Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

Given the geographic diversity of the various sub-regions within the Maritimes, policies to centralize the population and economy were not initially successful, thus Maritime factories closed while those in Ontario and Quebec prospered.

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

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 topological space X, the zero sets form the base for the closed sets of some topology on X.

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 closed system in an initial state, if work is done on the system in an adiabatic ( i. e. no heat transfer ) way, and given the final state after a process, the amount of work required to be transferred to the system is the same, irrespective of how this work is performed.

Given that no majority supported either a leader or a split, a registration campaign, enabling membership for only 20 euros, and a closed primary was organized, which Ségolène Royal won.

Given any closed spin network, a non-negative integer can be calculated which is called the norm of the spin network.

" Robinson also returned to the stage in 1993 with a Broadway production of Frank Gilroy's Any Given Day, but the play closed after only six weeks.

Given a closed loop, parameterized by where is a parameter and.

Given a sequence ( X < sub > n </ sub >, p < sub > n </ sub >) of locally compact complete length metric spaces with distinguished points, it converges to ( Y, p ) if for any R > 0 the closed R-balls around p < sub > n </ sub > in X < sub > n </ sub > converge to the closed R-ball around p in Y in the usual Gromov – Hausdorff sense.

Given the prominence of close-quarters boarding actions in later years, vessels were built as " cataphract " ships, with a closed hull to protect the rowers, and a full deck able to carry marines and catapults.

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 two subsets A and B of N and a set of functions F from N to N which is closed under composition, A is called reducible to B under F if

Given the unexpected size of German forces closed off in Stalingrad, on 23 November Stavka ( Soviet Armed Forces High Command ) decided to strengthen the outer encirclement preparing to destroy Axis forces in and around the city.

Given a point p in a manifold M, its closed tubular neighbourhood is diffeomorphic to, thus we have decomposed M into the disjoint union of and glued along their common boundary.

Given a topological space X and a sheaf O < sub > X </ sub > of local C-algebras, if for any point x in X there is an open subset V of X containing it and a subset D of C < sup > n </ sup > so that the restriction ( V, O < sub > V </ sub >) of ( X, O < sub > X </ sub >) is isomorphic to a closed complex subspace of D, O < sub > X </ sub > is also coherent, and we call it a holomorphic sheaf.

Given serious structural problems, the pavilion had been closed to the public for a number of years.

Given a linear operator, not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of, and we say that is closable.

Given the depopulation of Edinburgh's Old Town in the early part of the 20th century, many neighbouring church buildings were closed and their congregations united with Greyfriars, including the New North Church and Lady Yester's Church.

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 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 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 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 a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that ( x < sub > α </ sub >) is eventually in all members of the base containing this putative limit.

Given source XML containing at least

Given that any proposition containing conjunction, disjunction, and negation can be equivalently rephrased using conjunction and negation alone ( the conjunctive normal form ), we can now handle any compound proposition.

* Convex hull: Given a set of points, find the smallest convex polyhedron / polygon containing all the points.

# Playfair's axiom: Given a line and a point not on, there exists exactly one line containing such that

Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra A < sup > op </ sup > ( the opposite ring with the same action by K since the image of K → A is in the center of A ).

Given a model M of a Skolem theory T, the smallest substructure containing a certain set A is called the Skolem hull of A.

Given then a normal extension L of K, with automorphism group Aut ( L / K ) = G, and containing α, any element g ( α ) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates.

Given some pa formula containing N variables, this decision procedure requires simplifying 2 < sup > N </ sup > PA formulae.

Given a base for a topology and a point x, the set of all basis sets containing x forms a local base at x.

* Given a graph, is there a cycle containing a given edge?

Given a finite set X ( of elements called points ) and integers k, r, λ ≥ 1, we define a 2-design ( or BIBD, standing for balanced incomplete block design ) B to be a family of k-element subsets of X, called blocks, such that the number r of blocks containing x in X is not dependent on which x is chosen, and the number λ of blocks containing given distinct points x and y in X is also independent of the choices.

Given a list of n points, the following algorithm uses a median-finding sort to construct a balanced k-d tree containing those points.

Given the limited success in containing the Ghoul so far, he is charged with the difficult and thankless duty of simply holding out as long as he can.

Given a first-order logic formula containing a predicate, circumscribing this predicate amounts to selecting only the models of in which is assigned to true on a minimal set of tuples of values.