Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

Assuming the lower figure for the big blast and one shot estimated by the Japanese at 10 megatons, a conservative computation is that the 24 announced tests produced a total yield of at least 60 megatons.

Assuming the weather is halfway decent that day, hundreds of thousands of persons will mass along this thoroughfare as President John F. Kennedy and retiring President Dwight D. Eisenhower leave Capitol Hill following the oath-taking ceremonies and ride down this historic ceremonial route.

Assuming that the pixel color is expressed using straight ( non-premultiplied ) RGBA tuples, a pixel value of ( 0. 0, 0. 5, 0. 0, 0. 5 ) implies a pixel which has 50 % of the maximum green intensity and 50 % opacity.

Assuming that the ion is accelerated during a very short interval, the ion can be assumed to be travelling at constant velocity.

Assuming that 10 – 15 % of bread is thrown away as stale, if it could be made to stay fresh another 5 – 7 days then perhaps 2 million tons of flour per year would be saved.

Assuming average population densities of 0. 25 animals per km² in regions where it is known to be common or abundant, and 0. 02 per km² elsewhere, and with a total area of occupancy of 327, 000 km², a total population estimate of approximately 28, 000 is suggested.

Assuming a fixed sample size, the technique gives more accurate results when most of the variation in the population is within the groups, not between them.

:< u > The Copenhagen Interpretation </ u >: Assuming wave functions are not real, wave-function collapse is interpreted subjectively.

Assuming the Axiom of choice, is regular for each α.

Let e be the error in b. Assuming that A is a square matrix, the error in the solution A < sup >− 1 </ sup > b is A < sup >− 1 </ sup > e.

Assuming that the strength of the cable is proportional to its density per unit length, the weight, w, per unit length of the chain can be written T / c, where c is constant, and the analysis for nonuniform chains can be applied.

Assuming that the rule is known, one can derive the function that is used above in the classical heat calculation with respect to pressure.

Assuming that translation is not used transformations can be combined by simply multiplying the associated transformation matrices.

Assuming we can define at least one nonreal complex number, however, a complex number is definable if and only if both its real part and its imaginary part are definable.

Assuming that the visible material makes up only a small part of the cluster is the most straightforward way of accounting for this.

Assuming Chavez was approximately 10 years old in 1979, that places the events of Debt of Honor to be in or near 2000, although the overall technology presented in the novel and its successive novels Executive Orders and Rainbow Six ( especially in regards to the sophistication of internet technology ) are more consistent with the mid-1990's.

Assuming that the routing tables are consistent, the simple algorithm of relaying packets to their destination's next hop thus suffices to deliver data anywhere in a network.

Assuming the validity of Bell's theorem, any deterministic hidden-variable theory which is consistent with quantum mechanics would have to be non-local, maintaining the existence of instantaneous or faster-than-light relations ( correlations ) between physically separated entities.

Assuming you catch the balls at a consistent level, then the timing of when the balls land is already determined.

Assuming that the town would be in full foliage, the company showed up for outdoor shots on September 27, 1954.

Assuming that the town would be in full foliage, the company showed up for outdoor shots on September 27, 1954.

Assuming that the initial " N " may have been intended to stand as a negation, the name would mean " without order " or " no order ".

( Assuming the axiom of choice ), the following are equivalent.

Assuming the axiom of choice, every other infinite cardinal number is either regular or a ( weak ) limit.

Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe.

Assuming the axiom of choice, the union of a countable set of countable sets is itself countable.

Assuming the axiom of choice, is the first infinite cardinal which is singular ( the first infinite ordinal which is singular is ).

* Assuming that an individual had no choice in who, when and where to come into being: How are the choices of existence determined by what he is?

Assuming that the natural transformations for the adjunctions are given, all of these formulations allow for an explicit construction of the necessary data, and no choice principles are needed.

Assuming that everything is found to be acceptable, the FDA will decide if the NDA will get a standard or accelerated review and communicate the acceptance of the application and their review choice in another communication known as the 74-day letter.

Assuming that the variance is not infinite and that the central limit theorem applies to the sample then using the delta method, the variance is

Assuming that all atoms in a crystal vibrate with the same frequency ν, the average thermal energy can be estimated using the equipartition theorem as

Assuming that they are continuous, the order of differentiation does not matter ( Clairaut's theorem ).

Assuming that there are not multiple non-trivial roots of we have the " exact formula " by the residue theorem:

Assuming can be decided by a TM using cells, and following from the Immerman – Szelepcsényi theorem follows that can also be determined by a TM ( which we will call ) using cells.

Assuming that an interaction will occur ( with 100 % probability ) if the projectile hits the solid, and not at all ( 0 % probability ) if it misses, the total interaction probability for the single projectile will be the ratio of the area of the section of the solid ( the cross section, represented by ) to the total targeted area.

Assuming a compression algorithm that shrinks the input data by a factor of CF, the bit rate and video size would equal to:

Assuming Ignatius ' view was the Apostolic teaching and practice, the line of succession was unbroken and passed through the four ancient Patriarchal sees ( those local churches known to be founded by apostles ), Rome, Jerusalem, Antioch and Alexandria.

Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiation and multiply by the constant:

), the total human biomass is the average weight multiplied by the current human population of approximately 6. 5 billion ( see, e. g., ): Assuming 60 – 70 kg to be the average human mass ( approximately 130 – 150 lb on the average ), an approximation of total global human mass of between 390 billion ( 390 × 10 < sup > 9 </ sup >) and 455 billion kg ( between 845 billion and 975 billion lb, or about 423 million – 488 million short tons ).

For instance, raising the note B to B is equal to the note C. Assuming all such equivalences, the complete chromatic scale adds five additional pitch classes to the original seven lettered notes for a total of 12 ( the 13th note completing the octave ), each separated by a half-step.

Assuming that it was not abandoned it would certainly have been closed down in 425 by the Christian emperor Theodosius II when he prohibited the worship of the old Roman and Greek gods.

Assuming one has overcome the mind's initial flight response in those first torturous minutes, the body fights back by invoking a " blood rush ".

Assuming all references are to the same person, as some ( but by no means all ) of the Church fathers did, the earliest reference to him is the canonical Acts of the Apostles, this is his only appearance in the New Testament.

Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with as above by setting the interval length short enough to exclude any of the discrete masses.

Assuming Odo commissioned the tapestry, it was probably designed and constructed in England by Anglo-Saxon artists ( Odo's main power base being by then in Kent ); the Latin text contains hints of Anglo-Saxon ; other embroideries originate from England at this time ; and the vegetable dyes can be found in cloth traditionally woven there.

Assuming this was the case, the actual physical work of stitching was most likely undertaken by skilled seamsters.

Assuming that the transcriptions are not intended to be precise, only 11 of the 26 — Bravo, Echo, Hotel, Juliet ( t ), Kilo, Mike, Papa, Quebec, Romeo, Whiskey, and Zulu — are given English pronunciations by all these agencies, though not always the same English pronunciations.

Assuming night-cowl, and attended by a single maid,

Assuming colocation with the biblical Mount Zion, its southern section would have been walled at the beginning of the 2nd millennium BCE, in around 1850 BCE, by Canaanites who established a settlement there ( or in the vicinity ) named Jebus.

Assuming constant pressure and volume, the thermodynamic properties of the cell are related strictly to the behavior of its emf by:

Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, and, are related to the angle of deflection in the system of the center of mass by

Assuming that has an inverse in a neighbourhood of and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at and have a derivative given by the above formula.

Assuming that eye movements are closely linked to the current focus of attention, language processing can be studied by monitoring eye movements while a subject is presented auditorily with linguistic input.