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

Given the large number of U. S. military personnel and their dependents residing in Europe, it was expected that over 7 % of donors would be deferred due to the policy.

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given is each element's name, atomic number, year of first report, name of the discoverer, and some notes related to the discovery.

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

Given the increasing number of scientists who study the nervous system, several prominent neuroscience organizations have been formed to provide a forum to all neuroscientists and educators.

: Given any positive number ε, there is a sequence

Given the number of homes, it seems likely that no more than fifty people lived in Skara Brae at any given time.

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given that detecting a price cut becomes harder as the number of firms increases, the bigger are the gains from price cutting.

Given the ( period, cash flow ) pairs (, ) where is the total number of periods, the net present value is given by:

* 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 the ( period, cash flow ) pairs (, ) where is a positive integer, the total number of periods, and the net present value, the internal rate of return is given by in:

( Please note that Japan has a guideline of a limit of 1 % of GDP on defense spending, however, Japan defines a number of activities as non-defense spending ) Given these circumstances, some have viewed Article 9 as increasingly irrelevant.

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree.

Given that the Masoretes would not alter the sacred consonantal text, the Kethiv-Qere notes were a way of " correcting " or commenting on the text for any number of reasons ( grammatical, theological, aesthetic, etc.

Given the number of unpleasant, harmful or even fatal consequences of coming out in world societies, it is questionable to call being closeted a choice.

Given any input, GEN generates an infinite number of candidates, or possible realizations of that input.

Given an input consisting of any number of bytes, it produces as output a single byte that is strongly dependent on every byte of the input.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

Given the fact that the Lakeport area has an economy that is based somewhat on agricultural activities occurring in the Big Valley and Scotts Valley area which involve the growing and harvesting of food and nut crops as well as the increasing wine industry, there are a significant number of transient and seasonal farm workers.

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 scale of most commercial operations and the lack of mechanized record-keeping and recording before the industrial revolution, it made sense for most owners of enterprises in those times to carry out management functions by and for themselves.

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.

Given the longevity of CFC molecules, recovery times are measured in decades.

" or " Given that I have flipped a coin 100 times and it has landed heads-up 100 times, what is the likelihood that the coin is fair?

Given the choice between playing a game at an arcade three or four times ( perhaps 15 minutes of play for a typical arcade game ), and renting, at about the same price, exactly the same game — for a video game console — the console became the preferred choice.

Given the short distances of super special stages compared to the regular special stages and consequent near-identical times for the frontrunning cars, it is very rare for these spectator-oriented stages to decide rally results, though it is a well-known axiom that a team can't win the rally at the super special, but they can certainly lose it.

Given the wide variety of clientele at the club, which includes Willie the Giant and Elliott the Dragon, the amount of seating at the club could vary wildly at times.

Given that the landing package of NASA's Spirit rover mission rebounded off the surface in Gusev crater numerous times before coming to a standstill-taking much more time than anticipated-Beagle's timer may have been set to a too short time ( see Section 5. 4. 8 of the Inquiry Report );

Given the nature of the times, this was impossible, although their complaints were passed up the chain of command.

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

Given the conditions of the experiment, the Canterbury group conclude that if any such ' Tajmar ' effect exists, it is at least 22 times smaller than predicted by Tajmar in 2006.

Given an action that must be repeated, for instance, five times, different languages ' for loops will be written differently.

Internazionale scored four times past Given as Newcastle lost 1 – 4 in their opening match in the second group stage.

Given the murder rate in this town, it has about the same murder rate of a town 20 times its size.

Given the occasional suppression of the party and resultant lapses in its activity, the position was at times vacant.

Given the reduction in the rate of spread once screening for Hepatitis C in blood products was implemented in the 1990s it would seem that at least in recent times blood transfusion has been an important method of spreading for this virus.

Given the suspension of semicha, any beth din existing in medieval or modern times is in theory a court of laymen, acting as arbitrators.

Given that this mass is confined inside a 44 million km diameter sphere, this yields a density ten times higher than previous estimates.

Given the bitter and war-wrecked times through which they have lived, this is a flaw that is easy to understand.

Given the repression of the times, she tried to dodge an expected arrest, and frequently changed places of residence and took only temporary jobs.

* Given the extended recording times of data loggers, they typically feature a mechanism to record the date and time in a timestamp to ensure that each recorded data value is associated with a date and time of acquisition in order to produce a sequence of events.

Given a uniform distribution of backoff times, the expected backoff time is the mean of the possibilities.