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Given and other
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 and side
Given that spinning a Dyson Sphere would result in the atmosphere pooling around the equator, the Ringworld removes all the extraneous parts of the structure, leaving a spinning band landscaped on the sun-facing side, with the atmosphere and inhabitants kept in place through centrifugal force and 1000 mile high perimeter walls ( rim walls ).
Given a desired backbone conformation, side chain packing can be designed using variations of the dead-end elimination algorithm ; however, attempts to design proteins of novel folds have difficulty using this method due to an absence of plausible backbone models.
Given the results of Schweitzer ’ s study, the residents of nearby cities ( Buttonwillow, especially ) are at the highest risk of experiencing side effects from the spills.
Given: side a = 20, side c = 24, and angle C = 40 °
Given that the tank was reaching its viable limit, to avoid a corresponding weight increase, the appliqué steel plates were removed from its side armor, which instead had its base thickness increased to.
Given his expertise and background in understanding the voice and in understanding the wonders of the human voice, it is not surprising, maybe, that his approach in his early years was away from the theatrical side and more focused on singers.
Given was sent on loan again in January 1996, this time to Division One side Sunderland.
Given the 2 coefficients on one side of a classical turning point, the 2 coefficients on the other side of a classical turning point can be determined by using this local solution to connect them.
Given of a monoid M of every strings over some alphabet, one may define sets that consist of formal left or right inverses of elements in S. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating.
Given his adept use of angle, depth, and pace, his opponents often have difficulty breaking down one particular side.
Given the 2 coefficients on one side of the classical turning point, the 2 coefficients on the other side of the classical turning point can be determined by using this local solution to connect them.
Given that the angle of incidence is larger than the critical angle ( with respect to the normal of the surface ) and the refractive index is lower on the other side of the boundary relative to the incident ray, TIR will occur and no light will be able to pass through.
Given two groups of people, the interaction between them is such that a behavior X from one side elicits a behavior Y from the other side, The two behaviors complement one another, exemplified in the dominant-submissive behaviors of a class struggle.
Given that menudo is time and labor intensive to prepare — the tripe takes hours to cook ( or else it is extremely tough ), and many ingredients and side dishes ( such as salsa ) need to be cut and cooked — the dish is often prepared communally and eaten at a feast ; documents from the Works Progress Administration indicate that in the 1930s, among ( migrant ) workers in Arizona, menudo parties were held regularly to celebrate births, Christmas, and other occasions.
Given a square of 2 < sup > 2k </ sup > cells, 2 < sup > k </ sup > on a side, at the kth level of the tree, the hash table stores the 2 < sup > k-1 </ sup >- by-2 < sup > k-1 </ sup > square of cells in the center, 2 < sup > k-2 </ sup > generations in the future.
* Given two circles which will serve as top and bottom for a cup of given height, find the shape of the side wall of the cup so that the side wall has minimal area.
Given the orchestra plan, Welk, handed a pencil, drew a line on he right side of the plan, announcing, " loose them !".
Peter Mason writes: " Given the measures that the West Indian authorities had taken to ensure that no black man would ever captain a regional side, it was a great irony and a huge source of delight to Constantine that he should be the man to lead the team at the moment of their greatest achievement so far.
Given the fine ridges to either side, the summit of St Sunday Crag is surprisingly level and green.

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