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 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 a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given as a series of linked short stories, the book is also interspersed with brief commentaries on contemporary British politics.

" Given the fact that it's the third film in the series, you couldn't just end with them obtaining the object.

Given formal power series

Given a formal power series

Given a series with values in a normed abelian group G and a permutation σ of the natural numbers, one builds a new series, said to be a rearrangement of the original series.

Given his jazz and salsa conga playing experience and knowledge ( working as a sideman with such bands as salsa's Frankie Dante's Orquesta Flamboyan and jazz saxophonist Archie Shepp ), Dawson also created the long running " Salsa Meets Jazz " weekly concert series at the Village Gate jazz club where jazz musicians would sit in with an established salsa band, for example Dexter Gordon jamming with the Machito band.

" Given the chance to relive his life by switching bodies with " a talentless 20 year-old wannabe rock star with all the looks and charm Rocky used to have ," the series deals with the subsequent soul-searching and reflection over such a decision.

Given two complexes M < sub >*</ sub > and N < sub >*</ sub >, a chain map between the two is a series of homomorphisms from M < sub > i </ sub > to N < sub > i </ sub > such that the entire diagram involving the boundary maps of M and N commutes.

Given that only six vessels were constructed after the Coast Guard takeover, the " LV -" series numbers are most commonly used.

Given reinforcements and more equipment, Maude directed his force in a steady series of victories.

Given the innately rich and personal world of fantasy he has left himself locked into, Morgan goes off the deep end, enacting a series of bizarre stunts in a campaign to win Leonie back, including putting a skeleton in her bed and crashing her wedding dressed as a gorilla, for which scene Reisz borrows clips from from the original King Kong film to illustrate Morgan's fantasy world.

Given a ring R and an R-module M, a composition series for M is a series of submodules

Given a limited budget, the player must design a tank that can defeat a series of ever more challenging opponent tanks.

Given the publicity Trueman had received after his performances in the 1952 Test series, there was a lot of public interest in his football and Lincoln's attendances increased considerably when he was playing.

Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F ( S ) whose underlying set is N < sup > n </ sup > where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of N < sup > n </ sup >; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms.

Given their sales success, it was only natural that the Coupe de Ville and Sedan de Ville were moved to their own separate series in 1959, the Series 6300, being joined by a De Ville convertible in 1964.

:" Given that the structure of an econometric model consists of optimal decision rules of economic agents, and that optimal decision rules vary systematically with changes in the structure of series relevant to the decision maker, it follows that any change in policy will systematically alter the structure of econometric models.

Given a faster than light technology and weapons scaled similar to those of the Age of Sail, some of the battles described in the Honor Harrington series resemble real-life sea battles in our historical record.

This property is used to construct an arbitrary function from a series of Bessel functions by means of the Hankel transform.

The exponential function can be extended to a function which gives a complex number as e < sup > x </ sup > for any arbitrary complex number x ; simply use the infinite series with x complex.

More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

); arbitrary functions of expressions ; optimization ; derivatives, integrals, simplifications, sums, and products of expressions ; truncated series with expressions as coefficients, matrices of expressions, and so on.

DCTs are also closely related to Chebyshev polynomials, and fast DCT algorithms ( below ) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials, for example in Clenshaw – Curtis quadrature.

In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite ; this implies giving up the possibility to substitute arbitrary values for indeterminates.

The subject of Fourier series investigates the idea that an ' arbitrary ' periodic function is a sum of trigonometric functions with matching periods.

Since, the inequality could be satisfied for sufficiently small m. The sum of series first converges to the value of φ ( t ) with arbitrary high accuracy if m is small enough, and left part of the strong inequality is satisfied.

Although the difference between more than one sequel and a series is somewhat arbitrary, it is clear that some media franchises have enough sequels to become a series, whether originally planned as such or not.

Hence the definitions adopted in establishing rock nomenclature merely correspond to selected points ( more or less arbitrary ) in a continuously graduated series.

Scholars believe that Bach intended the works to be considered as a systematically conceived cycle, rather than an arbitrary series of pieces: Compared to Bach's other suite collections, the cello suites are the most consistent in order of their movements.

He became a pivotal figure in the Anahuac Disturbances, during which Mexico City's increasingly arbitrary and authoritarian rule led to a series of assemblies by Texans, with subsequent civil disturbances and outbreaks of violence precipitating the war.

Vieta accused Clavius, in a series of pamphlets ( 1600 ), of introducing corrections and intermediate days in an arbitrary manner, and misunderstanding the meaning of the works of his predecessor, particularly in the calculation of the lunar cycle.

In 1902, Hurwitz published a short proof using the Fourier series that applies to arbitrary rectifiable curves ( not assumed to be smooth ).

The Poincaré series generalizes the theta series to automorphic forms with respect to arbitrary Fuchsian groups.

and this series generalizes to arbitrary bases b by replacing 10 and 9 with b and b-1 respectively.

Among the most notable of these terms is the " Zord ", a term for the various building-sized mechanical or biomechanical robot machines used in the series, and the " Megazord ", a term for any arbitrary merger of Zords into a bigger, usually humanoid Zord.

In mathematics, the binomial series is the Taylor series at x = 0 of the function f given by f ( x ) = ( 1 + x )< sup > α </ sup >, where is an arbitrary complex number.

More than 99 % of them fall into one or the other of the six great groups which during the classic work of the Harvard College Observatory ... received designations ... by the rather arbitrary letters B A F G K M. That there should be so few types is noteworthy, but much more remarkable is the fact that they form a continuous series ... Russell is of the opinion that the principal differences in the stellar spectra arise in the main from variations in single physical variable in the stellar atmosphere ...".

Most 19th century histories of Gothic architectural style used a series of rather arbitrary categories based on the supposed evolution of the dominant patterns of window tracery.

A command line can have practically arbitrary length, and is a series of tokens.