Assuming that the image has been pre-smoothed by Gaussian smoothing and a scale space representation at scale has been computed, we can require that the gradient magnitude of the scale space representation, which is equal to the first-order directional derivative in the-direction, should have its first order directional derivative in the-direction equal to zero

* Assuming small vibration amplitudes ( which is generally true in MFM measurements ), to a first-order approximation, the resonance frequency can be related to the natural frequency and the force gradient.

), 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 ).

Using the quasi-stationary equilibrium approximation, that is, Assuming that the system is only slightly non-equilibrium, we have

Average growth approximation: Assuming that two stocks have the same earnings growth, the one with a lower P / E is a better value.

Assuming that her relationship with Edward and her marriage to John are over, Florence takes prussic acid – which she has carried for years in a vial that John thought held her heart medicine – and dies.

Assuming the radius of the first gear wheel is known as a function of z, and using the relationship, the above two equations can be combined to yield the differential equation:

Assuming that these documents are authentic, the attachment of these obligations to the King of England to ownership of land in Cornwall suggests that the area was under his direct rule and implies that the legal and administrative relationship between the king and his subjects was the same there as elsewhere in his kingdom.

Assuming that saving rises faster as a function of income than the relationship between investment and output, then an increase in the marginal propensity to save, ceteris paribus, will move the equilibrium point at which income equals output and investment equals savings to lower values.

Assuming Nabi is not interested in a relationship with her, Doki loses all hope.

Assuming thermal equilibrium between the anode surface and the transpiring argon, the gas enthalpy rise through the anode was calculated according to the relation Af whereby the specific heat of argon was taken as Af.

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.

Assuming enharmonicity, many accidentals will create equivalences between pitches that are written differently.

Assuming that the heat of fusion is temperature-independent and ignoring additional transition temperatures between different solid phases the equation

* Assuming no further changes in the copyright law, all sound recordings fixed between February 15 and December 31, 1972 will enter public domain in the United States:

Assuming that quantized values of L are equally spaced, the spacing between neighboring energies is

Assuming the responsibility of mediator between the two sides, and with Erasmus ' willing participation ( the robot being left as the leader and controlling-force of the thinking machines ), Duncan does something similar to Bene-Gesserit Sharing with the robot ; possibly along with what seems to be a physical-bonding ( similar to what Leto II did with the sandtrout ) into something of a human-machine hybrid.

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 bus commuters do not care about the color of the bus, consumers are expected to choose between bus and car still with equal probability, so the probability of car is still 0. 5, while the probabilities of each of the two bus types is 0. 25.

Assuming that all three concepts of mass are equivalent the gravitational interactions between masses of arbitrary sign can be explored.

* Assuming that the constitutional monarchy was in fact an ideal held throughout all ages of the past, despite the observed facts of British history and the several power struggles between monarchs and parliaments ;

Assuming that games and routine social acts have differentiated social positions, and that these positions create our cognitive perspectives, then it might be that by moving between roles in a game ( eg between hiding and seeking or buying and selling ) we come to learn about the perspective of the other.

Assuming the legs are recovered with a relative speed between leg and body which amounts to the same as the relative speed between water and body, the legs must be kicked back with five times the mean velocity of the swimmer.

Assuming a purely random distribution of wars between states, regardless of their democratic character, the predicted number of conflicts between democracies would be around ten.

Assuming the spacing between two ions is a, the potential in the lattice will look something like this:

Assuming that one gets the relation between the Townsend coefficients by putting ( 4 ) into ( 3 ) and transforming:

It is sometimes called a tickler oscillator because the feedback needed to produce oscillations is provided using a tickler coil ( T in the circuit diagram ) via magnetic coupling between coil L and coil T. Assuming the coupling is weak, but sufficient to sustain oscillation, the frequency is determined primarily by the tank circuit ( L and C in the illustration ) and is approximately given by.

Assuming Winston did not have the irradiation treatment done, it could have taken place any time between 1955 ( when Winston sold it ) and 1971, the year it was offered at auction.

Assuming that non-members take reasonable precautions and adjusting activities to the association's prohibition of their enforcing their own rights, the association is required to raise the non-member above his actual position by an amount equal to the difference between his position on an indifference curve he would occupy were it not for the prohibition, and his original position.

Assuming that the diameter of a choking pulley is twice the diameter of its shaft, which is a generous estimate, then the frictional resistance force at the outer diameter of a choking pulley is half the frictional resistance force at the shaft of a choking pulley.

Assuming that air resistance is not significant, there are only two forces exerted on the system as a whole: ( 1 ) gravity, and ( 2 ) the reaction force caused by the ignited gases being expelled from the rocket's nozzle at high speed.

Assuming perfect combustion conditions, such as complete combustion under adiabatic conditions ( i. e., no heat loss or gain ), the adiabatic combustion temperature can be determined.

Assuming that the radius of the star does not change over this interval, the long term variation in activity level appears to produce a temperature variation of 15 K, which corresponds to a variation in visual magnitude ( V ) of 0. 014.

Assuming the material expands with temperature ( enriched Uranium-235 at room temperature for example ), at an exactly critical state, it will become subcritical if warmed to lower density or become supercritical if cooled to higher density.

Assuming that the star is a supernova, the radiance of the light it emits corresponds ( by Planck's law ) to a temperature of several hundreds of kelvins, so that several frequencies may combine to produce the resonance frequencies of hydrogen atoms.

Assuming that the temperature T is an intensive quantity, i. e., a single-valued, continuous and differentiable function of three-dimensional space ( often called a scalar field ), i. e., that

Assuming the planet is perfectly grey with no greenhouse or tidal effects, and a Bond albedo of 0. 1, the temperature would be close to 1600 K. Although it has not been detected directly, it is certain that the planet is a gas giant.

Assuming an outdoor temperature of and an indoor temperature of, the above equation gives ( when temperatures are converted to Kelvin or Rankine ) a COP of 35. 98, or an EER of 122. 69.

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 ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

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 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.

:< 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.