* 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 the number of times tea is poured in a meal, the tapping is a timesaver in loud restaurants or lively company, as an individual being served might be speaking to someone else or have food in their mouth.

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 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 that even players with poor chess knowledge can use the strongest computer programs to analyse their games, the gap between the beginner and master player has narrowed in recent years.

Given his chance by the injuries to other players, he became the linchpin of the England team in the 1986 FIFA World Cup in Mexico.

Given that she had become the center of all the cooperative effort of the World Team ( as opposed to individual effort, however heroic ), even players of much greater strength began sharing their ideas with her, so that she would incorporate them into her analysis.

Given this complexity, early games using the system tended to be quite simple in layout, using the TIA to create simple symmetric playfields with players on top.

Then at UEFA Euro 2012, his team flopped leading to the retirement of star players Damien Duff and Shay Given.

Staunton was the first footballer to have reached a century of caps for the Republic of Ireland national team, and as of the end of Ireland's qualifying campaign for the 2010 FIFA World Cup he was still the joint record holder with former teammates Shay Given and Kevin Kilbane, though both players ( along with Robbie Keane ) have since emulated Staunton's appearance record.

Given the tuple of strategies chosen by the players, one is given an allocation of payments ( given as real numbers ).

; Condorcet winner: Given a preference ν on the outcome space, an outcome a is a condorcet winner if all non-dummy players prefer a to all other outcomes.

; Dominated outcome: Given a preference ν on the outcome space, we say that an outcome a is dominated by outcome b ( hence, b is the dominant strategy ) if it is preferred by all players.

Given a specific set of characters in each scenario, based on a scene from each of the movies, up to two players can control them, using their different abilities.

Given its size, Tunstall is the birthplace of an unusually large number of professional football players.

• b )" Given men " were players specially selected from other teams, including current opponents, to try and ensure that the teams were evenly matched for the purpose of gambling on the match result.

Given the great number of players who have played the game, players who have average career statistics but seemingly exceptional statistics in certain situations ( e. g. the playoffs or with the bases loaded ) are expected without providing proof that " clutch hitting " skills exist.

Given testable information, the maximum entropy procedure consists of seeking the probability distribution which maximizes information entropy, subject to the constraints of the information.

Given the existence of important public policy goals on both sides, a city seeking to maximize walkability should not favor cul-de-sacs over grids, but should also allow some cul-de-sacs as a legitimate residential option ’ ... ‘ In addition, there are “ middle ground ” alternatives between prohibiting cul-de-sacs and mandating them.

Given that the series was set in the relatively near future, Gridneff took great pains to ensure that the space scenes were depicted as accurately as possible, seeking advice from NASA and other space agencies as well as arranging a set visit by astronaut Pete Conrad, the third man to set foot on the Moon.

Given only ten minutes to decide, Laboa said he did not confer with the Vatican, but agreed to allow Noriega to enter the Vatican grounds-although from the very start, he confessed that he deceived Noriega, noting that he believed that Panamanian politics necessitated that his own role be to convince Noriega to surrender to the American army, not to grant him asylum within Vatican territory.