: Given any family of nonempty sets, their Cartesian product is a nonempty set.

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 a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

" Given that Thomas Aquinas died six years before Albertus Magnus ' death, this legend as stated is unlikely.

Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

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 its " blue-water " bias, the navy is even less inclined to become involved in counterdrug operations than the army or air force.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

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.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is

Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

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 then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.

Given some encrypted data (" ciphertext "), the goal of the cryptanalyst is to gain as much information as possible about the original, unencrypted data (" plaintext ").

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

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 such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.

Given its basal position in Tringa, it is fairly unsurprising that suspected cases of hybridisation between this species and the Common Sandpiper ( A. hypoleucos ) of the sister genus Actitis have been reported.

Given that syringes are not prohibited in the jurisdiction and their confiscation can only occur as part of an otherwise authorized arrest, almost 40 % of those who reported syringe confiscation were not arrested.

Given the University of Georgia School of Law's low in-state tuition of $ 14, 448, the New York Times recently completed a survey comparing starting salaries and degree costs of law schools and found " Georgia Law graduates earning some of the highest salaries in the country while their educational costs were reported among the very lowest, speaking to the quality of the education as well as the excellent return on investment provided at Georgia Law.

However, on 30 January, it was reported that Newcastle had given permission for Given to enter talks with his suitors.

After a transfer fee initially reported to be around £ 8 million was agreed between the clubs, Given signed for Manchester City on 1 February, with the transfer window set to close the following day.

Given that he had a number of strongholds in north-west Wales, it is not possible to know how much time he spent in the village, although it is reported that he preferred his lodge at Trefriw to his Palace at Aber.

Given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm.

Given the magnitude of the loans to Anglo Irish Bank the company was reported to be considering a € 700 million financial rescue of the QUINN group.

" Given the complete lack of citations to this research in GeoRef, JSTOR, Web of Knowledge, and other scientific bibliographic databases, neither the data collected from nor the interpretations and conclusions made as a result of this expedition very likely have been publicly reported in a scientific venue where their credibility can be openly evaluated.

* U. S. surveillance: the Los Angeles Times reported on 27 October 2004: " Given the size of the missing cache, it would have been difficult to relocate undetected before the invasion, when U. S. spy satellites were monitoring activity.

Given the fact that they are little-eroded, and that steam was reported in 1974, and an unconfirmed report of an eruption detected by satellite in 1985, the Hudson Mountains may be active.

Given Edith's reported unedifying performance as a witness, Weis does just about concede that her conviction was inevitable, as does Sir Henry Curtis Bennett KC, although he claims he could have saved her had she not rejected his advice not to take the witness stand.

Given England's failure to qualify for the World Cup that year, Ramsey's dismissal may have been justified, but the newspapers reported that " the whole episode was handled with brutal insensitivity.

Given that recorded vocalizations are an effective means to attract owls, and given the similarity of a distant accordion's tune to the call of the Laughing Owl as reported, it is apparent that the method might have worked.

Given that there was no evidence that either the U. S. or the U. S. S. R. had anything remotely like the UFOs reported, Sign personnel gradually began considering extraterrestrial origins for the objects.

On 19 February it was reported that Ireland had intimated to Manchester City team mate Shay Given that he would like to represent Ireland at the World Cup in South Africa in 2010 and that a return to the international fold was imminent, but when the Irish squad to face Bulgaria on 28 March and Italy on 1 April was announced, his name was once again omitted.

Given the conceptual context within which ancient thought thrived, how could anyone have questioned this principle??

Idealists are skeptics about the physical world, maintaining either: 1 ) that nothing exists outside the mind, or 2 ) that we would have no access to a mind-independent reality even if it may exist ; the latter case often takes the form of a denial of the idea that we can have unconceptualised experiences ( see Myth of the Given ).

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 first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

Given these changes, the classical terms " system ", " expansion " and " peripheral " no longer have the same connotations.

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 the variety of single stranded viruses that have been described this scheme-if it is accepted by the ICTV-will need to be extended.

Given his poor health history, during 1918 Munch felt himself lucky to have survived a bout of the Spanish Flu, the worldwide pandemic of that year.

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

Given enough of it, the inhabitants of planet Earth would have nothing to fight about.

Given that Gable and Cukor had worked together before, in Manhattan Melodrama and Gable had no objection to working with him then, and given Selznick's desperation to get Gable for Rhett Butler, if Gable had any objections to Cukor, certainly they would have been expressed before he signed his contract for the film.

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 the Greek fleet's unpreparedness resulting from the premature outbreak of the war, such an early Ottoman attack might well have been able to achieve a crucial victory.

Given that white is the most effective color in terms of facilitating the reflection of ground light, the hypothesis that straight hair could have been adaptively favorable, cannot be fully discounted in this regard.

We have seen a few cavities of the appropriate size and shape for ivory-bills, but these can be old, or exceptionally large Pileated Woodpecker cavities, or mammal-enlarged Pileated Woodpecker cavities .… Given the results, it is unlikely a population of any meaningful size of Ivory-billed Woodpeckers exists in south Florida.

Given earlier debates by Christian authors about the existence of Jesus, e. g. in Justin Martyr's 2nd century Dialogue with Trypho, it would have been expected that the passage from Josephus would have been used as a component of the arguments.

Given MIT's prominence, allegations of research misconduct or improprieties have received substantial press coverage.

Given the same set of verifiable facts, some societies or individuals will have a fundamental disagreement about what one ought to do based on societal or individual norms, and one cannot adjudicate these using some independent standard of evaluation.

Given that different groups in society have different beliefs, priorities, and interests, to which group would the media tailor its bias?

Given that in biology many phyla, families and genera have some species that live in the sea and others that live on land, marine biology classifies species based on the environment rather than on taxonomy.

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.