: 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 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 Switzerland's particularly strong opinions on who is " Swiss " and who is " other ", it is easy to see that groups such as Stoffunita make use of the fact that hip hop " is still considered a voice for the oppressed " as hip hop scholar and author Jeff Chang notes.

Given inconsistencies and the lack of reliable data and completely contradicting and unpredictable behavior, the true nature of ball lightning is still unknown and was often regarded as a fantasy or a hoax.

Given the well-publicized hoopla, some sources still consider this as a nomination for Davis ; however, the Academy does not officially record this as a nomination.

Given that the U. S. is still home to the world's largest economy, there's no reason it shouldn't have the most vibrant equity markets — unless regulation is holding back the creation of new public companies.

Given the circumstances, most commentators still rate Kramnik the stronger player in the match.

Given the lax copyright laws of the time, stage plays based on Uncle Tom's Cabin —" Tom shows "— began to appear while the novel was still being serialized.

The American Library Association issued a recommendation to libraries still holding Alms for Jihad: " Given the intense interest in the book, and the desire of readers to learn about the controversy first hand, we recommend that U. S. libraries keep the book available for their users.

Given that over 90 per cent of the area of kauri forest standing before 1000AD was destroyed by about 1900, it is not surprising that recent records are of smaller, but still very large trees.

Given inconsistencies and the lack of reliable data, the true nature of ball lightning is still unknown.

Given the way black music has been named by ( usually ) outsiders ever since the blues, the reaction to the name by artists who ostensibly fit into the ' neo-soul ' category represents a wonderful example of black self-determination in an industry that is still defiantly wedded to narrow definitions and images of black folks ".

Given Bluestein's algorithm, such a transform can be used, for example, to obtain a more finely spaced interpolation of some portion of the spectrum ( although the frequency resolution is still limited by the total sampling time ), enhance arbitrary poles in transfer-function analyses, etcetera.

Given a Vulcan's life expectancy of approximately 250 years, it is possible that she is still alive at the time of Star Trek: The Original Series and subsequent movies.

Given the technical limitations of daguerreotype photography which required long exposure times, this was one situation where the portrait subject remained quite still.

Weekly Standard Editor William Kristol said at the time, " Given that Reagan obviously couldn't be here, I think it was important to have Mrs. Thatcher because she and Reagan are really the great heroes of conservatism in the last few decades and still today.

Given similar doses of the same neuroleptic, differences among individuals still exist in the likelihood of developing tardive dyskinesia.

Given that G. atlantica is a small and G. stehlini a giant species, and that it is also not known when exactly the later evolved its large size, the ancestral lizards which colonized the central and western islands may have been still small or already large.

Given the option, most swimmers choose to use a dolphin kicking action, but there still is a small minority of swimmers who prefer the breaststroke kick, for recreational swimming and even for competition.

Given its harbors and fortifications, Saint Thomas still retained a strategic importance, and thus, in the 1860s, during the American Civil War and its aftermath, the United States government considered buying the island and its neighbors from Denmark for $ 7. 5 million, but at the time failed to find domestic legislative support for the bid.

Given the still developing nature of the East-Central European new market economies, the already inundated Commission anticipated a further significant increase in its workload.

Given its historical development, Lviv Oblast is one of the least Russified and Sovietized parts of Ukraine, with much of its Polish and Habsburg heritage still visible today.

Given Amtrak's own projected 41 hour travel time, the average speed is still in excess of 55 mph ( including stops ).

Given that the installed Mac base at the time still contained large numbers of 68000 based machines like the Mac Plus, these requirements restricted the number of machines it could run on.

Given a Hi of 0. 40, if the H_m is assumed to be 0. 25. then from the equation above RCM count is still high and ANH is not necessary, if BLs does not exceed 2303mL, since the hemotocrit will not fall below H_m.