* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G ( x, y ) ( i. e. the sets of morphisms from x to y ).

Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given the laws of exponents, f ( x )

Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if

Given a function f of a real variable x and an interval of the real line, the definite integral

Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

* Given a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨ a, b, c, d | a < sup > n </ sup > ba < sup > n </ sup > = c < sup > n </ sup > dc < sup > n </ sup >: n ∈ A ⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble

Given a topological space X, denote F the set of filters on X, x ∈ X a point, V ( x ) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x.

Given a vector x ∈ V and y * ∈ W *, then the tensor product y * ⊗ x corresponds to the map A: W → V given by

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Given a graph Λ ( for example, a d-dimensional lattice ), per each lattice site j ∈ Λ there is a discrete variable σ < sub > j </ sub > such that σ < sub > j </ sub > ∈

Given *- representations π, π ' each with unit norm cyclic vectors ξ ∈ H, ξ ' ∈ K such that their respective associated states coincide, then π, π ' are unitarily equivalent representations.

For a finite group G, the left regular representation λ ( over a field K ) is a linear representation on the K-vector space V whose basis is the elements of G. Given g ∈ G, λ ( g ) is the linear map determined by its action on the basis by left translation by g, i. e.

Given a function and an element g ∈ G,

* Given a maximal torus T in G, every element g ∈ G is conjugate to an element in T.

Given a ∈ A, one defines the function by.

Given a labelled state transition system ( S, Λ, →), a simulation relation is a binary relation R over S ( i. e. R ⊆ S × S ) such that for every pair of elements p, q ∈ S, if ( p, q )∈ R then for all α ∈ Λ, and for all p ' ∈ S,

Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic ; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf Ω < sup > 1 </ sup > of Kähler differentials.

Given a simply connected open subset D of C < sup > n </ sup >, there is an associated sheaf O < sub > D </ sub > of holomorphic functions on D. Throughout, U is any open subset of D. Then the set O < sub > D </ sub >( U ) of holomorphic functions from U to C has a natural ( componentwise ) C-algebra structure and one can collate sections that agree on intersections to create larger sections ; this is outlined in more detail at sheaf.

Given a topological space X and a sheaf O < sub > X </ sub > of local C-algebras, if for any point x in X there is an open subset V of X containing it and a subset D of C < sup > n </ sup > so that the restriction ( V, O < sub > V </ sub >) of ( X, O < sub > X </ sub >) is isomorphic to a closed complex subspace of D, O < sub > X </ sub > is also coherent, and we call it a holomorphic sheaf.

The exponential map is surjective on the stalks: Given a germ g of an holomorphic function at a point P such that g ( P ) ≠ 0, one can take the logarithm of g in a neighborhood of P. The long exact sequence of sheaf cohomology shows that we have an exact sequence

Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

The functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place.

Given a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

Given the representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h ( T ) is the operator of multiplication by the composition.

Given a strictly increasing integer sequence / function ( n ≥ 1 ) we can produce a faster growing sequence ( where the superscript n denotes the n < sup > th </ sup > functional power ).

Given that X, Y, and Z are sets of attributes in a relation R, one can derive several properties of functional dependencies.

Given a functional of the form

The general scenario is the following: Given a class S of computable functions, is there a learner ( that is, recursive functional ) which for any input of the form ( f ( 0 ), f ( 1 ),..., f ( n )) outputs a hypothesis ( an index e with respect to a previously agreed on acceptable numbering of all computable functions ; the indexed function should be consistent with the given values of f ).

Given a multilinear functional M < sub > n </ sub > of degree n ( that is, M < sub > n </ sub > is n-linear ), we can define a polynomial p as

Given a polynomially bounded functional F over the field configurations, then, for any state vector ( which is a solution of the QFT ), | ψ >, we have

Given an operator T, the range of the continuous functional calculus h → h ( T ) is the ( abelian ) C *- algebra C ( T ) generated by T. The Borel functional calculus has a larger range, that is the closure of C ( T ) in the weak operator topology, a ( still abelian ) von Neumann algebra.

Given the partition function Z in terms of the source field J, the energy functional is its logarithm.

Given a Banach space, a subset of, and a functional from to the dual space of the space, the variational inequality problem is the problem of solving respect to the variable ' belonging to the following inequality:

Given the unique mapping between densities and wave function, Runge and Gross then treated the Dirac action as a density functional,

Given a characteristic functional on a nuclear space A, the Bochner – Minlos theorem ( after Salomon Bochner and Robert Adol ' fovich Minlos ) guarantees the existence and uniqueness of the corresponding probability measure on the dual space, given by