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 absolute magnitude, for objects within our galaxy you can also calculate the apparent magnitude from any distance ( in parsecs ):

* Coreference resolution: Given a sentence or larger chunk of text, determine which words (" mentions ") refer to the same objects (" entities ").

Given their hunting instincts, they are prone to chasing moving objects.

Given the apparent magnitude (> 20 ) of all but the biggest trans-Neptunian objects, the physical studies are limited to the following:

Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category.

Given the rapid orbital decay of objects below approximately, the commonly accepted definition for LEO is between and above the Earth's surface.

* Given a category C with finite coproducts, a cogroup object is an object G of C together with a " comultiplication " m: G → G G, a " coidentity " e: G → 0, and a " coinversion " inv: G → G, which satisfy the dual versions of the axioms for group objects.

Given objects A < sub > 1 </ sub >,..., A < sub > n </ sub > in C, their biproduct is an object A < sub > 1 </ sub > ⊕ ··· ⊕ A < sub > n </ sub > together with morphisms

Given objects and in an additive category, we can represent morphisms as-by-matrices

* Ray tracing: Given a set of objects in space, produce a data structure that efficiently tells which object a query ray intersects first.

Given that the asteroid parent bodies of meteorites were some of the first solid material to condense from the early solar nebula, cosmochemists are generally, but not exclusively, concerned with the objects contained within the solar system.

* Given any manifold, there is a Lie groupoid called the pair groupoid, with as the manifold of objects, and precisely one morphism from any object to any other.

Given a subcategory S of C the inclusion functor I: S → C is both faithful and injective on objects.

Given the lack of surviving objects, we cannot know how common the techniques employed were, but the quality of the execution suggests that the binder was experienced in them.

Given a set of objects in d-dimensional space, preprocess them into a data structure so that for each query ray the first object hit by the ray can be found quickly.

Given any morphism between objects X and Y, if there is an inclusion map into the domain, then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain R → Y known as the range of f.

Given a partially ordered class P, we can form a category with P's elements as objects and a single arrow going from one object ( element ) to another if the first is less than or equal to the second.

Given a proper language, programs, including operating system servers, could import multiple interfaces and combine them as if they were objects native to that language — notably C ++.

Given that experience is so totally an affective, subjective, and personal process — not an abstract — it would be ironic, it's been argued, for experience designers, when designing experiences, to approach people merely as objects of commerce or cogs in a machine.

Given two objects A and B, their direct sum is written as.

Given a indexed family of objects A < sub > i </ sub >, indexed with i ∈ I from an index set I, one may write their direct sum as.

Additionally, there are composition operations: Given a sequence of sequences of objects, a sequence of objects, and an object Z: if

Given that there was no evidence that either the U. S. or the U. S. S. R. had anything remotely like the UFOs reported, Sign personnel gradually began considering extraterrestrial origins for the objects.