Under this assumption the space group must be Af and the following are the positions of the atoms in the unit cell.

Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number.

Under the decentralization plan, regulatory responsibilities were transferred from the central government to mayors and other municipal officials on the assumption that local officials would provide better oversight.

Under the monopsonistic assumption, an appropriately set minimum wage could increase both wages and employment, with the optimal level being equal to the marginal productivity of labor.

Under the assumption of perfect competition, supply is determined by marginal cost.

Under this assumption, half the round-trip delay is the propagation delay to be compensated.

Under an assumption of constant gravity, Newton's law of universal gravitation simplifies to F = mg, where m is the mass of the body and g is a constant vector with an average magnitude of 9. 81 m / s < sup > 2 </ sup >.

Under the assumption that the universe is homogeneous and isotropic, the curvature of the observable universe, or the local geometry, is described by one of the three " primitive " geometries ( in mathematics these are called the model geometries ):

Under the assumption of replica symmetry, the free energy is given by the expression:

Under this assumption, some American groups created programs which aimed to increase the self-esteem of students.

Under assumption, when U equals U * and λ equals unity, expected real wages would increase with labor productivity.

Under this assumption, the doctor, who is given n and m, wishes to estimate k, the number of children where both parents have agreed to participate.

Under the assumption that the Earth was composed of concentric ellipsoidal shells of uniform density, Clairaut's theorem could be applied to it, and allowed the ellipticity of the Earth to be calculated from surface measurements of gravity.

Under the assumption of economic rationality, it is the utility of its least urgent possible use from the best feasible combination of actions in which its use is included.

Under the open world assumption, if a statement cannot be proven to be true with current knowledge, we cannot draw the conclusion that the statement is false.

Under the assumption that the Ottoman viceroy wanted a Catholic Patriarch for the Coptic Catholics in 1824, the Pope erected the Patriarchate of Alexandria from the Apostolic Vicariate of Syria, Egypt, Arabia and Cyprus but it was basically titular.

Under this assumption, the reflected particles in the two-body case will not fully compensate the shadowing effect, because the reflected flux is weaker than the incident flux.

Under this assumption strings behave as one-dimensional objects and obey the Nambu-Goto action, which is classically equivalent to the Polyakov action that defines the bosonic sector of superstring theory.

Under the assumption of naturalism and indeterminism, where there only exists the natural world and that the natural world is indeterministic — events are not predetermined ( e. g., for quantum mechanical reasons ) and any event has a probability assigned to it — no event can be determined by a physical organism's perceived free will, and nor can any event be strictly determined by anything at all.

Under such an assumption, the Declaration of Independence would declare that government existed primarily for the reasons Locke gave, and some have extended that line of thinking to support a conception of limited government.

Under the assumption of logistic growth, resource limitation does not constrain individuals ’ reproductive rates when populations are small, but because there are few individuals, the overall yield is small.

Under this assumption the equation above can be rearranged into:

Under the assumption V = L, not all projective sets have the perfect set property or the property of Baire.

Under this assumption of " generic intersection ", the number of regions can also be determined in a non-inductive way, using the formula for the Euler characteristic of a connected planar graph ( viewed here as a graph embedded in the 2-sphere S < sup > 2 </ sup >).

Under the Riemann hypothesis, the error term can be reduced:

Under the Riemann hypothesis, the zeros of the eta function would be located symmetrically with respect to the real axis on two parallel lines, and on the perpendicular half line formed by the negative real axis.

Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points.

Under the null hypothesis of no autocorrelation, this statistic is

Under null hypothesis that the sample comes from the hypothesized distribution F ( x ),

Under a stronger hypothesis they showed that for infinitely many n at least two of n, n + 2, n + 4, and n + 6 are prime.

Under the Kurgan hypothesis, there are two possibilities for how the early Anatolian speakers could have reached Anatolia: from the north via the Caucasus, and from the west, via the Balkans, the latter of which is considered somewhat more likely by Mallory ( 1989 ) and Steiner ( 1990 ).

Under the contact hypothesis, several models have been developed.

Under this null hypothesis, the data are considered unlikely ( with a fair coin, the probability of this is 3 % and the result would be even more unlikely if the coin were biased in favour of tails ).

Under the null hypothesis that model 2 does not provide a significantly better fit than model 1, F will have an F distribution, with ( p < sub > 2 </ sub > − p < sub > 1 </ sub >, n − p < sub > 2 </ sub >) degrees of freedom.

Under the umbrella of the Plate hypothesis, the following sub-processes, all of which can contribute to permitting surface volcanism, are recognised:

Under this alternative hypothesis the disks and arms are modified version of the original stellar disk of the transformed spiral galaxy, and similarly, small remnants of disks and arms are embedded within " harassed " dEs.

# Under the null hypothesis the distributions of both groups are equal, so that the probability of an observation from one population ( X ) exceeding an observation from the second population ( Y ) equals the probability of an observation from Y exceeding an observation from X, that is, there is a symmetry between populations with respect to probability of random drawing of a larger observation.

# Under the alternative hypothesis the probability of an observation from one population ( X ) exceeding an observation from the second population ( Y ) ( after exclusion of ties ) is not equal to 0. 5.

Under this hypothesis, separate parts of reality would have wildly different characteristics.

Under the Mandibulata hypothesis, Myriapoda is the sister taxon to Pancrustacea, a group comprising the Crustacea and Hexapoda.

Under the Atelocerata hypothesis, Hexapoda is the closest, whereas under the Paradoxopoda hypothesis, Chelicerata is the closest.

Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create confidence intervals and conduct hypothesis tests about the population parameters.

Under the " packed planetary systems " hypothesis, which predicts that planetary systems form in such a way that the system could not support additional planets between the orbits of the existing ones, the gap would be expected to host a planet.

Under this hypothesis, both male and female Neanderthals participated in the single main occupation of hunting big game that flourished in Europe in the ice age like bison, deer, gazelles and wild horses.

Under the null hypothesis the expected distance ( r < sub > e </ sub > ) between the organisms ( measured as the nearest neighbour's distance ) with a known constant density ( ρ ) is

Under the null hypothesis of no difference between population means ( and assuming that standard ANOVA regularity assumptions are satisfied ) the sums of squares have scaled chi-squared distributions, with the corresponding degrees of freedom.

Under such an hypothesis, the aether should have been observed.

Under this interpretation, the physical symbol system hypothesis asserts merely that intelligence can be digitized.