Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

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

Given the objectivity and lowered costs of satellite-based land cover analysis, it appears likely that remote sensing technology will be an integral part of assessing the extent and damage of deforestation in the basin.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

Given a function w on U × Y, with finite integral of its modulus for any input function u and initial state x ( 0 ) over any finite time t, called the " supply rate ", a system is said to be dissipative if there exist a continuous nonnegative function V ( x ), with x ( 0 ) = 0, called the storage function, such that for any input u and initial state x ( 0 ) the difference V ( x ( t )) − V ( x ( 0 )) does not exceed the integral of the supply over ( 0, t ) for any t ( dissipation inequality ).

: Given that the particle begins at position x < sub > 1 </ sub > at time t < sub > 1 </ sub > and ends at position x < sub > 2 </ sub > at time t < sub > 2 </ sub >, the physical trajectory that connects these two endpoints is an extremum of the action integral.

Given an integral domain, let be an element of, the polynomial ring with coefficients in.

Using tensor calculus, proper time is more rigorously defined in general relativity as follows: Given a spacetime which is a pseudo-Riemannian manifold mapped with a coordinate system and equipped with a corresponding metric tensor, the proper time experienced in moving between two events along a timelike path P is given by the line integral

Having a measure on G allows us to define integration of bounded functions on G. Given a bounded function, the integral

Given an integral weight, one of two cases occur: ( 1 ) There is no such that is dominant, equivalently, there exists a nonidentity such that ; or ( 2 ) There is a unique such that is dominant.

Given a collection of differential 1-forms α < sub > i </ sub >, i = 1, 2, ..., k on an n-dimensional manifold M, an integral manifold is a submanifold whose tangent space at every point p ∈ M is annihilated by each α < sub > i </ sub >.

* Speech recognition: Given a sound clip of a person or people speaking, determine the textual representation of the speech.

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Given a context pertaining to a specific OSI layer, PDU is sometimes used as a synonym for its representation at that layer.

Given the hexadecimal representation 3FD5 5555 5555 5555 < sub > 16 </ sub >,

Given that, the Culture had intended all along to use him to discredit the brutality of the Azadian system by publicly giving the lie to the game's representation of social reality.

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 the mixed member proportional representation system used to elect its members, it is difficult for a single party to gain an overall majority of seats in the Scottish Parliament.

Given a state ρ of A, there is a *- representation π of A with distinguished cyclic vector ξ such that its associated state is ρ, i. e.

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.

A more general class are flat G-bundles with for a manifold F. Given a representation, the flat-bundle with monodromy is given by, where acts on the universal cover by deck transformations and on F by means of the representation.

Given an arbitrary category C, a representation of G in the category C is a functor from G to C. Such a functor selects an object of C and a subgroup of automorphisms of that object.

However, the modelling study also states that " Given the complex, three-dimensional nature of the real Pine Island glacier ... it should be clear that the [...] model is a very crude representation of reality.

Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.

Given a faithful irreducible representation ρ of G, the lattice Yang-Mills action is the sum over all lattice sites of the ( real component of the ) trace over the n links e < sub > 1 </ sub >, ..., e < sub > n </ sub > in the Wilson loop,

The Government believes that it must remain so .” Given the prospect of " incineration without representation ", Canadians seemed to feel that the doctrine which most encouraged restraint was the strategically soundest one to support.

Given this constraint, we can write in the following representation:

Given such a choice of scale, a central representation of the Heisenberg group is a map of-algebras which is the formal way of saying that it sends the center to a chosen scale.

Given a presentation of a group G by generators and relations and a subgroup H of G, the algorithm enumerates the cosets of H on G and describes the permutation representation of G on the space of the cosets.

Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the open elements of the interior algebra ( which form a base for a topology ).

Given an interior algebra, by replacing the topology of its Stone representation with the corresponding canonical preorder ( specialization preorder ) we obtain a representation of the interior algebra as a canonical preorder field.

Given the common view of knitting as work associated with middle-and lower-class women-typically older women-the groups hope that this representation will help to counter stereotypes.

Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.

Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line.

Given the principal stress state, we can use Mohr ’ s circle to solve for the maximum shear stresses our material will experience and conclude that the material will fail if:

Given the traceless character, only two of the principal components are independent.

Given that the Mother of God outranks St. Peter of Alcantara in the hierarchy of the saints, with the proclamation of the Virgin of Aparecida as Queen and Patroness of Brazil, the blessed Virgin Mary became the Nation's principal heavenly patron, and the Franciscan saint became the secondary patron.

Given the increasing seismic and volcanic activity, Johnston and the other volcanologists working for the USGS in its Vancouver branch prepared to observe any impending eruption.

Given the ( then highly unusual ) driver-only operation, the branch line was equipped with a two wire emergency telephone system at window level, a feature normally found only in tunnels on the London Underground.

Given a smooth curve γ on ( M, g ) and a vector field V along γ its derivative is defined by

Given a covariant derivative ∇, the parallel transport along a curve γ is obtained by integrating the condition.

Given a C < sup > n + 1 </ sup >- curve γ in R < sup > n </ sup > which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors

Let E be a rank k vector bundle over a smooth manifold M and let ∇ be a connection on E. Given a piecewise smooth loop γ: → M based at x in M, the connection defines a parallel transport map P < sub > γ </ sub >: E < sub > x </ sub > → E < sub > x </ sub >.