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 family of nonempty sets, their Cartesian product is a nonempty set.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G ( x, y ) ( i. e. the sets of morphisms from x to y ).

Given the widespread expansion in the tenth revision, it is not possible to convert ICD-9 data sets directly into ICD-10 data sets, although some tools are available to help guide users.

: Given two sets, there is a set whose members are exactly the two given sets.

Given a topological space X, a base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection of members of F.

Given any topological space X, the zero sets form the base for the closed sets of some topology on X.

Given that X, Y, and Z are sets of attributes in a relation R, one can derive several properties of functional dependencies.

Given a family of sets X < sub > i </ sub > the product is defined as

Given a presheaf, a natural question to ask is to what extent its sections over an open set U are specified by their restrictions to smaller open sets V < sub > i </ sub > of an open cover of U. A presheaf is separated if its sections are " locally determined ": whenever two sections over U coincide when restricted to each of V < sub > i </ sub >, the two sections are identical.

: Given a vector space V, any two linearly independent generating sets ( in other words, any two bases ) have the same cardinality.

Given this wide availabity of icon tools and icon sets, a problem can arise with custom icons which are mismatched in style to the other icons included on the system.

Given two sets we say is many-one reducible to and write

Given two partially ordered sets A and B, the lexicographical order on the Cartesian product A × B is defined as

Given an index set I, and open sets U < sub > i </ sub > contained in X, the nerve N is the set of finite subsets of I defined as follows:

Given of a monoid M of every strings over some alphabet, one may define sets that consist of formal left or right inverses of elements in S. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating.

Given a perfect graph G, Lovász forms a graph G * by replacing each vertex v by a clique of t < sub > v </ sub > vertices, where t < sub > v </ sub > is the number of distinct maximum independent sets in G that contain v. It is possible to correspond each of the distinct maximum independent sets in G with one of the maximum independent sets in G *, in such a way that the chosen maximum independent sets in G * are all disjoint and each vertex of G * appears in a single chosen set ; that is, G * has a coloring in which each color class is a maximum independent set.

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given a probability space, a filtration is a weakly increasing collection of sigma-algebras on,, indexed by some totally ordered set T, and bounded above by.

* Given the popularity of the radio show at the time many catch phrases appear frequently in Warner Bros. cartoon shorts from the 1930s and 1940s, such as Molly's " T ' ain't funny, McGee!

Given a core geometry, the B field needed for a given force can be calculated from ( 2 ); if it comes out to much more than 1. 6 T, a larger core must be used.

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

The isomorphism is not canonical ; it depends on a choice of basis in V. Given a basis ( e < sub > 1 </ sub >, ..., e < sub > n </ sub >) of V and an automorphism T in GL ( V ), we have

Given the time reversal operator T, it does nothing to the x-operator, TxT < sup >− 1 </ sup > =

Given the representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h ( T ) is the operator of multiplication by the composition.

Given Western Electric / AT & T Unix licensing, DEC ( and others ) were restricted to selling binary-only licenses.

Given a dynamical system ( T, M, Φ ) with T a group, M a set and Φ the evolution function

Given a linear differential operator T

Given a circle k, with a center O, and a point P outside of the circle, we want to construct the ( red ) tangent ( s ) to k that pass through P. Suppose the ( as yet unknown ) tangent t touches the circle in the point T. From symmetry, it is clear that the radius OT is orthogonal to the tangent.

* Pukui, Mary Kawena ; Elbert, Samuel Hoyt ; Mookini, Esther T. & Nishizawa, Yu Mapuana ( 1992 ): New Pocket Hawaiian Dictionary with a Concise Grammars and Given Names in Hawaiian.

* Given a maximal torus T in G, every element g ∈ G is conjugate to an element in T.

Given a torus T ( not necessarily maximal ), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is, Fix a maximal torus in G ; then the corresponding Weyl group is called the Weyl group of G ( it depends up to isomorphism on the choice of T ).