An open set U in X admits a local trivialization if and only if there exists a local section on U. Given a local trivialization one can define an associated local section by

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

Given, for countably many special values of, called characteristic values, the Mathieu equation admits solutions which are periodic with period.

Given any natural numbers l, m, n > 1 exactly one of the classical two-dimensional geometries ( Euclidean, spherical, or hyperbolic ) admits a triangle with the angles ( π / l, π / m, π / n ), and the space is tiled by reflections of the triangle.

Given the first four rolls turn up heads, the probability that the next toss is a head is in fact,

Given this assumption, there are four categories in which a firm's profit may be considered to be.

Given a year to complete the work, Gibson undertook the actual writing out of " blind animal terror " at the obligation to write an entire novel – a feat which he felt he was " four or five years away from ".

In the account books of Johanna, Duchess of Brabant and Wenceslaus I, Duke of Luxemburg, an entry dated May 14, 1379 reads: " Given to Monsieur and Madame four peters, two forms, value eight and a half moutons, wherewith to buy a pack of cards ".

Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):

Given that the four cups are in reference to the four expressions of redemption in Exodus 6: 6-7, some rabbis felt that it was important to include a fifth cup for the fifth expression of redemption in Exodus 6: 8.

# Given any elements a and a ' of G, if a ~ a ', then a < sup >− 1 </ sup > ~ a ' < sup >− 1 </ sup > ( this can actually be proven from the other four, so is strictly redundant ).

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 a sample of DNA, the instrument determines the order of the four bases — adenine, guanine, cytosine, and thymine — composing pieces of the sample.

Given that the four trials under consideration can indeed be consistently considered and experimentally realized, the assumptions concerning hidden variables which lead to the indicated mathematical contradiction are therefore collectively unsuitable to represent all experimental results ; namely the assumption of local hidden variables which occur equally in distinct trials.

Given a non-empty totally ordered set R with the following four properties:

The specific problems are: Given two, three or four points on a straight line, find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has a given ratio either ( 1 ) to the square on the remaining one or the rectangle contained by the remaining two or ( 2 ) to the rectangle contained by the remaining one and another given straight line.

Given how different the four foundational styles of Kajukenbo are it is impossible to fully incorporate everything and some specialization is inevitable.

Given four points in general linear position ( no three collinear ; in particular, no two coincident ), there are exactly three pairs of lines ( degenerate conics ) passing through them, which will in general be intersecting, unless the points form a trapezoid ( one pair is parallel ) or a parallelogram ( two pairs are parallel ).

Given this lengthy support it was reasonable for Tyrconnell and Tyrone to try to solicit help from Philip III, but Spanish policy was to maintain its recent ( 1604 ) peace with England, and its European fleet had anyway been destroyed by the Dutch over four months earlier.

Given four circles with curvatures k < sub > i </ sub > and centers z < sub > i </ sub > ( for i = 1 ... 4 ), the following equality holds in addition to equation ( 1 ):

Given the short distance ( 90 miles ) between Axis airfields on Sicily and the planned drop zones over Malta, it was possible for the motorized transports to make four round-trips per day.

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

James Berardinelli gave the film three stars out of four, writing: " Given the expectations that constrain it, Lethal Weapon 4 is probably the best motion picture that could possibly result from another teaming of Martin Riggs and Roger Murtaugh.

Given two actors, oneself and someone else, there are four types of possible behavior directly impacting the welfare of the actors: selfishness, altruism, spite, and cooperation.

Given four operators ( F, G, U, and V ) together with a product operation (∙) defined for any pair of these four operators, and given that the following four pairs of operators commute:

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

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 rigorous admission process it is not surprising that CUNY BA boasts a 70 % graduation rate within an average of 2. 2 years and that 60 % graduate with academic honors.

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

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

" Given that there's a gulf of 100, 000, 000 light years, he checks his diary to find out what he had been doing that day, and finds out that it was something he wished to hide.

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

Given proper nutrition, exercise, and veterinary care, most whippets live for 12 to 15 years.

Given that propensity to noble rot was a factor in Hungarian vineyard demarcations some 50 years before a messenger was supposedly mugged on his way to Schloss Johannisberg in Germany, the Hungarians probably have a better case.

Given this information, a timeline of roughly the years 15 – 98 CE can be established for his adult life.

" Given the situation in Spain for several years regarding terrorist attacks, these ties can be considered objectively as a threat to democracy ," the court said, also adding that these parties " contradicted the concept of a ' democratic society ' and presented a major danger to Spain's democracy ".

Given the variety of laws passed by Congress to disenfranchise US Citizens in Washington, DC or on US Bases, it is nearly impossible to thoroughly summarize two hundred years of history of disenfranchisement and alienation of in alienable rights, for Local, State and Federal Rights levels in these places.

" Given that declaration, along with seven years of bombings and the wording of their communiques throughout that time that strove to present an image of a powerful organization spread secretly throughout all sectors of society, the authorities took significant action.

Given the traditional account of the Kali Yuga lasting 432, 000 years and having started in 3102 BCE, which makes these claims problematic.

Given that under the External Relations Act the role of representing Ireland in the accreditation of ambassadors belonged to the King of Ireland on the advice of the Irish government, between those years the Irish head of state was unambiguously the King of Ireland.

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 the team's poor results in the last few years, which she attributes to Coach D ' Amato's " old-school methods ", she attempts to take a more hands-on approach to the team, including bringing in an innovative Offensive Coordinator Nick Crozier.

Given his military service, the selection of Arthur proved to be prudent, as he acted as the King's, and thus the Canadian Commander-in-Chief's, representative through the first years of World War I.

Given the value of the vehicle many " extra " cars have appeared over the years, even some sharing the same chassis number.

Given Dirichlet's young age ( he was 23 years old at the time ), Humboldt was only able to get him a trial position at the Prussian Military Academy in Berlin while remaining nominally employed by the University of Breslau.

Given the older ages of these stars, and the short lifetimes of micrometer-sized dust grains around stars due to Poynting Robertson drag, collisions, and radiation pressure ( typically hundreds to thousands of years ), it is thought that this dust is from the collisions of planetesimals ( e. g. asteroids, comets ).

Given that the Ashuruballit I synchronism with Akhenaten has become the crucial link between Egyptian and Mesopotamian history in recent years, this issue is a key area of focus and dispute.

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.