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 the generalized Riemann hypothesis, it can be shown to take time

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 meager point-set hypotheses, namely sobriety, this is completely accurate — it is possible to recover a sober space from its associated site.

Under this definition, plane tilings and space tilings ( honeycombs ) are considered to be polytopes, and are sometimes classed as apeirotopes because they have infinitely many cells ; tilings of hyperbolic spaces are also included under this definition.

Under both schemes the old rural intersection just outside the Potsdam Gate, and the Octagon ( Leipziger Platz ) just inside, were to be joined together to create a long rectangular space, with a gargantuan edifice standing in the middle of it.

Under liberal democratic systems, privacy creates a space separate from political life, and allows personal autonomy, while ensuring democratic freedoms of association and expression.

Under space law, the only internationally recognized authority for naming celestial bodies is the International Astronomical Union ( IAU ).

Under his leadership, the Examiner devoted 24 percent of its space to crime, presenting the stories as morality plays, and sprinkled adultery and " nudity " ( by 19th century standards ) on the front page.

Under the frequentist interpretation of probability, Bayes ' rule is a general relationship between and, for any events, and in the same event space.

Under the new regulations, smoking rooms are allowed, but are subjected to very strict conditions: they may occupy at most 20 % of the total floor space of the establishment and their size may not be more than 35 m² ; they need to be equipped with separate ventilation which replaces the full volume of air ten times per hour ; the air pressure of the smoking room must constantly be lower than the pressure in the contiguous rooms ; they have doors that close automatically ; no service can be provided in the smoking rooms ; cleaning and maintenance personnel may enter the room only one hour after it was last used for smoking.

Under Project Horizon in the late 1950s, use of spent missile stages for building a manned orbital space station had been proposed, and this general concept was picked up under AAP.

Under the EUL, St. John Properties has entered into a long-term ground lease of federally owned property for the development of commercial office space and research-and-development ( R & D ) space.

Under the current Russian space program Baikonur remains a busy space port, with numerous commercial, military and scientific missions being launched annually.

For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes dot product of vectors.

More space was needed to meet the academic demand, and so in 1948, Mr F. Bray, Under Secretary of the Ministry of Education, opened the converted Downings Warehouse.

" Under Florida law, a convicted murderer could not share cell space with another prisoner before his execution, but another convicted murderer was already awaiting execution at Raiford.

Under this concept, it has become a world leader in the development of low-noise receivers ; large parabolic-dish antennas ; tracking, telemetry, and command systems ; digital signal processing ; and deep space navigation.

Under normal conditions, medial and vestibular nuclei, cortical, thalamic, and cerebellar centers coordinate things such as head and eye movement, and orientation in space.

Under a common house rule, a player who lands on this space wins a jackpot consisting of a collection of fines and taxes that have been placed in the center of the board ( rather than having been paid to the Banker, as they would under the official rules ).

Under this definition, C is called a covering space of X.

Under normal conditions, one can thus calculate the volume of phase space occupied by the gas, by dividing Equation 1 by N !.

In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary ( or antiunitary ) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.

Under this duality, every compact Hausdorff space is associated with the algebra of continuous complex-valued functions on, and every commutative C *- algebra is associated with the space of its maximal ideals.