is the coproduct in the category of modules (i.e., a module direct sum). This general definition gives as a consequence the definition of the direct sum of Abelian groups and (since they are -modules, i.e., modules over the integers) and the direct sum of vector spaces (since they are modules over a field). Note that the direct sum of Abelian groups is the same as the group direct product, but that the term direct sum is not used for groups which are non-Abelian.Note that direct products and direct sums differ for infinite indices. An element of the direct sum is..
The Cartesian product of two sets and (also called the product set, set direct product, or cross product) is defined to be the set of all points where and . It is denoted , and is called the Cartesian product since it originated in Descartes' formulation of analytic geometry. In the Cartesian view, points in the plane are specified by their vertical and horizontal coordinates, with points on a line being specified by just one coordinate. The main examples of direct products are Euclidean three-space (, where are the real numbers), and the plane ().The graph product is sometimes called the Cartesianproduct (Vizing 1963, Clark and Suen 2000).
The direct sum of modules and is the module(1)where all algebraic operations are defined componentwise. In particular, suppose that and are left -modules, then(2)and(3)where is an element of the ring . The direct sum of an arbitrary family of modules over the same ring is also defined. If is the indexing set for the family of modules, then the direct sum is represented by the collection of functions with finite support from to the union of all these modules such that the function sends to an element in the module indexed by .The dimension of a direct sum is the sum of the dimensions of the quantities summed. The significant property of the direct sum is that it is the coproduct in the category of modules. This general definition gives as a consequence the definition of the direct sum of Abelian groups and (since they are -modules, i.e., modules over the integers) and the direct sum of vector spaces (since they are modules over a field). Note that the direct..
The tensor product of two vector spaces and , denoted and also called the tensor direct product, is a way of creating a new vector space analogous to multiplication of integers. For instance,(1)In particular,(2)Also, the tensor product obeys a distributive law with the directsum operation:(3)The analogy with an algebra is the motivation behind K-theory. The tensor product of two tensors and can be implemented in the Wolfram Language as: TensorProduct[a_List, b_List] := Outer[List, a, b]Algebraically, the vector space is spanned by elements of the form , and the following rules are satisfied, for any scalar . The definition is the same no matter which scalar field is used.(4)(5)(6)One basic consequence of these formulas is that(7)A vector basis of and of gives a basis for , namely , for all pairs . An arbitrary element of can be written uniquely as , where are scalars. If is dimensional and is dimensional, then has dimension .Using tensor products,..
The natural projection, also called the homomorphism, is a logical way of mapping an algebraic structure onto its quotient structures. The natural projection is defined formally for groups and rings as follows.For a group , let (i.e., be a normal subgroup of ). Then is defined by . Note (Dummit and Foote 1998, p. 84).For a ring, let be an ideal of a ring . is defined be . Note (Dummit and Foote 1998, p. 244).
An operation that takes two vector bundles over a fixed space and produces a new vector bundle over the same space. If and are vector bundles over , then the Whitney sum is the vector bundle over such that each fiber over is naturally the direct sum of the and fibers over .The Whitney sum is therefore the fiber for fiber direct sum of the two bundles and . An easy formal definition of the Whitney sum is that is the pull-back bundle of the diagonal map from to , where the bundle over is .
Exponentiation is the process of taking a quantity (the base) to the power of another quantity (the exponent). This operation most commonly denoted . In TeX, the Wolfram Language, and many other computer languages, exponentiation is denoted with a caret, i.e., as b^e. However, in FORTRAN, it is denoted b**e (Calderbank 1989, p. 29).
The substitution of(1)into the standard form cubic equation(2)The result reduces the cubic to the equation(3)which is easily turned into a quadratic equation in by multiplying through by to obtain(4)
A transformation of a polynomial equation which is of the form where and are polynomials and does not vanish at a root of . The cubic equation is a special case of such a transformation. Tschirnhaus (1683) showed that a polynomial of degree can be reduced to a form in which the and terms have 0 coefficients. In 1786, E. S. Bring showed that a general quintic equation can be reduced to the formIn 1834, G. B. Jerrard showed that a Tschirnhaus transformation can be used to eliminate the , , and terms for a general polynomial equation of degree .
The direct product of the rings , for some index set , is the setThe ring direct product is confusingly also called the complete direct sum (Herstein 1968).The ring direct product, like the group direct product, has the universal property that if any ring has a homomorphism to and a homomorphism to , then these homomorphisms factor through in a unique way.