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 the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

Given that the leadership of the EU is subject to a particularly low voter turnout, it may be termed an oligarchy of the European Council members.

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.

BONUS: Given the title of an Irving Stone biography, identify the subject.

The wavelets forming a continuous wavelet transform ( CWT ) are subject to the uncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event.

Given Binet's stance that intelligence testing was subject to variability and was not generalizable, it is important to look at the metamorphosis that mental testing took on as it made its way to the U. S.

Given the variety of approaches, Economic Geography has taken to many different subject matters, including: the location of industries, economies of agglomeration ( also known as " linkages "), transportation, international trade, economic development, real estate, gentrification, ethnic economies, gendered economies, core-periphery theory, the economics of urban form, the relationship between the environment and the economy ( tying into a long history of geographers studying culture-environment interaction ), and globalization.

Given testable information, the maximum entropy procedure consists of seeking the probability distribution which maximizes information entropy, subject to the constraints of the information.

Given the technical limitations of daguerreotype photography which required long exposure times, this was one situation where the portrait subject remained quite still.

" Given the subject matter of his poetry he was likely a " man in holy orders ," and the deep Christian knowledge conveyed through his verse implies that he was well learned in ecclesiastical and hagiographical literature, as well as the dogma and doctrine of the Catholic Church.

Given that investments are subject to revisions of future expectations the Nerbrand Z utilises uncertainty of consensus estimates to assess how much earnings forecasts can be revised in standard deviation terms before P / E rations return to normalised levels.

Given the subject of this film, pictures are so important because they transcend language.

Given that Montagu-Stuart-Wortley's orders prior to the attack had been " to occupy the ground that is won by the artillery " his dismissal remains a subject of controversy.

Given the nature of Usenet forums, parodic translation is prevalent in flame wars, where remarks such as " Translation: ' I do not have a clue and am throwing mud '" are used to imply — on very little ground — that another poster is not making any appreciable contribution to the subject.

Given that the subject was founded by structural functionalists ; concerned with organic cohesion and social solidarity, this view was somewhat groundless ( though it was Parsons who had introduced Durkheim to American audiences, and his interpretation has been criticized for a latent conservatism ).

Given the rules from NEC, it was a pretty sensitive subject, so direct references had to be changed ", explained Vic Ireland, who worked closely with the game's localization.

Given that the subject is one that has only emerged very recently, bionanotechnology and nanobiotechnology serve as blanket terms for various related technologies.

Given the right combination of hypnotist and subject, hypnosis can be a lethal weapon.

Given the details with which Keyes writes about her subject matter, it is easy to forget her books are novels.