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 ).

Similarly, Hare refers to “ the crude caricature of act utilitarianism which is the only version of it that many philosophers seem to be acquainted with .” Given what Bentham says about second order evils it would be a serious misrepresentation to say that he and similar act utilitarians would be prepared to punish an innocent person for the greater good.

Given the costs associated with such policies ( e. g., global climate disruption, widespread habitat loss and species extinctions, consumption of natural resources, pollution, urban congestion, intensifying competition for remaining resources, and increasing disparity between the wealthy and the poor ), some economists, scientists, and philosophers have questioned the biophysical limits to growth, and the desirability of continuous growth.

Given that this etymology does not appear in the usual scholarly works on etymology, this claim is usually dismissed as a false etymology.

Given its lack of a D ' Oyly Carte or Savoy connection, Thespis has a tenuous claim to be a " Savoy Opera.

Given this inherent distortion, it is not unusual for a television or video projector to truthfully claim a linear contrast ratio ( difference between darkest and brightest output levels ) of 1000: 1 or greater, equivalent to 10 bits of audio precision even though it may only accept signals with 8-bit precision and use an LCD panel that only represents 6 or 7 bits per channel.

Given the failure of Walter Raleigh to establish a colony at Roanoke Island in 1584 and the successful settlement at Jamestown in 1607 and on learning that Samuel de Champlain had sailed into the St. Lawrence to initiate the settlement of New France, pressure was mounting to lay claim to the resource rich New World.

Given the lack of evidence of many Freudian hypotheses, some scientific researchers proposed the existence of unconscious mechanisms that are very different from the Freudian ones.

Given two numbers in scientific notation,

Given the many available models, it appears that the material removal mechanism in EDM is not yet well understood and that further investigation is necessary to clarify it, especially considering the lack of experimental scientific evidence to build and validate the current EDM models.

Given the age of the institution, many of the collections have great historical as well as scientific value, such as specimens collected by Darwin.

Given the possible relationship between awareness and cognition, and consciousness, this theory contributes an interesting perspective in the philosophical and scientific dialogue of awareness and living systems theory.

Given historical animosity between science and religion ( Haque, 2001 ), controversy surrounding the scientific study of religion is expected.

Given this shortfall, incel does not appear to be taken seriously by scientific and academic communities.

Given its toxicity, this may be folklore rather than scientific fact.

Given that she is a scientific genius, some or all of her powers may be generated by her costume ( despite having shown, during her Dark Avengers tenure, the ability to project concussive blasts while in civilian clothing ).

Given the meaninglessness of theistic claims, Ayer opined that there was " no logical ground for antagonism between religion and natural science ", as theism alone does not entail any propositions which the scientific method can falsify.

Given that he's more of a military man than a scientific one, his interests tend to conflict with the other main characters.

" Given the complete lack of citations to this research in GeoRef, JSTOR, Web of Knowledge, and other scientific bibliographic databases, neither the data collected from nor the interpretations and conclusions made as a result of this expedition very likely have been publicly reported in a scientific venue where their credibility can be openly evaluated.

Given the fact that no science can today be mastered by a single individual, Lonergan advocated sub-division of the scientific process in all fields.

Given the serious methodological flaws in their survey studies and obituary study, it is reasonable to conclude that the Cameron group's papers would have been rejected by more prestigious scientific journals.

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.