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 the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

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 the Saint Lawrence's heavy traffic and years of mis-use, the ecoregion is negatively effected by high pollution which in turn has resulted in diminished local populations of native fauna.

Given that the oldest known waterfowl, Vegavis iaai, dates from the Late Cretaceous, galliform ancestors must also have roamed the Earth contemporaneously with animals such as Tyrannosaurus rex.

Given that the Characiformes originated slightly more than 115 mya, the Anostomoidea probably diverged from their characiform relatives the Late Cretaceous and diversified thereafter ; a Paleogene origin of the Anostomidae is thus most likely, with present-day subfamiilies ( whatever eventually will be accepted ) having diverged by the start of the Neogene.

Given the bias of the Victorian age, female students were denied direct study in anatomy and could not attend drawing classes with live models ( who were often prostitutes ) until a decade later.

Given the military strength of the Great Powers, these interventions were nearly always decisive and quickly ended the civil wars.

Given that the ColecoVision could produce near arcade-quality ports, industry magazines like Electronic Games were unanimous in their enthusiasm over the console.

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 their speed and maneuverability, clippers frequently mounted cannon or carronades and were used for piracy, privateering, smuggling, or interdiction service.

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.

Along with a hand-picked escort led by Given Campbell, Davis and his wife were captured on May 10, 1865, at Irwinville in Irwin County, Georgia.

Given the realities of the Cold War, German reunification and recovery of lost territories in the east were not realistic goals as both of Stalin's notes specified the retention of the existing " Potsdam "- decreed boundaries of Germany.

Given that Fine Gael normally received 25 % of the election result, and were reduced to third place this was an achievement in itself.

Given that most French supporters of phrenology were liberal, left-wing or socialist, it was an objective of the social elite of France who held a restrained vision of social change that phrenology remain on the fringes.

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 medieval medicine's main theory of humorism, spices and herbs were indispensable to balance " humors " in food, a daily basis for good health at a time of recurrent pandemics.

Given that there were 25, 000 people working on Shuttle operations, the payroll alone was the Shuttle's single biggest operating cost.

Given the versatility of both vegetables, they were usually boiled or steamed, and eaten by itself or as ingredients in soups and strews.

Given two school classes, one with 20 students, and one with 30 students, the grades in each class on a test were:

Given inefficiencies in the heat transfer, the actual NERVA designs were planned to produce about 5 GW, which would make them the largest nuclear reactors in the world.

Given the difference between their roles and emphases, some scholars have speculated that the interactions between the Æsir and the Vanir reflect the types of interaction that were occurring between social classes ( or clans ) within Norse society at the time.

Given that many businessmen and bankers were terrified of Bryan's populist rhetoric and support for ending the gold standard, Hanna had few problems in raising record amounts of money.

Given Rickover's single-minded focus on naval nuclear propulsion, design and operations, it came as a surprise to many when in 1982, near the end of his career, he testified before the U. S. Congress that, were it up to him, he " would sink them all.

Given their long range, high survivability and ability to carry many medium-and long-range nuclear missiles, submarines were credible and effective means for full-scale retaliation even after a massive first strike.

BONUS: " Given a pair of resistors, give their equivalent resistance if they were connected in parallel with each other.

Given the biogeography and appearance of the subgenus Olor it seems likely that these are of a more recent origin, as evidence shows by their modern ranges ( which were mostly uninhabitable during the last ice age ) and great similarity between the taxa.