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 a smooth Φ < sup > t </ sup >, an autonomous vector field can be derived from it.

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 field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

" Given that science continually seeks to adjust its theories structurally to fit the facts, i. e., adjusts its maps to fit the territory, and thus advances more rapidly than any other field, he believed that the key to understanding sanity would be found in the study of the methods of science ( and the study of structure as revealed by science ).

Given the currently keen interest in biotechnology and the high levels of funding in that field, attempts to exploit the replicative ability of existing cells are timely, and may easily lead to significant insights and advances.

Given the union's commitment to international solidarity, its efforts and success in the field come as no surprise.

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.

Given a bounded sequence, there exists a closed ball that contains the image of ( is a subset of the scalar field ).

Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.

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 a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

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.

Given such a field, an absolute value can be defined on it.

Given a locally compact topological field K, an absolute value can be defined as follows.

Given a grid point field of geopotential height, storm tracks can be visualized by contouring its average standard deviation, after the data has been band-pass filtered.

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 separable extension K ′ of K, a Galois closure L of K ′ is a type of splitting field, and also a Galois extension of K containing K ′ that is minimal, in an obvious sense.

Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

Given a class function G: V → V, there exists a unique transfinite sequence F: Ord → V ( where Ord is the class of all ordinals ) such that

Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.

Given a natural transformation Φ from h < sup > A </ sup > to F, the corresponding element of F ( A ) is.

Given a topological space X, denote F the set of filters on X, x ∈ X a point, V ( x ) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and that converge to x.

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

* Shay Given – former goalkeeper for both the Republic of Ireland and Newcastle United F. C ..

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 subset F =

Given a discrete-time stationary ergodic stochastic process on the probability space, AEP is an assertion that

" Given the Rules ' permissive backdrop and their inclusion of a specific rule on expert testimony that does not mention ' general acceptance ,' the assertion that the Rules somehow assimilated Frye is unconvincing.

Given that both A and not-A are seen to be “ true ,” Kant concludes that it ’ s not that “ God doesn ’ t exist ” but that there is something wrong with how we are asking questions about God and how we have been using our rational faculties to talk about universals ever since Plato got us started on this track!

Similarly, Hare refers to “ the crude caricature of act utilitarianism which is the only version of it that many philosophers seem to be acquainted with .” Given what Bentham says about second order evils it would be a serious misrepresentation to say that he and similar act utilitarians would be prepared to punish an innocent person for the greater good.

" As Shelly Kagan says, “ Given the parameters of the actual world, there is no question that …( maximally )… promoting the good would require a life of hardship, self-denial, and austerity … a life spent promoting the good would be a severe one indeed .”

The inscription reads: “ Belonging to Ahaz ( son of ) Yehotam, King of Judah .” Given the process that created and preserved bullae, they are virtually impossible to forge.

With regard to African Americans, Gilman wrote in the American Journal of Sociology: “ The problem, is this: Given: in the same country, Race A, progressed in social evolution, say, to Status 10 ; and Race B, progressed in social evolution, say, to Status 4..

Denis MacShane, a former Foreign Office minister, said: “ Given the appalling suffering and repression of the Bahraini people, it would be a fitting gesture for the Countess of Wessex to auction these trinkets and distribute the proceeds to the victims of the regime .”

Given the Nazi predilection for “ racial purity ”, the Roma were among their first victims.

* “ Comment on Richard Schantz, ‘ The Given Regained ’”, in Philosophy and Phenomenological Research lxii ( 2001 ), 181-5

* “ Hegel and the Myth of the Given ”, in Wolfgang Welsch und Klaus Vieweg, Herausg., Das Interesse des Denkens: Hegel aus heutiger Sicht ( München: Wilhelm Fink Verlag, 2003 ), pp. 75 – 88.

Given that Unferth shows his dislike for Beowulf early in the story, Unferth ’ s choice to award Beowulf with Hrunting, which means “ thrusting ,” can be interpreted as a sign of peace and acceptance.

* Not being automatically granted the luxury of a “ diagnostic period .” Given that they know the company, COO ’ s turned CEO ’ s are often expected to hit the ground running when in actuality they too need to enter diagnostic mode to fully understand their new role and to see the Company from a new perspective.

* Given “ Medalla de la Universidad de Alcala de Henares ” and “ Medalla de Oro de la Ciudad de Zamora ”, Spain, 2005

Given the existence of important public policy goals on both sides, a city seeking to maximize walkability should not favor cul-de-sacs over grids, but should also allow some cul-de-sacs as a legitimate residential option ’ ... ‘ In addition, there are “ middle ground ” alternatives between prohibiting cul-de-sacs and mandating them.

Another partition of the given interval, Q, is defined as a refinement of the partition, P, when it contains all the points of P and possibly some other points as well ; the partition Q is said to be “ finer ” than P. Given two partitions, P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P and Q, re-numbered in order.

Given a set of data and the desire to produce some kind of “ model ” of those data ( model, in this case, meaning a function fitted through the data ), there are a variety of functions that can be chosen for the fit.

Apart from the very rare “ Kirill of Turov ’ ( even when this designation occurs, it tends to be just one of several variant readings ), the headings in ‘ his ’ manuscripts include “ Kirill the monk ,” “ Kirill the philosopher ,” ‘ Saint Kirill ,” “ The Blessed father Kirill ,” “ the blessed monk Kirill ,” “ Kirill the unworthy monk ,” “ the venerable Kirill .” Given this variety of labels, ‘ Kirill ’ s ’ texts invite several candidates for being their more likely authors ( writers whose existence has been substantiated with historical facts ).