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 a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as

Given enough time, a hypothetical monkey typing at random would, as part of its output, almost surely produce all of William Shakespeare | Shakespeare's plays.

* Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.

* Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.

The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f ( x ) = a < sup > x </ sup > mod N. Given N, find Q = 2 < sup > q </ sup > such that < math > N ^ 2

A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval ( 0, 1 ), the variate

Given any random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >..., X < sub > n </ sub >, the order statistics X < sub >( 1 )</ sub >, X < sub >( 2 )</ sub >, ..., X < sub >( n )</ sub > are also random variables, defined by sorting the values ( realizations ) of X < sub > 1 </ sub >, ..., X < sub > n </ sub > in increasing order.

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

Given a probability space with, the indicator random variable is defined by if otherwise

Given N independent and identically distributed Rayleigh random variables with parameter, the maximum likelihood estimate of is

Given a random variate U drawn from the uniform distribution in the interval < nowiki >( 0, 1 )</ nowiki >, then the variate

Given a random variate U drawn from the uniform distribution in the interval < nowiki ></ nowiki >, the variate

: Given a random galaxy in a location, the correlation function describes the probability that another galaxy will be found within a given distance.

Given a random graph of n nodes and an average degree < math >< k ></ math >.

Given two column vectors and of random variables with finite second moments, one may define the cross-covariance to be the matrix whose entry is the covariance.

Fallacious evaluation: " Given that the counterattacks against Germany occurred only after they had conquered the greatest amount of territory under their control, regression to the mean can explain the retreat of German forces from occupied territories as a purely random fluctuation that would have happened without any intervention on the part of the USSR or the Western Allies.

Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value.

Given the model and realizations ( samples ) of the random vector, the task is to estimate both the mixing matrix and the sources.

Given a random sample ( X < sub > i </ sub >, Y < sub > i </ sub >), i = 1, ..., n, each pair ( X < sub > i </ sub >, Y < sub > i </ sub >) satisfies

Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?

Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:

Given discrete random variable with support and with support, the conditional entropy of given is defined as:

Given the null hypothesis that the observed frequencies result from random sampling from a distribution with the given expected frequencies, the distribution of G is approximately a chi-squared distribution, with the same number of degrees of freedom as in the corresponding chi-squared test.