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 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 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 any expression involving complex numbers, bras, kets, inner products, outer products, and / or linear operators ( but not addition ), written in bra-ket notation, the parenthetical groupings do not matter ( i. e., the associative property holds ).

* Given any combination of complex numbers, bras, kets, inner products, outer products, and / or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

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 the poorly understood nature of the endocannabinoid system, it will take many more generations of progress to even scratch the surface of the extraordinarily complex mechanisms that control neurological development / degeneration.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Given the irregular shape of the nucleus, Halley's rotation is likely to be complex.

Given any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations ( to avoid ambiguity, we can specify the square root with complex argument less than π ).

Given the complex nature of human smuggling and trafficking operations, the difference between these two criminal operations is not always readily apparent.

Given that the site envisioned for this museum in Phillipsburg has been sold for development as a townhouse complex and college campus annex, it is unclear what role Phillipsburg will play in this museum.

* Given two complex vectors x and y, multiplication by U preserves their inner product ; that is,

* Given that Historic U. S. Route 66 runs through Elk City, a sprawling museum complex has developed, which includes the National Route 66 Museum, the Old Town Museum, the Transportation Museum, the Farm and Ranch Museum, and the Blacksmith Museum.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generated by the roots of p. If K is a subfield of the complex numbers, the existence is automatic.

Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is " above " the plane.

Given two normed vector spaces V and W ( over the same base field, either the real numbers R or the complex numbers C ), a linear map A: V → W is continuous if and only if there exists a real number c such that

Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M,

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

Given the often complex badge-engineering that BMC undertook, it is common amongst enthusiasts to use the ADO number when referring to vehicles as a single design ( for example, saying ' The ADO15 entered production in 1959 '- this encompasses the fact that when launched, the ADO15 was marketed as both the Morris Mini Minor and the Austin Seven ).

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 the complex macro-rheological behavior of blood, it is not surprising that a single equation fails to completely describe the effects of various rheological variables ( e. g., hematocrit, shear rate ).

Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

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

Without doubt, the single most significant paper concerning the distribution of prime numbers was Riemann's 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject.

Given basic sound blocks, that a machine digitized, one has a bunch of numbers which describe a wave and waves describe words.

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 that France maintained a strong standing army, and that Prussia and the other German states would need weeks to mobilise their conscript armies, the French held the initial advantage of troop numbers and experience.

Given two metric spaces ( X, d < sub > X </ sub >) and ( Y, d < sub > Y </ sub >), where d < sub > X </ sub > denotes the metric on the set X and d < sub > Y </ sub > is the metric on set Y ( for example, Y might be the set of real numbers R with the metric d < sub > Y </ sub >( x, y )

: Given: a function f: A R from some set A to the real numbers

Given two numbers in scientific notation,

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

Given an ordered sequence of real numbers: the first difference is defined as

Given these numbers, the VW 166 is the most mass-produced amphibious car in history.

Given that the Austrians, with their superior numbers and overwhelming artillery firepower, were now intent upon seizing the opportunity to launch a counterattack of their own, retreat was most difficult.

Given the large numbers involved and the nature of the requirements, they argued, it was unlikely to find any members of Al Qaeda.

Given von Spee ’ s superiority in speed, firepower, efficiency and numbers, why Cradock chose to engage puzzles historians.

Given a real dynamical system ( R, M, Φ ), I ( x ) is an open interval in the real numbers, that is.