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 the integral representation of a principal branch of γ, the following equation holds for all positive real s, x:

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 >.

Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

Given a function f: X → Y, the set X is the domain of f ; the set Y is the codomain of f. In the expression f ( x ), x is the argument and f ( x ) is the value.

) Given a domain D we define a tuple over D as a partial function

Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range:

Given on the periodic domain

Given any set of points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian.

Given a fundamental domain for the group action ( for the full, orientation-reversing symmetry group, a ( 2, 3, 7 ) triangle ), the reflection domains ( images of this domain under the group ) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group ( reflections in the lines of a given fundamental triangle give a set of 3 generating reflections ).

Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain ( also called fundamental region ) for this action is a set D of representatives for the orbits.

Given any morphism between objects X and Y, if there is an inclusion map into the domain, then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain R → Y known as the range of f.

Given an element a and a non-zero element b in a Euclidean domain R equipped with a Euclidean function d, there exist q and r in R such that and either or.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

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 a topological space X, let G < sub > 0 </ sub > be the set X.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given two points P and Q on C, let s ( P, Q ) be the arc length of the portion of the curve between P and Q and let d ( P, Q ) denote the length of the line segment from P to Q.

Given a linearly recursive sequence, let C be the transpose of the companion matrix of its characteristic polynomial, that is

Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")

Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively.

Given that the police had no evidence against " Thomas ", either at Leatherslade Farm or connection with either of the two gangs, Butler was prepared to let him go.

Given a concrete category ( C, U ) and a cardinal number N, let U < sup > N </ sup > be the functor C → Set determined by U < sup > N </ sup >( c ) = ( U ( c ))< sup > N </ sup >.

Given a unit vector n in R < sup > 3 </ sup > and an angle φ, let R ( φ, n ) represent a counterclockwise rotation about the axis through n ( with orientation determined by n ).

Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

) Given a section σ of E let the corresponding equivariant map be ψ ( σ ).

Given an integer n > 1, let H be any subgroup of the multiplicative group

" Given the massive undertaking in creating 11 volumes over 50 years, errors and incompleteness were inevitable by Durant's own reckoning ; but no other historical survey matches let alone exceeds the breadth and depth of his project, he claimed.

Given the military-industrial requirements of the Cold War, the authors of the then secret National Security Council Report 68 ( NSC-68 ) proposed the US government undertake a massive permanent national economic expansion that would let it “ siphon off ” a part of the economic activity produced to support an ongoing military buildup to contain the Soviet Union.

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 >.

Given a compact subset K of X and an open subset U of Y, let V ( K, U ) denote the set of all functions such that Then the collection of all such V ( K, U ) is a subbase for the compact-open topology on C ( X, Y ).

Given a set of primitive types, let

Given a semiautomaton, let T < sub > a </ sub >: X → X, for a ∈ Σ, denote the transformation of X defined by T < sub > a </ sub >( x ) = T ( a, x ).

Given a triangle ABC and its three Malfatti circles, let D, E, and F be the points where two of the circles touch each other, opposite vertices A, B, and C respectively.

: Given such a tangent vector v, let f be the curve given in the x < sup > i </ sup > coordinate system by.