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 any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations ( to avoid ambiguity, we can specify the square root with complex argument less than π ).

Given a permutation π of m elements,

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 a state ρ of A, there is a *- representation π of A with distinguished cyclic vector ξ such that its associated state is ρ, i. e.

Given any r on this sphere, and an angle – π < a ≤ π, the versor is on the 3-sphere of H.

Given a fiber bundle π: E → B and a continuous map f: B ′ → B one can define a " pullback " of E by f as a bundle f * E over B ′.

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.

In his 1859 paper On the Number of Primes Less Than a Given Magnitude Riemann found an explicit formula for the normalized prime-counting function π < sub > 0 </ sub >( x ) which is related to the prime-counting function π ( x ) by

Given a series with values in a normed abelian group G and a permutation σ of the natural numbers, one builds a new series, said to be a rearrangement of the original series.

Given a total interaction cross section σ ( typically measured in barns ), the mean free path of a prompt neutron is where n is the nuclear number density.

Given a graph Λ ( for example, a d-dimensional lattice ), per each lattice site j ∈ Λ there is a discrete variable σ < sub > j </ sub > such that σ < sub > j </ sub > ∈

Given a signature σ, a σ-structure M

) Given a section σ of E let the corresponding equivariant map be ψ ( σ ).

Given a local smooth frame ( e < sub > 1 </ sub >, …, e < sub > k </ sub >) of E over U, any section σ of E can be written as ( Einstein notation assumed ).

Given two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ.

Given two structures A and B of the same signature σ, A is said to be a weak substructure of B, or a weak subalgebra of B, if

Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

Given m real values t < sub > i </ sub >, called knots, with

# Composition operator ( also called the substitution operator ): Given an m-ary function and m k-ary functions:

" Given their 3: 00 a. m. start time and non-inclusion ( at frontman John Fogerty's insistence ) in the Woodstock film, members have expressed bitterness over their experiences at the famed festival.

Given the requirements of weight, and fragment density required this dictated using approximately 700 fragments, and being able to aim the mine with an accuracy of around two feet ( 0. 6 m ) at the center of the target zone.

Given a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

* 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 two bodies, one with mass m < sub > 1 </ sub > and the other with mass m < sub > 2 </ sub >, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass

Given local and transient influences on surface height, the values defined below are based on a " general purpose " model, refined as globally precisely as possible within 5 m of reference ellipsoid height, and to within 100 m of mean sea level ( neglecting geoid height ).

Given a vector bundle V over M, the corresponding field concept is called a section of the bundle: for m varying over M, a choice of vector

Given an m × n matrix with real entries ( or entries from any other field ) and rank r, then there exists at least one non-zero r × r minor, while all larger minors are zero.

Given that the eigenstates of the Hamiltonian always from a set of orthonormal basis, both the cases of m

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 central simple algebras A ~ M ( n, S ) and B ~ M ( m, T ) over the same field F, A and B are called similar ( or Brauer equivalent ) if their division rings S and T are isomorphic.

*: Given a hash h it should be difficult to find any message m such that h = hash ( m ).

*: Given an input m < sub > 1 </ sub > it should be difficult to find another input m < sub > 2 </ sub > such that and.

Given the interpretation via linear mappings and direct sums, there is a special type of block matrix that occurs for square matrices ( the case m = n ).