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 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 constants C, D and V, there are only finitely many ( up to diffeomorphism ) compact n-dimensional Riemannian manifolds with sectional curvature | K | ≤ C, diameter ≤ D and volume ≥ V.

Given constants C, D and V, there are only finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D and volume ≥ V.

Given two differentiable manifolds

is also true in the case of compact manifolds, due to Yau's proof of the Calabi conjecture: Given a compact, Kähler, holomorphically symplectic manifold ( M, I ), it is always equipped with a compatible hyperkähler metric.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given an evaluation e of variables by elements of M < sub > w </ sub >, we

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

Given two complexes M < sub >*</ sub > and N < sub >*</ sub >, a chain map between the two is a series of homomorphisms from M < sub > i </ sub > to N < sub > i </ sub > such that the entire diagram involving the boundary maps of M and N commutes.

Given an SVD of M, as described above, the following two relations hold:

Given a set M of molecules, chemical reactions can be roughly defined as pairs r =( A, B ) of subsets from M.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given the morphological distinctness of the Cape Verde birds and the fact that the Cape Verde population was isolated from other populations of Red Kites, it cannot be conclusively resolved at this time whether the Cape Verde population was not a distinct subspecies ( as M. migrans fasciicauda ) or even species that frequently absorbed stragglers from the migrating European populations into its gene pool.

Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Given a smooth curve γ on ( M, g ) and a vector field V along γ its derivative is defined by

Given a group G, a factor group G / N is abelian if and only if ≤ N.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given control by the U. N. to Magneto until the E Is for Extinction story.

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

Given that the unified atomic mass unit is one twelfth the mass of one atom of carbon-12, meaning the mass of such an atom is 12 u, it follows that there are N < sub > A </ sub > atoms of carbon-12 in 0. 012 kg of carbon-12.

Informally it solves the following problem: Given an integer N, find its prime factors.

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

Given a set A of N people, it has 2 < sup > N </ sup > possible subsets.

Given a concrete category ( C, U ) and a cardinal number N, let U < sup > N </ sup > be the functor C → Set determined by U < sup > N </ sup >( c ) = ( U ( c ))< sup > N </ sup >.