* 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.

Idealists are skeptics about the physical world, maintaining either: 1 ) that nothing exists outside the mind, or 2 ) that we would have no access to a mind-independent reality even if it may exist ; the latter case often takes the form of a denial of the idea that we can have unconceptualised experiences ( see Myth of the Given ).

Given the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.

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 this popular understanding of the term, it can also be used illegitimately and inappropriately, as a means to dismiss what are in fact substantial and well-evidenced accusations.

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 the rugged terrain of Ngazidja and Nzwani, and the dedication of extensive tracts to agriculture on all three islands, population pressures on Comoros are becoming increasingly critical.

Given this basic definition, further criteria are needed to establish a dividing line between postlexical clitics and morphological affixes, since both are characterized by a lack of prosodic autonomy.

Given that a father's age is 1 less than twice that of his son, and that the digits AB making up the father's age are reversed in the son's age ( i. e. BA ), leads to the equation 19B-8A

Given its different forms, there are various ways of representing uncertainty and modelling economic agents ' responses to it.

* Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition ‡;

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

Raymond also coined an aphorism he dubbed " Linus ' Law ", inspired by Linus Torvalds: " Given enough eyeballs, all bugs are shallow ", that first appeared in The Cathedral and the Bazaar.

Given the great historical link between Persian and Jewish history, modern day Persian Jews are called " Esther's Children ".

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 the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given that genes are universal to living organisms, genetics can be applied to the study of all living systems, from viruses and bacteria, through plants and domestic animals, to humans ( as in medical genetics ).

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.

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 that the coiled texture is the natural state of some African Americans ' hair, or perceived as being more " African ", this simple style is now often seen as a sign of self-acceptance and an affirmation that the beauty norms of the ( eurocentric ) dominant culture are not absolute.

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 the mean arrival and service rates, then actual rates vary around these average values randomly and hence have to be determined using a cumulative probability distribution function.

Given that large-scale, mechanised production could not be eliminated once invented, the Party arranges the destruction of surplus goods – before that makes " the masses too comfortable, and hence, in the long run, too intelligent ".

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 that data dredging efforts typically examine large datasets with many variables, and hence even larger numbers of pairs of variables, spurious but apparently statistically significant results are almost certain to be found by any such study.

; Dominated outcome: Given a preference ν on the outcome space, we say that an outcome a is dominated by outcome b ( hence, b is the dominant strategy ) if it is preferred by all players.

Given a non-negative eigenvector v, and that at least one of its components say j-th is strictly positive, the corresponding eigenvalue is strictly positive, indeed, given n such that ( A < sup > n </ sup >)< sub > ii </ sub > > 0, hence: r < sup > n </ sup > v < sub > i </ sub >

Given that he was writing in 1919, Keynes ' prediction that the next war would begin twenty years hence had an uncanny accuracy.

Given the dimensions of the ellipsoid, the conversion from lat / lon / height-above-ellipsoid coordinates to X-Y-Z is straightforward — calculate the X-Y-Z for the given lat-lon on the surface of the ellipsoid and add the X-Y-Z vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid.

Given the ability to update the token bucket every S milliseconds, the number of tokens to add every S milliseconds

Given the slim dimensions of the 2892, it provides a good platform on which to add or build a chronograph complication.

Given the contribution margin, a manager can easily compute breakeven and target income sales, and make better decisions about whether to add or subtract a product line, about how to price a product or service, and about how to structure sales commissions or bonuses.

Given the large amount of auto and farm machinery traffic the road carried, plus the high number of fatalities from vehicular accidents, the Commonwealth of Kentucky Department of Transportation opted to add two more lanes.

Given the scarcity of the two MGB EPs Lo-Fi B-Sides and Loser Anthems, Good decided to add the two discs to the latter half of the second disc as a nod to fans.

Given a linearly recursive sequence, let C be the transpose of the companion matrix of its characteristic polynomial, that is

Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define

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

Given a condition number, the relative error of compensated summation is effectively independent of n. In principle, there is the O ( nε < sup > 2 </ sup >) that grows linearly with n, but in practice this term is effectively zero: since the final result is rounded to a precision ε, the nε < sup > 2 </ sup > term rounds to zero unless n is roughly 1 / ε or larger.

Given a convex polytope P defined by the matrix inequality, if each row in A corresponds with a bounding hyperplane and is linearly independent of the other rows, then each facet of P corresponds with exactly one row of A, and vice versa.

: Given any n complex numbers z < sub > 1 </ sub >,..., z < sub > n </ sub > which are linearly independent over the rational numbers Q, the extension field Q ( z < sub > 1 </ sub >,..., z < sub > n </ sub >, exp ( z < sub > 1 </ sub >),..., exp ( z < sub > n </ sub >)) has transcendence degree of at least n over Q.

: Given any n formal power series f < sub > 1 </ sub >,..., f < sub > n </ sub > in tC < nowiki ></ nowiki > t < nowiki ></ nowiki > which are linearly independent over Q, then the field extension C ( t, f < sub > 1 </ sub >,..., f < sub > n </ sub >, exp ( f < sub > 1 </ sub >),..., exp ( f < sub > n </ sub >)) has transcendence degree at least n over C ( t ).

Given an ordered choice of linearly independent eigenvectors, especially an eigenbasis, they can be indexed by eigenvalues, i. e. using a double index, with x < sub > i, j </ sub > being the j < sup > th </ sup > eigenvector for the i < sup > th </ sup > eigenvalue.