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 the character's unprecedented popularity and the fact that his name was partially derived from the Latin word for " lucky ", some rather notable individuals and organizations adopted Felix as a mascot.

Given two partially ordered sets A and B, the lexicographical order on the Cartesian product A × B is defined as

Given Andrew's hopes for Toowoomba's career, Andrew had attempted to partially lead Harry off track until he was sent back to Norway.

Given a probability space, a filtration is a weakly increasing collection of sigma-algebras on,, indexed by some totally ordered set T, and bounded above by.

Given the extremely disadvantaged position in which the English were, Vernon ordered a retreat but not before throwing some 300 bombs on the city.

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

Given the distance that these troops had to march in order to make contact with the enemy, it was clear to the two Corps commanders that they would be unable to attack at 4: 00 as ordered.

Given intelligence that the rebels had been stockpiling weapons at Concord, Massachusetts, he ordered a troop of British regulars to march there on the night of 18 April to confiscate them.

Given a non-empty totally ordered set R with the following four properties:

Given an orthonormal basis ordered such that, we see that

Given 25, 000 men, he was ordered to clear Confederate obstacles on the Mississippi River.

Given points a, b, c, and d on a line, if it is known that the points are ordered as ( a, b, c ) and ( b, c, d ), then it is also true that ( a, b, d ).

#: Given three ordered periods,,, the price index for periods and times the price index for periods and should be equivalent to the price index for periods and.

Given that the documentary had aroused much tension among the Party ’ s top leaders, the Ministry of Radio, Film and Television ordered CCTV to revise the original version of the film before it could be broadcast again.

Given an ordered choice of linearly independent eigenvectors, especially an eigenbasis, they can be indexed by eigenvalues, i. e. using a double index, with x < sub > i, j </ sub > being the j < sup > th </ sup > eigenvector for the i < sup > th </ sup > eigenvalue.

Given its inner-city roots, chicken and food is quite often ordered a la carte.

Given a specific task to solve, and a class of functions, learning means using a set of observations to find which solves the task in some optimal sense.

Given a class function G: V → V, there exists a unique transfinite sequence F: Ord → V ( where Ord is the class of all ordinals ) such that

Given two school classes, one with 20 students, and one with 30 students, the grades in each class on a test were:

Given a general statement such as all ravens are black, a form of the same statement that refers to a specific observable instance of the general class would typically be considered to constitute evidence for that general statement.

Given that Rothko had known in advance about the luxury decor of the restaurant and the social class of its future patrons, the exact motives for his abrupt repudiation remain mysterious.

Given a case for which the class is unknown, guess that it belongs to the same class as the majority in its immediate neighborhood.

Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k:

Given an oriented manifold M of dimension n with fundamental class, and a G-bundle with characteristic classes, one can pair a product of characteristic classes of total degree n with the fundamental class.

A more general class are flat G-bundles with for a manifold F. Given a representation, the flat-bundle with monodromy is given by, where acts on the universal cover by deck transformations and on F by means of the representation.

Given a real vector bundle E over M, its k-th Pontryagin class is defined as

Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra A < sup > op </ sup > ( the opposite ring with the same action by K since the image of K → A is in the center of A ).

The general scenario is the following: Given a class S of computable functions, is there a learner ( that is, recursive functional ) which for any input of the form ( f ( 0 ), f ( 1 ),..., f ( n )) outputs a hypothesis ( an index e with respect to a previously agreed on acceptable numbering of all computable functions ; the indexed function should be consistent with the given values of f ).

Given a perfect graph G, Lovász forms a graph G * by replacing each vertex v by a clique of t < sub > v </ sub > vertices, where t < sub > v </ sub > is the number of distinct maximum independent sets in G that contain v. It is possible to correspond each of the distinct maximum independent sets in G with one of the maximum independent sets in G *, in such a way that the chosen maximum independent sets in G * are all disjoint and each vertex of G * appears in a single chosen set ; that is, G * has a coloring in which each color class is a maximum independent set.

Given a pair of spaces ( X, A ) the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of ( X, A ) is defined as an automorphism of X that preserves A, i. e. f: X → X is invertible and f ( A )

Given that we are only interested in what happens on shell, we would often take the quotient by the ideal generated by the Euler-Lagrange equations, or in other words, consider the equivalence class of functionals / flows which agree on shell.

Given two groups of people, the interaction between them is such that a behavior X from one side elicits a behavior Y from the other side, The two behaviors complement one another, exemplified in the dominant-submissive behaviors of a class struggle.

Given the dominance of West Indian bowling at the time, and the fragility of the Australian batting line-up, Wessels ' performance during that series was world class.

Given special funding by the Provincial Government, the school hired George Curtis from Dalhousie's Faculty of Law to serve as their first Dean and within two months the Faculty was educating its first incoming class.

Given that disadvantaged students generally have fewer opportunities to learn academic content outside of school, wasted class time due to an unfocused lesson presents a particular problem.

Given that the North Shore portions of the riding were largely affluent and upper middle class in character and normally a Liberal bastion, Douglas ' strong showing is not so surprising given the working-class and labour background of much of even the better-off parts of the riding in Burnaby.

Given that Canada had a relatively poor track record at producing world class soccer talent, Montreal fans were likely put off by the prospect that the quality of the team's play would instantly diminish for the 1984 season.

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

:: Given a recursive presentation P

Given a finite presentation P =

Given the fact that the period P of an object in circular orbit around a spherical object obeys

Given the abundance of such optimization problems in everyday life, a positive answer to the " P vs. NP " question would likely have profound practical and philosophical consequences.

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 an eclipse, then there is likely to be another eclipse after every period P. This remains true for a limited time, because the relation is only approximate.

: Given any x and y, x = y if, given any predicate P, P ( x ) if and only if P ( y ).

Given to films dealing with science and technology by the Alfred P. Sloan Foundation each year at the Sundance Film Festival.

Given two permutations π and σ of m elements and the corresponding permutation matrices P < sub > π </ sub > and P < sub > σ </ sub >