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

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 CIA's purposeful destruction of most records, its failure to follow informed consent protocols with thousands of participants, the uncontrolled nature of the experiments, and the lack of follow-up data, the full impact of MKUltra experiments, including deaths, will never be known.

Given this uncertainty, between April – May 2006 at the JINR, a FLNR – PSI team conducted experiments probing the synthesis of this isotope as a daughter in the nuclear reaction < sup > 242 </ sup > Pu (< sup > 48 </ sup > Ca, 3n )< sup > 287 </ sup > Fl.

Given that there was no tradition of writing poetry in the vernacular, these experiments were more radical than those in fiction writing and also less easily accepted by the reading public.

Given the results from Venera 4, the Venera 5 and Venera 6 landers contained new chemical analysis experiments tuned to provide more precise measurements of the atmosphere's components.

Given the results from Venera 4, the Venera 5 and Venera 6 landers contained new chemical analysis experiments tuned to provide more precise measurements of the atmosphere's components.

Given that it seems to require extraordinary visuo-spatial abilities and memory, this form of chess has led to considerable research in psychology, starting with the research of Alfred Binet in 1893, continuing with the work of chess grandmaster and psycho-analyst Reuben Fine in 1965, and culminating in the last two decades with several scientific articles describing experiments on the psychology of blindfold chess.

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 a torus T ( not necessarily maximal ), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is, Fix a maximal torus in G ; then the corresponding Weyl group is called the Weyl group of G ( it depends up to isomorphism on the choice of T ).

Given the popularity of the first-person shooter genre, it was assumed that all tactical-level military gameplay necessarily involved individual combat action.

Given a 3-manifold ( not necessarily compact ) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups.

Given the previous two declarations, the following triple requires that < tt > ex: John </ tt > is necessarily a < tt > foaf: Person </ tt >, and < tt > ex: CompanyX </ tt > is necessarily a < tt > foaf: Organization </ tt >:

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 a timelike unit vector field ( which can be interpreted as a family or congruence of nonintersecting world lines, not necessarily geodesics ), Raychaudhuri's equation can be written

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

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 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 " mappings " of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take, based on a set of " rules ", each of the form:

Given that they were illegally confined, the Africans were entitled to take whatever legal measures necessary to secure their freedom, including the use of force.

Given the quick development of his illness, Alexander's doctors considered amputating his leg, but none wished to take responsibility for the act.

Given a rebbe's physical awareness of God, and the Rebbe's transcendent perception of Godliness, many hasidim take special care to observe the specific and sometimes minute practices of their Rebbe.

Given the team's poor results in the last few years, which she attributes to Coach D ' Amato's " old-school methods ", she attempts to take a more hands-on approach to the team, including bringing in an innovative Offensive Coordinator Nick Crozier.

Given this difference, it is necessary that logistic regression take the natural logarithm of the odds ( referred to as the logit or log-odds ) to create a continuous criterion.

Given any set of points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian.

Given the state of the pharmacopoeia, and patients ' demands for something to take, physicians began making " blunderbuss " concoctions of various drugs, proven and unproven.

Given that manufacturers are regarded as authorities and that most sales do not happen on manufacturer Web sites, best-in-class manufacturers take their best content and enable other retailers or dealers to publish the information on their sites.

Given that Emmerich only spoke the Westphalian dialect, Brentano could not transcribe her words directly, and often could not even take notes in her presence.

Given the abnormalities of the microvasculature and other side effects of diabetes, these wounds take a long time to heal and require a specialized treatment approach for proper healing.

Given the available manpower, Anderson could not take advantage of all of his 60 guns.

Given to Irina in 2009 as part of he divorce settlement, she was approached on David Geffen's behalf by broker Merle Wood, with Geffen paying US $ 300 million to take ownership in 2011.

Given the small number of NAION events with PDE5 use ( less than 1 in 1 million ), the large number of users of PDE5 inhibitors ( millions ) and the fact that this event occurs in a similar population to those who do not take these medicines, the FDA concluded that they were not able to draw a cause and effect relationship, given these patients underlying vascular risk factors or anatomical defects.

Given the chaotic situation in China, the corruption Doihara methodically spread didn't take long to reach the very top.

Given a choice of new classes she could take, she chose communications, in part because her father was a journalist and her mother an actress.

Given reasonable estimates of the time needed to compute the hash, this would take about 1 second to find.

Given SRK's performance, it is all the more important to take note of Saif, whose Rohit Patel will be remembered just as much as his Sameer in Dil Chahta Hai.

Given Japan had surrendered to the Allies at the end of World War II, the Japanese forces in Indochina stepped aside and allowed nationalist groups to take over public buildings in most of the major cities.

Given the attention lavished on the Kuomintang ( KMT ) by both U. S. President Franklin D. Roosevelt and the media, especially Henry Luce's Time Magazine, the U. S. public was slow to take notice of the Communists ' rise in importance in China.

Given Fascism, Communism and ethnic cleansing in the Balkans, it's not hard to foresee that the neo-nationalist resurgence already under way in parts of Europe will at some point take a violent form.

Given the absence of any progress with the UK, Mauritius has decided to " internationalize " the dispute and take up the matter at all appropriate legal and political forums.