Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given unlimited resources, a classical computer can simulate an arbitrary quantum algorithm so quantum computation does not violate the Church – Turing thesis.

Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that

Given that hearing impairments can vary by frequency and that audiograms are plotted with a logarithmic scale, the idea of a percentage of hearing loss is somewhat arbitrary, but where decibels of loss are converted via a recognized legal formula, it is possible to calculate a standardized " percentage of hearing loss " which is suitable for legal purposes only.

Given a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers.

Given an inertial frame of reference and an arbitrary epoch ( a specified point in time ), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit.

Given Bluestein's algorithm, such a transform can be used, for example, to obtain a more finely spaced interpolation of some portion of the spectrum ( although the frequency resolution is still limited by the total sampling time ), enhance arbitrary poles in transfer-function analyses, etcetera.

Given an arbitrary ( n ; m ; p ) machine S, such that every two of its states are distinguishable from one another, then there exists an experiment of length

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.

Given an arbitrary series

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

Given an arbitrary window in a spatial file manager, it must be possible to determine with complete certainty which folder that window represents.

Given the precision of Brown's calculations, it must have come as a great disappointment to have to introduce this arbitrary adjustment.

Given an arbitrary direction z ( usually determined by an external magnetic field ) the spin z-projection is given by

Given an arbitrary square matrix, the elementary divisors used in the construction of the Jordan normal form do not exist over F, so the invariant factors a < sub > i </ sub >( x ) as given above must be used instead.

Given an arbitrary point on a torus, four circles can be drawn through it.

Given arbitrary complex numbers A, B, C, D such that AD − BC ≠ 0, define the quantities

Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g ′ compatible with the almost complex structure J in an obvious manner:

Given any Euclidean triangle ABC and an arbitrary point P let d ( P ) = PA + PB + PC.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

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 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 topological space X, a base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection of members of F.

Given any topological space X, the zero sets form the base for the closed sets of some topology on X.

Given a locally compact topological field K, an absolute value can be defined as follows.

Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X.

Given a complex vector bundle V over a topological space X,

Given a point x of a topological space X, and two maps f, g: X → Y ( where Y is any set ), then f and g define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal ;

Given any unital ring R, the set of singular n-simplices on a topological space can be taken to be the generators of a free R-module.

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.

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 a compact topological space X, the topological K-theory K < sup > top </ sup >( X ) of ( real ) vector bundles over X coincides with K < sub > 0 </ sub > of the ring of continuous real-valued functions on X.

Given a topological group G, the Bohr compactification of G is a compact Hausdorff topological group Bohr ( G ) and a continuous homomorphism

Given any topological space X we can define a ( possibly ) finer topology on X which is compactly generated.

Given a topological space and a subset of, the subspace topology on is defined by

Given a topological space X = 〈 X, T 〉 one can form the power set Boolean algebra of X:

Given a field of sets the complexes form a base for a topology, we denote the corresponding topological space by.

Given a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interior and closure.

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 any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

Given a set of training examples of the form, a learning algorithm seeks a function, where is the input space and

Given an operator on Hilbert space, consider the orbit of a point under the iterates of.

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

Given the date of his publication and the widespread, permanent distribution of his work, it appears that he should be regarded as the originator of the concept of space sailing by light pressure, although he did not develop the concept further.

Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.

Given a completely regular space X there is usually more than one uniformity on X that is compatible with the topology of X.

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system