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 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 a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category.

* Given a category C with finite coproducts, a cogroup object is an object G of C together with a " comultiplication " m: G → G G, a " coidentity " e: G → 0, and a " coinversion " inv: G → G, which satisfy the dual versions of the axioms for group objects.

Given objects and in an additive category, we can represent morphisms as-by-matrices

Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples into one category or the other.

Given a concrete category ( C, U ) and a cardinal number N, let U < sup > N </ sup > be the functor C → Set determined by U < sup > N </ sup >( c ) = ( U ( c ))< sup > N </ sup >.

Given the way black music has been named by ( usually ) outsiders ever since the blues, the reaction to the name by artists who ostensibly fit into the ' neo-soul ' category represents a wonderful example of black self-determination in an industry that is still defiantly wedded to narrow definitions and images of black folks ".

Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that predicts whether a new example falls into one category or the other.

Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category C < sup > op </ sup >.

Given a concrete category C, it is often the case that the opposite category C < sup > op </ sup > per se is abstract.

Given a category C and a morphism

The prime mechanism then appeared to be this: Given a space X carrying a vector bundle, that implied in the homotopy category a mapping from X to a classifying space BG, for the relevant linear group G. For the homotopy theory the relevant information is carried by compact subgroups such as the orthogonal groups and unitary groups of G. Once the cohomology H *( BG ) was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in H *( X ) in the same dimensions.

Recent findings suggest to differentiate between three types of hoarding, that is: pure hoarding, hoarding plus OCD ( i. e., comorbid OCD ), and OCD-based hoarding Given the aforementioned distinction, it was proposed to increase coverage of compulsive hoarding in the forthcoming Diagnostic and Statistical Manual of Mental Disorders, fifth edition ( DSM-V ), both by creating a distinct category for compulsive hoarding, provisionally named, Hoarding Disorder ( either in the main manual under " obsessive-compulsive spectrum disorders " or in the appendix ), and by including hoarding as a potential symptoms of OCD.

Given a partially ordered class P, we can form a category with P's elements as objects and a single arrow going from one object ( element ) to another if the first is less than or equal to the second.

Given and, a morphism in the coslice category is a map making the following diagram commute:

Given the inherent technical limitations of the visible and palpable signs from these sacroiliac joint mobility maneuvers another broad category of clinical signs have been described called provocative maneuvers.

Given a category C, an idempotent of C is an endomorphism

Given the smaller size of pickup trucks in the Tacoma's category, crash testing for these sized trucks lags with how well most full size trucks perform.

Given a continuous map there is a map defined by This makes into a functor from the category of topological spaces into itself.

Given the size and nature of the mortgage market, many observers classify mortgage lending as a separate category of personal borrowing, and consequently residential mortgages are excluded from some definitions of consumer credit-such as the one adopted by the Federal Reserve in the US.

Ironically, also nominated in the category of Contemporary Male Horse was Point Given who won under Baffert's training the Belmont Stakes, Preakness Stakes, Haskell Invitational Handicap, Travers Stakes and ultimately the Eclipse Award for Horse of the Year.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.

* Shay Given – former goalkeeper for both the Republic of Ireland and Newcastle United F. C ..

: Given any set A and any set B, if for every set C, C is a member of A if and only if C is a member of B, then A is equal to B.

: Given any set A and any set B, if A is a nonempty set ( that is, if there exists a member C of A ), then if A and B have precisely the same members, then they are equal.

: Given any set A and any set B, there is a set C such that, given any set D, D is a member of C if and only if D is equal to A or D is equal to B.

Given objects A < sub > 1 </ sub >,..., A < sub > n </ sub > in C, their biproduct is an object A < sub > 1 </ sub > ⊕ ··· ⊕ A < sub > n </ sub > together with morphisms

Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined to be the curvature of that circle or line.

Given two points P and Q on C, let s ( P, Q ) be the arc length of the portion of the curve between P and Q and let d ( P, Q ) denote the length of the line segment from P to Q.

Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition

Given the discriminator PARTY TYPE ( which could be individual, partnership, C Corporation, Sub Chapter S Association, Association, Governmental Unit, Quasi-governmental agency ) the two subtype entities are PERSON, which contains individual-specific information such as first and last name and date of birth, and ORGANIZATION, which would contain such attributes as the legal name, and organizational hierarchies such as cost centers.

Given: side a = 20, side c = 24, and angle C = 40 °

: Given two sets, A and T, of equal size, together with a weight function C: A × T → R. Find a bijection f: A → T such that the cost function: