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 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.

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.

" 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 these changes, the classical terms " system ", " expansion " and " peripheral " no longer have the same connotations.

Given the system ( corresponding to )

Given the limited budget for schools, it is claimed that a voucher system would weaken public schools while at the same time not necessarily providing enough money for people to attend private schools.

Given that the bulk of family chores fell to women, and that they were five times more likely than men to head a single-parent family, the shortcomings of Finland's child day-care system affected women more than it did men.

Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.

Given a syscall that does nothing, a full round-trip under BSD would require about 40μs, whereas on a user-space Mach system it would take just under 500μs.

Given the diversity of functions performed by neurons in different parts of the nervous system, there is, as expected, a wide variety in the shape, size, and electrochemical properties of neurons.

Given the increasing number of scientists who study the nervous system, several prominent neuroscience organizations have been formed to provide a forum to all neuroscientists and educators.

Given the complexity of NLP problems, it is often difficult to predict performance only on the basis of glass-box evaluation, but this type of evaluation is more informative with respect to error analysis or future developments of a system.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

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.

Given that the authors were the primary developers of the Unix operating system, it is natural that QED had a strong influence on the classic UNIX text editors ed, sed and their descendants such as ex and sam.

Given the opportunity, for example during an illness that suppresses the immune system, the virus is reactivated and travels to the end of the nerve cell, where it causes the symptoms described above.

: Given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates.

Given Ω microstates at a particular energy, the probability of finding the system in a particular microstate is p = 1 / Ω.

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system

) Given that the telephone system nominal voiceband ( including guardband ) is 4, 000 Hz, the required digital sampling rate is 8, 000 Hz ( see Nyquist rate ).

Given the structural similarities between bacterial flagella and bacterial secretory systems, it is thought that bacterial flagella may have evolved from the type three secretion system ; however, it is not known for certain whether these pores are derived from the bacterial flagella or the bacterial secretory system.

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.

Given a testing procedure E applied to each prepared system, we obtain a sequence of values