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

) Given the exclusionary and ancestral conception of citizenship held by Greek city-states, a relatively large portion of the population took part in the government of Athens and of other radical democracies like it.

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

Given MLB's refusal to consider expansion, Kirksey, Cullinan, Smith, and Hofheinz joined forces with would-be owners from other cities and announced the formation of a new league to compete with the established National and American Leagues.

Given that non-violence has priority, all other principles yield to it whenever there is a conflict.

Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given his record, it is not surprising that the other Party leaders were suspicious of Beria's motives.

Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.

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 this reliance on literary sources, it is doubtful whether Nostradamus used any particular methods for entering a trance state, other than contemplation, meditation and incubation.

" Given, a certain population, with various needs and powers of production, in possession of certain lands and other sources of material: required, the mode of employing their labour which will maximize the utility of their produce.

Given that Dada created its own means for interpreting its products, it is difficult to classify alongside most other contemporary art expressions.

Given this and other evidence, many scholars agree that Peter was martyred in Rome under Nero, although some scholars argue that he may have been martyred in Palestine.

" 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 role that many social groups have in the development of public policy and the extensive connections between state bureaucracies and other institutions, it has become increasingly difficult to identify the boundaries of the state.

Given two points ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) and ( x < sub > 2 </ sub >, y < sub > 2 </ sub >), the change in x from one to the other is ( run ), while the change in y is ( rise ).

Given any two cards from the deck, there will be one and only one other card that forms a set with them.

Given the symmetry of circularly polarized light, we could have in fact selected any other two orthogonal components and found the same phase relationship between them.

Given that France maintained a strong standing army, and that Prussia and the other German states would need weeks to mobilise their conscript armies, the French held the initial advantage of troop numbers and experience.

Given an initial allocation of goods among a set of individuals, a change to a different allocation that makes at least one individual better off without making any other individual worse off is called a Pareto improvement.

Given the circumstances, the Ghent Altarpiece is a difficult work to use for comparison when assessing other attributions, especially as several other artists from the brothers ' workshops probably worked on it as well.

# Given any elements a and a ' of G, if a ~ a ', then a < sup >− 1 </ sup > ~ a ' < sup >− 1 </ sup > ( this can actually be proven from the other four, so is strictly redundant ).

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.

Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category.

* Given a category C with finite coproducts, a cogroup object is an object G of C together with a " comultiplication " m: G → G G, a " coidentity " e: G → 0, and a " coinversion " inv: G → G, which satisfy the dual versions of the axioms for group objects.

Given some " primitive formulas " such as PM's primitives S < sub > 1 </ sub > V S < sub > 2 </ sub > OR, ~ S ( negation ) one is forced to define the axioms in terms of these primitive notions.

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

That is, no exception is known to the following: Given two large cardinal axioms A1 and A2, one of three ( mutually exclusive ) things happens:

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms.

** Closure axiom for addition: Given two integers a and b, their sum, a + b is also an integer.

** Closure axiom for multiplication: Given two integers a and b, their product, a · b is also an integer.

Given the short distances of super special stages compared to the regular special stages and consequent near-identical times for the frontrunning cars, it is very rare for these spectator-oriented stages to decide rally results, though it is a well-known axiom that a team can't win the rally at the super special, but they can certainly lose it.

#( Affine axiom of parallelism ) Given a point A and a line r, not through A, there is at most one line through A which does not meet r.

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

Given the Cantor – Dedekind axiom, this algorithm can be regarded as an algorithm to decide the truth of any statement in Euclidean geometry.

#( TR 3 ) Given a map between two morphisms, there is a morphism between their mapping cones ( which exist by axiom ( TR 1 )), that makes everything commute.

: Given any family of nonempty sets, their Cartesian product is a nonempty set.

Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

" Given that Thomas Aquinas died six years before Albertus Magnus ' death, this legend as stated is unlikely.

Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

Given its " blue-water " bias, the navy is even less inclined to become involved in counterdrug operations than the army or air force.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

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.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is

Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given the then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.

Given some encrypted data (" ciphertext "), the goal of the cryptanalyst is to gain as much information as possible about the original, unencrypted data (" plaintext ").

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

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 such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.