Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.

# Given an array of values to be sorted, set up an auxiliary array of initially empty " pigeonholes ," one pigeonhole for each key through the range of the original array.

Given their long range, high survivability and ability to carry many medium-and long-range nuclear missiles, submarines were credible and effective means for full-scale retaliation even after a massive first strike.

Given a stream of symbols and their probabilities, a range coder produces a space efficient stream of bits to represent these symbols and, given the stream and the probabilities, a range decoder reverses the process.

Given the small temperature range kinetic studies occur in, it is reasonable to approximate the activation energy as being independent of the temperature.

* Toys: Given their ability to counteract the force of gravity at close range, magnets are often employed in children's toys, such as the Magnet Space Wheel and Levitron, to amusing effect.

Given the rapidity with which its last population was eliminated, it is likely that aboriginal hunting caused its extinction over the rest of its original range ( aboriginal peoples apparently never inhabited the Commander Islands ).

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.

Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range:

Given their limited range, they can only meet the needs of 90 percent of the population.

Given this find, the UN Inspectors have requested access to the Al-Aziziyah weapons range to verify that all 155 R-400 bombs can be accounted for and proven destroyed.

Given a hill angle and launch angle as before, it can be shown that the range along the hill forms a ratio with the original range along the imaginary horizontal, such that:

Given the varied climatic and ecological areas regions, the park supports a range of animal species including over 360 bird species.

Given the range of diversity in soil type, climate and occupations, these cuisines vary significantly from each other and use locally available spices, herbs, vegetables, and fruits.

Given that lupin seeds have the full range of essential amino acids and that they, contrary to soy, can be grown in more temperate to cool climates, lupins are becoming increasingly recognized as a cash crop alternative to soy.

Given any morphism between objects X and Y, if there is an inclusion map into the domain, then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain R → Y known as the range of f.

Given the origins of the cardinal tetra, namely blackwater rivers whose chemistry is characterised by an acidic pH, low mineral content and the presence of humic acids, the species is adaptable to a wide range of conditions in captivity, though deviation from the soft, acidic water chemistry of their native range will impact severely upon breeding and fecundity.

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 Brittany's rich musical heritage, contemporary performers produce a range of music sung in Gallo ( see Music of Brittany ).

Given its wide range of shop mix and entertainment facilities, including theaters / cinemas, the centre became a very important destination for visitors in Denizli and neighbouring cities.

Given the remarkable range of Aristotelian philosophy, it was possible to discuss all kinds of issues in medieval and Renaissance philosophy.

Given a surface through which a metal wire passes, electrons move in both directions across the surface at an equal rate.

Given this basic definition, further criteria are needed to establish a dividing line between postlexical clitics and morphological affixes, since both are characterized by a lack of prosodic autonomy.

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

Given these connotations, dandyism can be seen as a political protestation against the rise of levelling egalitarian principles, often including nostalgic adherence to feudal or pre-industrial values, such as the ideals of " the perfect gentleman " or " the autonomous aristocrat ", though paradoxically, the dandy required an audience, as Susann Schmid observed in examining the " successfully marketed lives " of Oscar Wilde and Lord Byron, who exemplify the dandy's roles in the public sphere, both as writers and as personae providing sources of gossip and scandal.

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

Given the post-war shortages, both parties agreed not to publish it.

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

Given the dual status of science as objective knowledge and as a human construct, good historiography of science draws on the historical methods of both intellectual history and social history.

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 a binary operation, an idempotent element ( or simply an idempotent ) for the operation is a value for which the operation, when given that value for both of its operands, gives the value as the result.

Given the realities of the Cold War, German reunification and recovery of lost territories in the east were not realistic goals as both of Stalin's notes specified the retention of the existing " Potsdam "- decreed boundaries of Germany.

Given that both A and not-A are seen to be “ true ,” Kant concludes that it ’ s not that “ God doesn ’ t exist ” but that there is something wrong with how we are asking questions about God and how we have been using our rational faculties to talk about universals ever since Plato got us started on this track!

# 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 any of the above definitions of " myth ", the myths of many religions, both ancient and modern, share common elements.

Given the versatility of both vegetables, they were usually boiled or steamed, and eaten by itself or as ingredients in soups and strews.

Given the current value of the port of Toledo to Ohio, it can be reasonably suggested that both sides benefitted from the conflict.

Given that the right to be politically active has granted them the opportunity to expand their knowledge and influence in current affairs, they are now considered one of the main components in the country's decision-making in both politics and economy.

* Shay Given – former goalkeeper for both the Republic of Ireland and Newcastle United F. C ..

" Given their penchant for blacksmithing, many scholars believe the legend of the Cyclopes ' single eye arose from an actual practice of blacksmiths wearing an eyepatch over one eye to prevent flying sparks from blinding them in both eyes.

Given the type of warfare that had been carried out in the conflict — wherein highly mobile groups of soldiers and rebels continued to move to and from on the desert terrain, retreating at a time and then suddenly attacking to regain control of small towns and villages in the Eastern rebel held parts of Libya — had led to the technical becoming a vehicle of choice for both sides.

Pope Saint Adeodatus I or Deodatus I ( which is Given by God in Latin, also called Deusdedit, which is God Has Given ; both are now considered variants of the same name ) ( died 8 November 618 ) was Pope from 13 November 615 to his death.

Given the tense nature of racial tension in America during the time of the film's production, Poitier felt he was " under close observation by both " Tracy and Hepburn during their first dinner meetings prior to production.