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 topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).

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 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 a point ( in terms of its coordinates ) and the direction ( azimuth ) and distance from that point to a second point, determine ( the coordinates of ) that second point.

Given the differences of opinion within the Lord Aberdeen cabinet over the direction of foreign policy with regard to relations between Britain and the French under Napoleon III, it is not surprising that debate raged within the government as Louis Bonaparte, now assuming the title of Emperor Napoleon III of France.

Given their original purpose, military marching bands typically march in a forward direction with ( no variation of ) straight lines.

Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.

Given an embedding of a rooted tree in the plane, if one fixes a direction of children ( starting from root, then first child, second child, etc.

About the new markets, Ergen said, " Given the assets we've been accumulating, I don't think it's hard to see we're moving in a different direction from simply pay-TV, which is a market that's becoming increasingly saturated.

Given the similarities with readings from people living on loess soils, the general direction of the local movement according to Price et al., is from the northeast to the southwest.

Given the direction of the current, it is clear that the buoys would be dragged in the opposite direction to that claimed by Mr Jim Brooks.

Given estimates of the image gradients, a search is then carried out to determine if the gradient magnitude assumes a local maximum in the gradient direction.

Given vector fields and lying in the tangent bundle, the affine connection describes how to differentiate the vector field along the direction.

Given an incident direction from the surface to the light source and the surface normal direction the specularly reflected direction ( all unit vectors ) is:

Given Bernal's Marxist and pro-Soviet sympathies it is perhaps remarkable that there has never been any suggestion that he fed any information in that direction.

Given the change in direction apparent in his poetry, scholars assume only his death a short time after publication prevented him from further exploration ; the collection may in fact have been published posthumously.

Given a unit vector u = ( u < sub > x </ sub >, u < sub > y </ sub >, u < sub > z </ sub >), where u < sub > x </ sub >< sup > 2 </ sup > + u < sub > y </ sub >< sup > 2 </ sup > + u < sub > z </ sub >< sup > 2 </ sup > = 1, the matrix for a rotation by an angle of θ about an axis in the direction of u is

Given charge of policy direction for the government in the latter role, he was often dubbed the " Minister for Thought " by the Press.

Given that v < sup > 2 </ sup > ignores direction, it is logical to assume that the formula can be extended to the entire sample, replacing m with the entire sample's mass, equal to the molar mass times the number of moles n yielding

Given a differential equation of the form ( for example, when F has zero slope in the x and y direction at F ( x, y ) ):

Given the abundance of these different guidance cues it was previously believed that growth cones integrate various information by simply summing the gradient of cues, in different valences, at a given point in time, to making a decision on the direction of growth.

Given the knowledge of the direction of the market we would ( potentially ) make a different investment vehicle decision.

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 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 an element φ of H *, the orthogonal complement of the kernel of φ is a one-dimensional subspace of H. Take a non-zero element z in that subspace, and set.

Given two dual numbers p, and q, they determine the set of z such that the Galilean angle between the lines from z to p and q is constant.

Given the sphere defined by the points ( x, y, z ) such that

Given ω = ( xθ, yθ, zθ ), with v = ( x, y, z ) a unit vector, the correct skew-symmetric matrix form of ω is

Given a ( countable ) set of variables x, y, z, etc.

Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that

Given the rapid growth in absolute value of Γ ( z + k ) when k → ∞, and the fact that the reciprocal of Γ ( z ) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex s and x.

Given a modular form f ( z ) of weight k, the mth Hecke operator acts by the formula

Given four circles with curvatures k < sub > i </ sub > and centers z < sub > i </ sub > ( for i = 1 ... 4 ), the following equality holds in addition to equation ( 1 ):

Given a real matrix M and vector q, the linear complementarity problem seeks vectors z and w which satisfy the following constraints:

Given M ( a, b ; z ), the four functions M ( a ± 1, b, z ), M ( a, b ± 1 ; z ) are called contiguous to M ( a, b ; z ).

Given a rotation axis z represented by a unit vector k