: 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 that John of Worcester wrote his chronicle after the eruption of the Canterbury – York supremacy struggle, the story of Ealdred renouncing any claims to Worcester needs to be considered suspect.

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

* 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 any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

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 any expression involving complex numbers, bras, kets, inner products, outer products, and / or linear operators ( but not addition ), written in bra-ket notation, the parenthetical groupings do not matter ( i. e., the associative property holds ).

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

Given a left neutral element and for any given then A4 ’ says there exists an such that.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given that Gable and Cukor had worked together before, in Manhattan Melodrama and Gable had no objection to working with him then, and given Selznick's desperation to get Gable for Rhett Butler, if Gable had any objections to Cukor, certainly they would have been expressed before he signed his contract for the film.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.

We have seen a few cavities of the appropriate size and shape for ivory-bills, but these can be old, or exceptionally large Pileated Woodpecker cavities, or mammal-enlarged Pileated Woodpecker cavities .… Given the results, it is unlikely a population of any meaningful size of Ivory-billed Woodpeckers exists in south Florida.

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Given the above-mentioned problems, regulators face the challenging task of regulating a market that is changing very rapidly, without stifling any type of innovation, and without improperly disadvantaging any competitor.

Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.

: Given any positive number ε, there is a sequence

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis.

Given that Aleppo was to far off from Jerusalem for an attack and Damascus, recently allied with the Kingdom of Jerusalem against Zengi, had entered into an alliance with Nur ad-Din, the crusaders decided to attack Damascus, the conquest of which would preclude a combination of Jerusalem's enemies.

* Given to senior citizens and the disabled as a combination photo ID and MetroCard.

Given to MTA employees as a combination photo ID and MetroCard pass.

* AGSP ( Any Given Screen Position ) is the combination of VSP and Linecruncher, f. ex.

Given the operational name Storm-333 and involving a combination of Spetsnaz, GRU, KGB, and Soviet paratroopers were a total of 24 men from the « Гром » (" Thunder ") detachment of Alpha Group.

Given a finite set of probability density functions p < sub > 1 </ sub >( x ), …, p < sub > n </ sub >( x ), or corresponding cumulative distribution functions P < sub > 1 </ sub >( x ), …, P < sub > n </ sub >( x ) and weights w < sub > 1 </ sub >, …, w < sub > n </ sub > such that and the mixture distribution can be represented by writing either the density, f, or the distribution function, F, as a sum ( which in both cases is a convex combination ):

Given the microcosm complex that CIA service tends to cultivate and develop within the organization's comparatively small ranks in combination with the military background of such a large contingency of the agency's active members ; receipt of either of these awards is viewed with utmost admiration almost universally across the " family " of US intelligence services.

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 the combination of employees at the plant and secondary jobs created by Indian Point ’ s economic activity, the plant is responsible for 2, 500 jobs in Westchester, Orange, Rockland, Putnam and Dutchess counties .”

: Given that there are almost eight times as many data points in this sample as in the previous analysis by Burbidge & Napier ( 2001 ), we must conclude that the previous detection of a periodic signal arose from the combination of noise and the effects of the window function.

Given the right combination of hypnotist and subject, hypnosis can be a lethal weapon.

Given by subcutaneous injection, it is used in combination with human chorionic gonadotropin ( hCG ) to assist in ovulation and fertility.

: Given positive integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub > such that gcd ( a < sub > 1 </ sub >, a < sub > 2 </ sub >, ..., a < sub > n </ sub >) = 1, find the largest integer that cannot be expressed as an integer conical combination of these numbers, i. e., as a sum

Given a Gödel numbering of recursive functions, there is a primitive recursive function s of two arguments with the following property: for every Gödel number p of a partial computable function f with two arguments, the expressions and are defined for the same combinations of natural numbers x and y, and their values are equal for any such combination.

One vice-president and associate publisher said, " Given the success of The Da Vinci Code, everybody around town knows how popular the combination of thriller and history can be and what a phenomenon it can become.

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

Given the poorly understood nature of the endocannabinoid system, it will take many more generations of progress to even scratch the surface of the extraordinarily complex mechanisms that control neurological development / degeneration.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Given the irregular shape of the nucleus, Halley's rotation is likely to be complex.

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 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 the complex nature of human smuggling and trafficking operations, the difference between these two criminal operations is not always readily apparent.

Given that the site envisioned for this museum in Phillipsburg has been sold for development as a townhouse complex and college campus annex, it is unclear what role Phillipsburg will play in this museum.

* Given two complex vectors x and y, multiplication by U preserves their inner product ; that is,

* Given that Historic U. S. Route 66 runs through Elk City, a sprawling museum complex has developed, which includes the National Route 66 Museum, the Old Town Museum, the Transportation Museum, the Farm and Ranch Museum, and the Blacksmith Museum.

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 an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generated by the roots of p. If K is a subfield of the complex numbers, the existence is automatic.

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 two normed vector spaces V and W ( over the same base field, either the real numbers R or the complex numbers C ), a linear map A: V → W is continuous if and only if there exists a real number c such that

Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M,

Given a complex vector bundle V over a topological space X,

Given the often complex badge-engineering that BMC undertook, it is common amongst enthusiasts to use the ADO number when referring to vehicles as a single design ( for example, saying ' The ADO15 entered production in 1959 '- this encompasses the fact that when launched, the ADO15 was marketed as both the Morris Mini Minor and the Austin Seven ).

Given the rapid growth in absolute value of Γ ( z + k ) when k → ∞, and the fact that the reciprocal of Γ ( z ) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex s and x.

Given the complex macro-rheological behavior of blood, it is not surprising that a single equation fails to completely describe the effects of various rheological variables ( e. g., hematocrit, shear rate ).