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

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 two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.

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 the opportunity, adults eat hatchlings, which may be protected by separating the two groups with a net, breeding box or separate tank.

Given that Bozizé accuses Sudan of supporting the UFDR rebels who are actively fighting the Central African Government, the relation between the two countries has remained good.

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 two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given that the vast majority of all emeralds are treated as described above, and the fact that two stones that appear visually similar may actually be quite far apart in treatment level and therefore in value, a consumer considering a purchase of an expensive emerald is well advised to insist upon a treatment report from a reputable gemological laboratory.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.

: Given two points, determine the azimuth and length of the line ( straight line, arc or geodesic ) that connects them.

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

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

Given this understanding, the Board withdrew the allegation of unsportsmanlike behaviour two days before the fourth Test, thus saving the tour.

Given the above commonalities there appear to be only two string theories: the heterotic string theory ( which is also the type I string theory ) and the type II theory.

Given the notion of a lexeme, it is possible to distinguish two kinds of morphological rules.

Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.

Given barely two months to assemble a large sea-going invasion fleet, the Kriegsmarine opted to convert inland river barges into makeshift landing craft.

In the account books of Johanna, Duchess of Brabant and Wenceslaus I, Duke of Luxemburg, an entry dated May 14, 1379 reads: " Given to Monsieur and Madame four peters, two forms, value eight and a half moutons, wherewith to buy a pack of cards ".

# Given any two distinct points, there is exactly one line incident with both of them.

# Given any two distinct lines, there is exactly one point incident with both of them.

Given Poseidon's connection with horses as well as the sea, and the landlocked situation of the likely Indo-European homeland, Nobuo Komita has proposed that Poseidon was originally an aristocratic Indo-European horse-god who was then assimilated to Near Eastern aquatic deities when the basis of the Greek livelihood shifted from the land to the sea, or a god of fresh waters who was assigned a secondary role as god of the sea, where he overwhelmed the original Aegean sea deities such as Proteus and Nereus.

" Given the ubiquity of keyboards and the growing expectation that secondary school pupils and university students will type their essays and coursework, I think it ( touch typing ) is one the most useful skills a child can learn-at any age from seven upwards.

Given this function, a transformer sold for isolation is often built with special insulation between primary and secondary, and is tested, specified, and marked to withstand a high voltage between windings, typically in the 1000 to 5000 volt range.

Given a 3D audio " picture " of the scene, a specialised algorithm generates secondary sound waves necessary to recreate it in the particular room and then instructs a large number of speakers ( 300-400 ) to generate the " audio hologram ".

Given the propensity for cardiovascular and cerebrovascular complications, minimizing vascular risk factors and implementing therapy for primary or secondary prevention of stroke and myocardial infarction seems prudent.

Given the combination of employees at the plant and secondary jobs created by Indian Point ’ s economic activity, the plant is responsible for 2, 500 jobs in Westchester, Orange, Rockland, Putnam and Dutchess counties .”

Given that the Mother of God outranks St. Peter of Alcantara in the hierarchy of the saints, with the proclamation of the Virgin of Aparecida as Queen and Patroness of Brazil, the blessed Virgin Mary became the Nation's principal heavenly patron, and the Franciscan saint became the secondary patron.

Given the terrain of New York City after the initial invasion, the Resistance usually stations its soldiers in covered positions like craters and buildings.

Given a fertile mind and boundless energy, he pursued other fields of interest: growing vegetables in Puerto Rico ; owning a beautician school, restaurants, lounges, a marina, marine dredging, data processing services, multi-level marketing of vitamins and cosmetics, cement block manufacturing, country-western music publishing and recording studios, telecommunications companies, a high-end boutique shop, general insurance agency, internet connection and data center, a walk-in medical clinic, wholesale gasoline and diesel fuel distribution operation, a travel agency, home security monitoring, mail order medical and diabetic supply company, a Big & Tall men's clothing store, an AM radio station, a mobile medical screening company, a book publishing company, a warehouse fulfillment and distribution company, an advertising media consulting firm, television stations, mobile home parks, a 225 room hotel in Nassau, Bahamas, a 500, 000 square foot commercial warehouse complex in Las Vegas, and a five-star 23 story resort hotel and residences on Ft. Lauderdale Beach.

Given Nexstar's termination of the Fox affiliations of several of its stations during the past two years, it is ironic that KLRT is one of two Fox affiliates that are being picked up by either Nexstar or Mission ; the other is WJKT in Jackson, Tennessee ( although that station is controlled by and based out of Memphis, Tennessee's ABC affiliate WPTY-TV, which is also being sold to Nexstar ).

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

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

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 the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.

Given sufficient travel time and engineering work, both unmanned and generational interstellar travel seem possible, though present considerable technological and economic challenges are unlikely to be met in the near future, particularly for manned probes.

Given the repeated failures and frustrations of workers ' revolutions and movements, Marx also sought to understand capitalism, and spent a great deal of time in the reading room of the British Museum studying and reflecting on the works of political economists and on economic data.

Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.

Given fixed resources, it is often better to spend more time collecting additional training data and more informative features than it is to spend extra time tuning the learning algorithms.

Given enough time and repeated exposure to the antibiotic, a population of antibiotic-resistant bacteria will emerge.

Given time t, the source produces ωt oscillations.

Given prevailing views on homosexuality at the time, circulation of the diaries helped undermine support for clemency for Casement.

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 sufficient time, oxygen, and water, any iron mass will eventually convert entirely to rust and disintegrate.

Given the medieval medicine's main theory of humorism, spices and herbs were indispensable to balance " humors " in food, a daily basis for good health at a time of recurrent pandemics.

Given the number of homes, it seems likely that no more than fifty people lived in Skara Brae at any given time.

" Given enough time, these formations can meet and fuse to create columns of calcium carbonate.