: 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 any combination of complex numbers, bras, kets, inner products, outer products, and / or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

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 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 the ability to knock down, in essence, any gene of interest, RNAi via siRNAs has generated a great deal of interest in both basic and applied biology.

Given any gene in the body of a person, we can trace a single chain of human ancestors back in time, following the lineage of this one gene.

Given the inherent variability of the technology, a new gene could insert into the targeted genome in a way that could lead to deleterious interactions with other genes.

Given that C-values were assumed to be constant because DNA is the stuff of genes, and yet bore no relationship to presumed gene number, this was understandably considered paradoxical ; the term " C-value paradox " was used to describe this situation by C. A.

The question concerned the educational value of pre-Christian classical thought: " Given that the body of knowledge of the pre-Christian Romans was heathen in origin, was it safe to teach it to Christian children?

Given by the Georgia Historical Society, in conjunction with the Governor of Georgia, to individuals whose accomplishments and community service reflect the ideals of the founding body of Trustees, which governed the Georgia colony from 1732 to 1752.

Given the distance from the main body, our problem is to determine the form of the surface that satisfies the equipotential condition.

Given a constant density and volume the self-potential of such body depends only on the eccentricity ε of the ellipsoid:

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 the lack of concrete historical knowledge about one of the most potent figures in British mythology, it is unlikely that any definitive conclusions about the claims for these places will ever be established ; nevertheless it is both interesting and important to try to evaluate the body of evidence which does exist and examine it critically.

Given by the Georgia Historical Society, in conjunction with the Governor of Georgia, to individuals whose accomplishments and community service reflect the ideals of the founding body of Trustees, which governed the Georgia colony from 1732 to 1752.

Given by the Georgia Historical Society, in conjunction with the Governor of Georgia, to individuals whose accomplishments and community service reflect the ideals of the founding body of Trustees, which governed the Georgia colony from 1732 to 1752.

* 1957 Given the status of a ‘ public legal body ’.

Given the recency with which spoken language has emerged, it is very likely that human body language does include some more or less involuntary responses that have a similar origin to the communication we see in other animals.

Given Poole's exposure to the vacuum of outer space beginning in 2001, his body had been flash-frozen and freeze-dried so that it was mostly intact after 1, 000 years.

Given the excellent state of preservation of the body and the artifacts recovered from it, the absence of the photograph suggests that he may have reached the summit and deposited it there.

Given the proportions of his body, and the fact that in adulthood he was sometimes mistaken for a small child, it is likely that Joseph was a pituitary dwarf, not a dwarf of the achondroplasic variety.

Given that the prominent Uranians were trained Classicists, I consider ludicrous the view, widely held, that ‘ Uranian ’ derives from the German apologias and legal appeals written by Karl-Heinrich Ulrichs in the 1860s, though his coinage Urning — employed to denote ‘ a female psyche in a male body ’ — does indeed derive from the same Classical sources, particularly the Symposium.

Given that breast cancer is a large killer of women and a very high percentage of young women use underarm deodorants, I think we should be carrying out properly funded, further investigations into parabens and where they are found in the body ," says Philip Harvey, an editor of the Journal of Applied Toxicology, which published the research.

Given the total mass and the scalars r and v at a single point of the orbit, one can compute the specific orbital energy, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being " suborbital " ( a ballistic missile, for example ), having enough energy to be " orbital ", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and / or go to infinity ( as a meteor, for example ).

Given the choice to fight or pursue other sports, the young Andre chose boxing and has poured his mind, body and soul into the sport & became a world champion.

# Given the Heir-A woman re-shapes her body to perfection.

Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium ( such as a rocket or aircraft, missile or bullet ), an important problem is the determination of the nose cone geometrical shape for optimum performance.

Given the long half-life of the drug in the body, a mildly sedated army would be unlikely to be an effective fighting force.

Given their underfocused nature, wide fields of view, and high light collecting ability, the ocelli are superbly adapted for measuring changes in the perceived brightness of the external world as an insect rolls or pitches around its body axis during flight.

Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium ( such as a rocket or aircraft, missile or bullet ), an important problem is the determination of the nose cone geometrical shape for optimum performance.