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 a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by

* Given a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨ a, b, c, d | a < sup > n </ sup > ba < sup > n </ sup > = c < sup > n </ sup > dc < sup > n </ sup >: n ∈ A ⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble

Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

Given a grammar in GNF and a derivable string in the grammar with length n, any top-down parser will halt at depth n.

Given objects A < sub > 1 </ sub >,..., A < sub > n </ sub > in C, their biproduct is an object A < sub > 1 </ sub > ⊕ ··· ⊕ A < sub > n </ sub > together with morphisms

* Aggregate signature-a signature scheme that supports aggregation: Given n signatures on n messages from n users, it is possible to aggregate all these signatures into a single signature whose size is constant in the number of users.

Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).

Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).

Given some training data, a set of n points of the form

Given a codeword, there are roughly 2 < sup > n H ( p )</ sup > typical output sequences.

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 that the bulk of family chores fell to women, and that they were five times more likely than men to head a single-parent family, the shortcomings of Finland's child day-care system affected women more than it did men.

Given the preponderance of men among the colonists in the early years, generally European men married American Indian women.

Given that specialized phutball boards are hard to come by, the game is usually played on a 19 × 19 Go board, with a white stone representing the football and black stones representing the men.

Given that Constantine had already crushed his rival Maxentius in 312, the two men decided to divide the Roman world between them.

Given their sting, however, they must always be treated with caution, and the discovery of men o ' war washed up on a beach may lead to the closure of the whole beach.

Given the test scores of two random samples of men and women, does one group differ from the other?

Given that the Persian fleet appears to have been undermanned in the aftermath of Salamis, 200-300 ships would indeed give this number of naval personnel ( using Herodotus's standard complement of 200 men per ship ).

Given the profession, some masnade were less mercenaries than bandits and desperate men.

Given that the darker-skinned Calormenes differed in appearance from northern peoples ( like Telmarines and Archenlanders ), it is possible that they are descended from another group from the world of men ; however, this is not stated in the books.

Given the demands of the Siege of Calais, no further men could be summoned for the defence of Northern England.

Given its small staff of only 24 men, the Einsatzgrupppe Ägypten would have needed as much help as possible from the local Arabs and from the Afrika Korps to kill the roughly 50, 000 Jews of Egypt, to say nothing of the half million or so Jews of the Yishuv.

Given the operational name Storm-333 and involving a combination of Spetsnaz, GRU, KGB, and Soviet paratroopers were a total of 24 men from the « Гром » (" Thunder ") detachment of Alpha Group.

Given 25, 000 men, he was ordered to clear Confederate obstacles on the Mississippi River.

Given clear that the army under his command was all that Paraguay with maximum effort could have led a successful military campaign against successive Bolivian army, superior in men and materials, making back to the Rio Parapiti.

Given the buildup of tensions with Great Britain, the US Navy established a shipyard for warships and rapidly built eleven ships at the facility, employing 3, 000 men at the yard, many recruited from New York City.

Given that men who had sworn fealty to James were now casting about for a way of getting a new king, this insistence on fidelity must have struck a chord.

Given the bulk of the data, it is likely that, for most men, midlife is a time of achievement and satisfaction.

Given the morphological diversity of the earlobes found among men, women, and children, some earlobes are large, some earlobes are pendulous, and some earlobes are large and pendulous, but some are prominent because of the structure and form of the dense, interlacing connective tissue fibers that shape the earlobe anatomy independent of the tail of the helix ( cauda helicis ).

Given that the building of the extensive Necropolis that abuts it on the hillside above involved removing a huge amount of the original overlaying rock, it is also possible that this ' canyon ' extended even further before men began quarrying there.

Given this security, she can proceed to do what she really wants to do — fall in love with men who are weak and irresponsible ,” won him a following across Canada.

Given that Ferengi had no objection to doing business with women of other species, it can be reasonably assumed that their attitudes toward females had evolved over time.

" Given his fantasies involving the busty tobacconist, the sensual math teacher, the fat-bottomed peasant women on bicycles, Volpina the man-eater and Gradisca whom he tried to grope at the Cinema Fulgor, Titta complains that it can ’ t be helped.

Given that the number of women needing to travel has doubled since 1995, there is a very strong demand for these kinds of services.

*" Given the significant dangers that varieties of cultural essentialism pose to feminist agendas, I believe that the development of a feminist perspective that is committed to antiessentialism both about ' women ' and about ' cultures ' is an urgent and important task for a postcolonial feminist perspective.

Given the lyrics of the song Dickson plays, which describes life at sea without women, it is more plausible that he intended a joke about the virtually all-male environment of the Black Maria.

Given that breast cancer is a large killer of women and a very high percentage of young women use underarm deodorants, I think we should be carrying out properly funded, further investigations into parabens and where they are found in the body ," says Philip Harvey, an editor of the Journal of Applied Toxicology, which published the research.

Cafferty courted controversy on September 23, 2004 while discussing terrorist demands for the release of two female scientists from an Iraqi prison and remarking: " Given the way these mutants treat women in their societies, the women are probably better off in U. S. custody.

Given their regulations, colleges are enforcing statistical proportionality .... Common sense and poll data suggest that men are more interested in playing sports than women.

Given his passionate opposition to discrimination, he wanted to break into the man's world of river sport, by building a club for women.

Given that diabetic women show a significant variability in their response to this medical disorder, it is not surprising that the disease ’ s influence on arousal is also highly variable.

The poem sold extremely well and prompted a verse epistle battle from pretended " Sylvia " s (" Sylvia " having spurned the poet, he vows to be quit of love ) who would offer to defend women from Gould's cruelty and pretended " answers " from the author of Love Given O ' er ( even though few of the " Sylvia " poems were by women, and only one of the " answer " poems was by Gould ( a year after the publication of the Love Given O ' er )) ( Sloan ).