Applying a result of MacIntyre on the model theory of p-adic integers, one deduces again that ζ < sub > G </ sub >( s ) is a rational function in p < sup >− s </ sup >.

Applying rational basis review the U. S. Supreme Court struck down the ordinance as applied to CLC.

Applying reduction of order in case of a multiple root m < sub > 1 </ sub > will yield expressions involving a discrete version of ln,

Applying multiple, successive gaussian blurs to an image has the same effect as applying a single, larger gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied.

Applying the completeness theorem to this result, gives the existence of a model of T where the formula C < sub > T </ sub > is false.

Applying Fubini's theorem and integrating on the fiber, one gets

Applying this theorem when R is itself a PID gives us a way of building Dedekind domains out of PIDs.

* Applying the divergence theorem to the product of a scalar function g and a vector field F, the result is

* Applying the divergence theorem to the cross-product of two vector fields, the result is

* Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector, the following theorem can be proven:

* Applying the divergence theorem to the cross-product of a vector field F and a non-zero constant vector, the following theorem can be proven:

Applying the fundamental theorem of calculus twice gives

Proof: Applying the binomial theorem,

Applying Van Kampen's theorem gives the result.

Applying again Bayes ' theorem, and assuming the classification between spam and ham of the emails containing a given word (" replica ") is a random variable with beta distribution, some programs decide to use a corrected probability:

Applying the binomial theorem to the last factor,

Applying the spectral theorem, or Borel functional calculus for infinite dimensional systems, we see that it generalizes the classical entropy.

Applying Birkhoff's theorem, this is sufficient to tell us that the groups form a variety, and so it should be defined by a collection of identities.

Applying Wick's theorem and doing calculation with the usual normal ordering in this thermal context is possible but computationally impractical.

Applying divergence theorem to the right hand side yields:

The ancient geometers are not done yet, for if the fifth vertex of the pentagon is marked as E and FE and BF are joined ( with FE = BF = z ), then cyclic quadrilateral EFBA will be formed with diagonals length d ( diameter ) and b. Applying the ' Almagest ' theorem yet again:

Applying Bayesian probability in practice involves assessing a prior probability which is then applied to a likelihood function and updated through the use of Bayes ' theorem.

Applying the convolution theorem allows fast calculation of the bispectrum.

Applying a negative DC voltage (" bias ") to the control grid will repel some of the electron stream back towards the cathode, thus isolating the plate from the cathode ; full bias will turn the tube off by blocking all current from the cathode.

Applying heat to a food usually, though not always, chemically transforms it, thus changing its flavor, texture, consistency, appearance, and nutritional properties.

Applying the right amount of inputs in the right place and at the right time benefits crops, soils and groundwater, and thus the entire crop cycle.

Applying this concept to human society, Kropotkin presented mutual aid as one of the dominant factors of evolution, the other being self assertion, and concluded that In the practice of mutual aid, which we can retrace to the earliest beginnings of evolution, we thus find the positive and undoubted origin of our ethical conceptions ; and we can affirm that in the ethical progress of man, mutual support not mutual struggle – has had the leading part.

Applying a pressure is not benificial, thus not recommended.

Applying again the matrix operator on both side yields

Applying the above formula to the 25-meter-diameter antennas often used in radio telescope arrays and satellite ground antennas at a wavelength of 21 cm ( 1. 42 GHz, a common radio astronomy frequency ), yields an approximate maximum gain of 140, 000 times or about 50 dBi ( decibels above the isotropic level ).

Applying both sides of this matrix equation to the component vector v < sub > 1 </ sub >( φ < sub > 1 </ sub >< sup >− 1 </ sup >( P )) and invoking ( 1 ) and ( 2 ) yields

Applying on both sides yields.

Applying the above bundle map ( pointwise ) to this section yields the pullback of α by φ, which is the 1-form φ < sup >*</ sup > α on M defined by

Applying the Theil index to allocation processes in the real world does not imply that these processes are stochastic: the Theil yields the distance between an ordered resource distribution in an observed system to the final stage of stochastic resource distribution in a closed system.

Applying the Cauchy-Bunyakowski-Schwartz inequality to each term of the last expression yields

Applying equation ( 1 ) to a subsystem yields:

Applying the Lagrange interpolation formula to do quadratic interpolation on the inverse of f yields

Applying the method hierarchically ( according to the data structure at hand ), optionally in both increasing and decreasing direction, yields highly efficient multidimensional range search which is important in both commercial and technical applications, e. g. as a procedure underlying nearest neighbour searches.

Applying the more detailed form of those calculations to the Ivy Mike device yields vaporized pusher gas expansion velocity of 290 kilometers per second and an implosion velocity of perhaps 400 kilometers per second if 3 / 4 of the total tamper / pusher mass is ablated off, the most energy efficient proportion.

Applying the scheme discussed above on a fairly modern Windows based system typically results in the following drive letter assignments:

Applying the conservation of energy, for steady-state conditions, and Newton ’ s law of cooling to the temperature nodes shown in Figure 2 gives the following set of equations.

Applying the inverse system to gives the following.

Applying the most commonly cited definition of systemic risk, that of the Financial Stability Board ( FSB ), to the core activities of insurers and reinsurers, the report concludes that none are systemically relevant for at least one of the following reasons:

Applying these distinctions to the categories in German wine classification gives the following overview.

Applying this value successively to the tones of a chromatic scale, starting from A above middle C with a frequency of 440 Hz, produces the following sequence of pitches:

Applying an Euler transform to the latter series, the following is obtained:

Applying linearization techniques to the equation of motions for both species, to the equation of continuity, and Poisson's equation, and introducing the spatial and temporal harmonic operators, we can get the following expression: