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 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.

The stained glass window depicts a black man, arms outstretched, reminiscent of the Crucifixion of Jesus, and is inscribed: " Given by The People of Wales ".

Given the width and definition of the window function w ( t ), we initially require the height of the window function to be scaled so that

After a transfer fee initially reported to be around £ 8 million was agreed between the clubs, Given signed for Manchester City on 1 February, with the transfer window set to close the following day.

Hughes was very active in the January 2009 transfer window, signing Wayne Bridge from Chelsea, Craig Bellamy from West Ham United, Shay Given from Newcastle United as well as Nigel de Jong from Hamburg.

* Given the weather, anyone appearing at the window and doing something horrifying enough to instantly kill someone would have had to trample the flowerbed, which is still intact, and come right up to the window ;

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.

During the transfer window of the same month, Given moved to Manchester City, elevating Harper's position within the club and so after nearly 16 years, Harper was finally first-choice goalkeeper at 33 years old.

: Given that there are almost eight times as many data points in this sample as in the previous analysis by Burbidge & Napier ( 2001 ), we must conclude that the previous detection of a periodic signal arose from the combination of noise and the effects of the window function.

Given the ability to geocode street addresses and other features, one can create a lot of interesting spatial analysis, location-based service, political campaigning apps and localised search services.

Given the similarity in the spatial relationship between the Diyar Planitia and Harran Sulci, it is likely that the formation of Sarandib Plantia and Samarkand Sulci are related.

Given the similarity in the spatial relationship between the Sarandib Planitia and Samarkand Sulci, it is likely that the formation of Diyar Planitia and Harran Sulci are related.

Given their typically short dispersal distance, the spatial arrangement of habitat is important to the conservation of the Karner blue butterfly.

:: See Committee Note to Interim Rule 8001 ( f ) (" Given the short time limit to file the petition with the circuit clerk, subdivision ( f )( 1 ) provides that entry of a certification on the docket does not occur until an effective appeal is taken under Rule 8003 ( a ) or ( b ).

Given Armstrong's concerns, the administration employed general counsel J. Patrick Abell to file a friendly test case to determine the constitutionality of the incentive package.

Given this " semantic view " of the user's file system, Magellan not only exposed " hidden meaning " from disparate data, but also facilitated the actual movement of files and directories into a better physical organization.

Given his years of promoting baseball in Canada while a member of the Blue Jays, Whitt remained active as an ambassador of Canadian baseball Indeed, Whitt's coaching career first came to prominence in his capacity as manager of the Canadian national baseball team.

His performances on loan impressed Kenny Dalglish, who had recently been appointed manager of Newcastle United, and prompted him to sign Given again.

Given was one of the first signings of then-newly appointed manager Dalglish.

Newcastle manager Joe Kinnear had already stated that the club was " not interested at all in selling him ", and described the amount on offer for Given as " insulting ".

While team captain Kolo Touré was on international duty, Given was given the captain's armband by manager Roberto Mancini, who described the Donegal-born player as " one of the five best goalkeepers in the world ".

Given the contribution margin, a manager can easily compute breakeven and target income sales, and make better decisions about whether to add or subtract a product line, about how to price a product or service, and about how to structure sales commissions or bonuses.

However he lost his place again when Given was fit, though another injury to Given later in the season allowed Harper to gain more first team appearances under the new boss, and the manager who first brought him to Newcastle, Kevin Keegan.

Given that, Yankee manager Billy Martin decided to sit out Reggie Jackson on the grounds that he was 1-for-15 so far in the series and, in his own words, " can't hit Splittorff.

The signing of Republic of Ireland international Shay Given in January 2009 meant that Schmeichel fell even further down the pecking order at City, and in August 2009, he was allowed to link up with former manager Sven-Göran Eriksson at Notts County.