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 the range of his office-holding both regionally and at court, he did not need to draw ruinously on his own resources to dispense patronage on a grand scale and he was active in the arbitration of local disputes ; even state matters were regularly referred for his personal adjudication.

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 that the oldest known waterfowl, Vegavis iaai, dates from the Late Cretaceous, galliform ancestors must also have roamed the Earth contemporaneously with animals such as Tyrannosaurus rex.

Given the dependence of Hittite chronology on Egyptian chronology, a lowering of Egyptian dates would result in a lowering of the end of the Hittite New Kingdom and a resulting reduction ( or complete removal ) of the Anatolian Dark Age.

Given the known dates in the life of Scylax, the date should be about 490 B. C.

Given dates are by official Gregorian calendar.

Given the fact that these gestation lengths are only estimates of an average, it is helpful to consider gestation time as a range of dates rather than a single " due date ".

Given the fact that these meeting dates are well-established, it is a normal practice for senior DoD personnel interested in the particular study topics for that year to come to the last day of the meeting and be briefed in person on the study findings to-date.

Given Larry ’ s well-established reputation and strengths as a DJ, the release of the album was followed by a deluge of bookings for an album world tour that continued for the rest 2005 ; two U. S. tours, two Australian tours, and countless European dates ensured that the Calyx sound spread far and wide.

Given these, a new term Anthropocene, is specifically proposed and used informally for the latest part of this epoch since approximately synchronous lithospheric evidence, or more recently atmospheric evidence, of human impacts have been found on the Earth and its ecosystems ; these impacts may be considered of global significance for future evolution of living species.

Given this and other evidence, many scholars agree that Peter was martyred in Rome under Nero, although some scholars argue that he may have been martyred in Palestine.

Given that assumption, the evidence for the one-seed theory is the fact that a woman can conceive without coming to orgasm ( though this was still a matter of dispute in the ancient world and the Middle Ages ).

Given the difficulty in treating scleroderma, treatments with a smaller evidence base are often tried to control the disease.

Given Jerome's conservative methods, and that manuscript evidence from outside Egypt at this early date is very rare ; these Vulgate readings have considerable critical interest.

Given a general statement such as all ravens are black, a form of the same statement that refers to a specific observable instance of the general class would typically be considered to constitute evidence for that general statement.

Given the lack of evidence of many Freudian hypotheses, some scientific researchers proposed the existence of unconscious mechanisms that are very different from the Freudian ones.

Given the biogeography and appearance of the subgenus Olor it seems likely that these are of a more recent origin, as evidence shows by their modern ranges ( which were mostly uninhabitable during the last ice age ) and great similarity between the taxa.

Given overwhelming evidence of Louis's collusion with the invaders, the verdict was a foregone conclusion – with 693 deputies voting guilty, none for acquittal, with 23 abstaining.

Given the many available models, it appears that the material removal mechanism in EDM is not yet well understood and that further investigation is necessary to clarify it, especially considering the lack of experimental scientific evidence to build and validate the current EDM models.

Given the lack of concrete historical knowledge about one of the most potent figures in British mythology, it is unlikely that any definitive conclusions about the claims for these places will ever be established ; nevertheless it is both interesting and important to try to evaluate the body of evidence which does exist and examine it critically.

Given that the police had no evidence against " Thomas ", either at Leatherslade Farm or connection with either of the two gangs, Butler was prepared to let him go.

Given the influence of the temperature of water, as well as the lack of complete adjustment for age, alcohol consumption and smoking, the study concludes that there is limited evidence, but hot mate drinking is " probably carcinogenic to humans ".

Given the amount of evidence suggesting a link between antipsychotic medications and metabolic syndrome, several agencies have recommended regular clinical monitoring of weight, symptoms of hyperglycemia, glucose, hypertension and signs and symptoms of cardiovascular disease in chronically medicated patients with schizophrenia.

Given that the conduct of conclaves is strictly confidential and that any cardinal revealing such details would face automatic excommunication, no documentary evidence has ever substantiated or disproved the widely claimed rumour.

Given the prevalence of-assa place-names and words scattered around all sides of the Aegean Sea, this possessive suffix was sometimes considered evidence of a shared non-Indo-European language or an Aegean Sprachbund preceding the arrivals of Luwians and Greeks.

Given the evidence of A. C. Elias about the acrimony of Swift's departure from the Temple household, evidence from Swift's Journal to Stella about how uninvolved in the Temple household Swift had been, and the number of repeated observations about himself by the Tale's author, it seems reasonable to propose that the digressions reflect a single type of man, if not a particular character.

Given the context in which it is used, as well as the evidence of Swift's writing, the possibility that " bloody " is also a minced oath ( or more precisely, a slang usage of an otherwise legitimate word masquerading as a minced oath ) cannot be lightly dismissed.

Given this dismal track record there is no evidence that had Taichang reign lasted any longer than it did, he could have turned around the fortunes of the beleaguered Ming Dynasty after the long steady decline of the latter years of Wanli's reign.

Given enough time, natural change in isolated language can obliterate any meaningful linguistic evidence of a known common genetic source for the languages.

Given Margaret Gibson's thoroughly documented background, the report of her dying confession attracted the attention of film historians, but aside from circumstantial evidence, no independent confirmation has emerged.

Given the unstable state of affairs in Massachusetts, Hillsborough instructed Governor Bernard to try to find evidence of treason in Boston.

Given the limits of the existing evidence, some of these questions may be unresolvable with currently-available techniques.