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Given and is
: Given any family of nonempty sets, their Cartesian product is a nonempty set.
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 a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:
* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.
" Given that Thomas Aquinas died six years before Albertus Magnus ' death, this legend as stated is unlikely.
Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.
* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.
* 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 the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.
After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.
Given its " blue-water " bias, the navy is even less inclined to become involved in counterdrug operations than the army or air force.
Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).
Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.
While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.
Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is
Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.
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 then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.
Given some encrypted data (" ciphertext "), the goal of the cryptanalyst is to gain as much information as possible about the original, unencrypted data (" plaintext ").
Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.
Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.
Given such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.

Given and plane
Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:
Given an embedding of a rooted tree in the plane, if one fixes a direction of children ( starting from root, then first child, second child, etc.
Given a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers.
* Given n points in the plane, find the two with the smallest distance from each other.
Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is " above " the plane.
Given a point in the plane, draw a straight line connecting it with the north pole on the sphere.
The original problem was stated in the form that has become known as the Euclidean Steiner tree problem or geometric Steiner tree problem: Given N points in the plane, the goal is to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either directly or via other points and line segments.
# Given a subdivision of the plane into vertical slabs, determine which slab contains a given point.
The problem in more mathematical terms is: Given a needle of length dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will cross a line?
Given a fixed oriented line L in the Euclidean plane R < sup > 2 </ sup >, a meander of order n is a non-self-intersecting closed curve in R < sup > 2 </ sup > which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane.
Given three points in a plane as shown in the figure, the point is a convex combination of the three points, while is not.
Given any four points and in the compactified complex plane, the cross-ratio is defined by
Given a line in a plane, there exists at least one point in the plane that is not on the line.
Given a plane in space, there exists at least one point in space that is not in the plane.
Given any analytic function in the upper half plane, the function where is real will also be analytic in the upper half of the plane.
: Given five points in the plane in general position, prove that four of them form a convex quadrilateral.
Given a spread of, the André / Bruck-Bose construction < sup > 1 </ sup > produces a translation plane of order q < sup > 2 </ sup > as follows: Embed as a hyperplane of.

Given and wave
Given basic sound blocks, that a machine digitized, one has a bunch of numbers which describe a wave and waves describe words.
Given the derivatives, wave – particle duality can be assessed as follows.
Given the right overall conditions of the compressed fuel — high enough density and temperature — this heating process will result in a chain reaction, burning outward from the center where the shock wave started the reaction.
Given a revisionist makeover, the implied sexuality of the vintage films was enhanced and made explicit in a new wave of erotic thrillers.
Given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm.
Given the unique mapping between densities and wave function, Runge and Gross then treated the Dirac action as a density functional,
Given the right overall conditions of the compressed fuel – high enough density and temperature – this heating process can result in a chain reaction, burning outward from the center where the shock wave started the reaction.
Given the right overall conditions of the compressed fuel — high enough density and temperature — this heating process can result in a chain reaction, burning outward from the center where the shock wave started the reaction.

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