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

: Given any family of nonempty sets, their Cartesian product is a nonempty set.

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

Given the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.

Given the large number of U. S. military personnel and their dependents residing in Europe, it was expected that over 7 % of donors would be deferred due to the policy.

Given their great antiquity, where and when they first existed is not known and perhaps unknowable.

Given that the ColecoVision could produce near arcade-quality ports, industry magazines like Electronic Games were unanimous in their enthusiasm over the console.

Given their speed and maneuverability, clippers frequently mounted cannon or carronades and were used for piracy, privateering, smuggling, or interdiction service.

Given their strategic location in the North Atlantic, the Faroes could have proved useful to Germany in the Battle of the Atlantic, possibly as a submarine base.

Given limited defense budgets, air forces tended to be conservative in their aircraft purchases, and biplanes remained popular with pilots because of their agility, and remained in service long after they had ceased to be competitive.

Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).

Given that the Qur ' an is vague regarding the punishment of homosexual sodomy, Islamic jurists turned to the collections of the hadith and seerah ( accounts of Muhammad's life ) to support their argument for Hudud punishment ; these are perfectly clear but particularly harsh.

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.

" Given, a certain population, with various needs and powers of production, in possession of certain lands and other sources of material: required, the mode of employing their labour which will maximize the utility of their produce.

Given that the authors were the primary developers of the Unix operating system, it is natural that QED had a strong influence on the classic UNIX text editors ed, sed and their descendants such as ex and sam.

Given their wide historical distribution, red wolves probably utilized a large suite of habitat types at one time.

Given that by this time it was common for castles to be built in stone, and that many barons had expanded or refortified their castles, this was not an easy task.

Given that the decree was issued at Memphis, the ancient capital of Egypt, rather than Alexandria, the centre of government of the ruling Ptolemies, it is evident that the young king was anxious to gain their active support.

Given its exposed spot on the top of Stow Hill, the town is often referred to with the couplet " Stow on the Wold, where the winds blow cold and the cooks can't roast their dinners ", but there is no source for this.

Given their hunting instincts, they are prone to chasing moving objects.

Given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city.

: Given any positive number ε, there is a sequence

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 recursively enumerable set A of positive integers that has insoluble membership problem, ⟨ a, b, c, d | a < sup > n </ sup > ba < sup > n </ sup > = c < sup > n </ sup > dc < sup > n </ sup >: n ∈ A ⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble

Given a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

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 an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.

Given the ( period, cash flow ) pairs (, ) where is a positive integer, the total number of periods, and the net present value, the internal rate of return is given by in:

Given that the absolute vorticity is positive ( negative ) in the Northern ( Southern ) hemisphere, the threshold value should be taken as positive ( negative ) north ( south ) of the equator.

Given any neighborhood of, we can choose a small positive and a small of either sign to get values both greater and less than.

Given names are chosen based on a range of factors, including possession of pleasing sound and tonal qualities, as well as bearing positive associations or a beautiful shape.

" Dietrich Bonhoeffer of the German Confessing Church framed the same characterization in less positive terms when he called Pietism the last attempt to save Christianity as a religion: Given that for him religion was a negative term, more or less an opposite to revelation, this constitutes a rather scathing judgment.

Given a sequence of positive integers, the Gödel encoding of the sequence is the product of the first n primes raised to their corresponding values in the sequence:

Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesized logic program which entails all the positive and none of the negative examples.

* Given a positive real number ε, an ε-isometry or almost isometry ( also called a Hausdorff approximation ) is a map between metric spaces such that

Given the prevalence of disapproval as a tool of government, including the criminal law and diplomatic relations, some fail to see voting as a positive and voluntary choice of a desirable outcome.

* Given a positive integer n, the set of all positive divisors of n forms a distributive lattice, again with the greatest common divisor as meet and the least common multiple as join.

Given an appropriately designed spam filtering algorithm, outbound spam filtering can be implemented with a near zero false positive rate, which keeps customer related issues with blocked legitimate email down to a minimum.

Given a positive integer n, it is not at all a routine matter to determine how many isomorphism types of groups of order n there are.

Given that bankruptcies and real estate prices did not fare as negatively in Central Canada as in the rest of Canada and the United States during the NEP, it is possible that the NEP had a positive effect in Central Canada.

Given the integral representation of a principal branch of γ, the following equation holds for all positive real s, x:

Given two nondecreasing positive functions a and b one can say that

Given that a positive power lens will magnify an object and a negative power lens will minify it, it is often possible to tell whether a lens is positive or negative by looking through it.