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 India's historic position that the original intent of the line was to separate the two nations by the highest mountains in the world, in these locations India extended its forward posts northward to the ridges, regarding this move as compliant with the original border proposal, although the Simla Convention did not explicitly state this intention.

Given the great depth of Lake Tahoe, and the locations of the normal faults within the deepest portions of the lake, modeling suggests that earthquakes on these faults can trigger tsunamis.

Given the market chain's premium locations, the costs of rent for tenants are generally high, but other Chinese businesses, such as Sam Woo Restaurants, Chinese traditional medicine shops, and gift stores, have been known to follow 99 Ranch Market to its new locations, with 99 Ranch market becoming the anchor tenant for the smaller stores and restaurants within developing Asian suburban shopping areas.

Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts ; the result is a pa formula.

Given the Wyld's inconstant nature, however, concrete locations associated with Flux are few ; more commonly, the energies of Flux pass over an area, causing rapid change and general chaos, then move on.

Given so few bits to work with, one major design challenge of RDS-TMC was to find a way of using 16 bits ( about 65000 locations ) to describe an entire state or country.

The realization problem for Euclidean minimum spanning trees is stated as follows: Given a tree T = ( V, E ), find a location D ( u ) for each vertex u ∈ V so that T is a minimum spanning tree of D ( u ): u ∈ V, or determine that no such locations exist.

Given the level of surveillance, and consistent with the conspiratorial traditions of Irish political movements, the union proceeded clandestinely, forming small groups of trusted friends in order to keep informers at bay, meeting in isolated locations and in subway tunnels.

: Given two sets, P (" facilities ") and L (" locations "), of equal size, together with a weight function w: P × P → R and a distance function d: L × L → R. Find the bijection f: P → L (" assignment ") such that the cost function:

Given appropriate transmitting equipment, GPS is capable of pinpointing locations to the thickness of a pencil lead, enabling scientists to gather data to complement what is available from seismographs.

* 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 " mappings " of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take, based on a set of " rules ", each of the form:

Given that no majority supported either a leader or a split, a registration campaign, enabling membership for only 20 euros, and a closed primary was organized, which Ségolène Royal won.

Given three Americans who self-identify and are socially accepted as white, black and Hispanic, and given that they have precisely the same Afro-European mix of ancestries ( one " mulatto " grandparent ), there is no objective test that will identify their US endogamous group membership without an interview.

Given Alfonso's maternal membership in Spain's titled aristocracy ( his mother, María de la Piedad de Yturbe y Scholtz-Hersmendorff, was the Marquesa de Belvís de las Navas, and his paternal kinship to the royal courts of Europe, the hotel quickly proved to be popular with vacationing members of Europe's social elites for its casual but discreet luxury.

Given the legal focus of the ICJ's work, membership of these sections is predominantly drawn from the legal profession: lawyers, judges, legal academics and law students.

Given the new revenues which many tribes are realizing from gambling casinos and other economic development, or from settlement of 19th-century land claims, some have established more restrictive rules to limit membership.

Given a small ball or marbles, he would invent games and play by himself for hours.

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 the absolute magnitude, for objects within our galaxy you can also calculate the apparent magnitude from any distance ( in parsecs ):

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Love and Erlandson continued to pursue with the band, and released the single " Be A Man "— an outtake from the Celebrity Skin sessions — for the soundtrack of the Oliver Stone film Any Given Sunday ( 1999 ).

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given more time to prepare for trial, Kidd likely would have been able to find the deposition Palmer gave when he was captured in Rhode Island two years earlier.

Given Chrétien's known tendency to create new stories and characters, being the first to mention the hero Lancelot and his love affair with Queen Guinevere for example, the name might also be entirely invented.

Given that the dispute with Pizarro over Cuzco had kept intensifying, Almagro spent a great deal of time and money equipping a company of 500 men for a new exploration south of Peru.

Given a particular query, the DBMS ' query optimizer is responsible for devising the most efficient strategy for finding matching data.

Given the limited success of the Charity School, however, Wheelock intended his new College as one primarily for whites.

Given a particular query, the DBMS ' query optimizer is responsible for devising the most efficient strategy for finding matching data.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given the magnitude of the disaster, reconstruction and economic recovery will remain the primary focus of the Salvadoran Government for some time to come.

Given a left neutral element and for any given then A4 ’ says there exists an such that.

Given the limited budget for schools, it is claimed that a voucher system would weaken public schools while at the same time not necessarily providing enough money for people to attend private schools.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given the wide disparity of concepts and methods across domains, 4GL limitations lead to recognition of the need for the 5GL.

Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").