Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

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 two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.

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 opportunity, adults eat hatchlings, which may be protected by separating the two groups with a net, breeding box or separate tank.

Given that Bozizé accuses Sudan of supporting the UFDR rebels who are actively fighting the Central African Government, the relation between the two countries has remained good.

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 two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given that the vast majority of all emeralds are treated as described above, and the fact that two stones that appear visually similar may actually be quite far apart in treatment level and therefore in value, a consumer considering a purchase of an expensive emerald is well advised to insist upon a treatment report from a reputable gemological laboratory.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.

: Given two points, determine the azimuth and length of the line ( straight line, arc or geodesic ) that connects them.

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

Given this understanding, the Board withdrew the allegation of unsportsmanlike behaviour two days before the fourth Test, thus saving the tour.

Given two secondary stations, the time difference ( TD ) between the primary and first secondary identifies one curve, and the time difference between the primary and second secondary identifies another curve, the intersections of which will determine a geographic point in relation to the position of the three stations.

Given the above commonalities there appear to be only two string theories: the heterotic string theory ( which is also the type I string theory ) and the type II theory.

Given the notion of a lexeme, it is possible to distinguish two kinds of morphological rules.

Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.

Given barely two months to assemble a large sea-going invasion fleet, the Kriegsmarine opted to convert inland river barges into makeshift landing craft.

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 any two distinct points, there is exactly one line incident with both of them.

# Given any two distinct lines, there is exactly one point incident with both of them.

Given the small population of the region ( compared with the Central Canadian provinces or the New England states ), the regional economy is a net exporter of natural resources, manufactured goods, and services.

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 France maintained a strong standing army, and that Prussia and the other German states would need weeks to mobilise their conscript armies, the French held the initial advantage of troop numbers and experience.

Given events, and, Bayes ' rule states that the conditional odds of given are equal to the marginal odds of multiplied by the Bayes factor:

The law states that " given enough eyeballs, all bugs are shallow "; or more formally: " Given a large enough beta-tester and co-developer base, almost every problem will be characterized quickly and the fix will be obvious to someone.

Given an isolated physical system, the allowed states of this system ( i. e. the states the system could occupy without violating the laws of physics ) are part of a Hilbert space H. Some properties of such a space are

Dr. Tennison states: " Given the account of Elliot Paul, and given that Lead Belly witnessed boogie-woogie in 1899 in the Arklatex ; and given the North to South migration of the Thomas family ; and given the Texas & Pacific headquarters in Marshall in the early 1870s ; and given that Harrison County had the largest slave population in the state of Texas ; and given the fact that the best-documented and largest-scale turpentine camps in Texas did not occur until after 1900 in Southeast Texas, it is most probable that boogie-woogie spread from Northeast to Southeast Texas, rather than from Southeast to Northeast Texas, or by having developed diffusely with an even density over all of the Piney Woods of East Texas.

Given the observation space, the state space, a sequence of observations, transition matrix of size such that stores the transition probability of transiting from state to state, emission matrix of size such that stores the probability of observing from state, an array of initial probabilities of size such that stores the probability that. We say a path is a sequence of states that generate the observations.

Linus's law states, " Given enough eyeballs all bugs are shallow.

Given *- representations π, π ' each with unit norm cyclic vectors ξ ∈ H, ξ ' ∈ K such that their respective associated states coincide, then π, π ' are unitarily equivalent representations.

Given an arbitrary ( n ; m ; p ) machine S, such that every two of its states are distinguishable from one another, then there exists an experiment of length

Given two states and in, is bisimilar to, written, if there is a bisimulation such that is in.

Given that Euler's rotation theorem states that there is always an instantaneous axis of rotation, the angular velocity,, can be given by a vector over this axis

However, the modelling study also states that " Given the complex, three-dimensional nature of the real Pine Island glacier ... it should be clear that the [...] model is a very crude representation of reality.

Given the above definitions, Frobenius ' theorem states that a subbundle E is integrable if and only if it arises from a regular foliation of M.

Given an assignment of every variable to the Marked or Unmarked states, this pa translation reduces to a PA expression, which can be simplified.

Given a particular input, it will always produce the same output, and the underlying machine will always pass through the same sequence of states.

Given that the Republic of Indonesia did not want the RIS parliament to become the legislature of the unitary republic, in May 1950, Hatta and representatives from the federal states agreed to establish a new parliament comprising the 150 members of the RIS parliament, 46 members of the KNIP Working Committee, 13 from the Republic of Indonesia Supreme Advisory council and 32 RIS senators, making 241 members.

Given that the ESCP is a regulation and not a directive, it is arguable whether it has left too many aspects to the discretion of member states, which could call into question the legal certainty expected from a European regulation.

Given that 1 John 2: 1 views Jesus as a paraclete, the reference in John 14: 16 states that Jesus sends a second paraclete to continue the life of the Church after his departure.

Given a twice continuously differentiable function f of one real variable, Taylor's theorem for the case n = 1 states that

Given that the vast majority of criminal sentencing is done at the state level, the American Law Institute and the American Bar Association have each recommended such systems for all the states, and nearly half the states presently have such systems, although significant variations exist among them.