Applying the induction rule repeatedly, starting from the independently proved base case, proves many, often infinitely many, other cases.

Applying the principle of indifference incorrectly can easily lead to nonsensical results, especially in the case of multivariate, continuous variables.

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

Applying the doctrine of judicial immunity adopted by the U. S. Supreme Court in Bradley v. Fisher in 1871 and held applicable to § 1983 actions in Pierson v. Ray in 1967, Judge Luther M. Swygert, writing for himself and Judges Harlington Wood, Jr., and William G. East, found that immunity is available only when a judge has jurisdiction over the subject-matter of a case and that it is not available when he acts in " clear absence of all jurisdiction.

Applying Seifert's algorithm to this diagram, as expected, does produce a Seifert surface ; in this case, it is a punctured torus of genus g = 1, and the Seifert matrix is

Applying again the matrix operator on both side yields

Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field f or a vector field v ; the use of the scalar Laplacian and vector Laplacian gives two more:

Applying separation of variables again to the second equation gives way to the pair of differential equations

Applying the axiom of choice again we get a function from the first order formulas to such functions

Applying auxin to the cut stem again inhibits lateral dominance.

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 Itō's lemma once again to, it follows that

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 >.

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 Lagrange's formula with the observation that Ω • r ( t ) = 0 at all times,

Applying higher pressure of the order of hundreds or thousands of kilobars induces a series of phase transformations, in particular with a tetragonal phase appearing at about 900 kbar.

Applying a conception of such illocutionary acts according to which they are ( roughly ) acts of saying something with the intention of communicating with an audience, he describes indirect speech acts as follows: " In indirect speech acts the speaker communicates to the hearer more than he actually says by way of relying on their mutually shared background information, both linguistic and nonlinguistic, together with the general powers of rationality and inference on the part of the hearer.

Applying the rules and customs of prize law, the courts decided whether the Letter of Marque was valid and current, and whether the captured vessel or its cargo in fact belonged to the enemy ( not always easy, when flying false flags was common practice ), and if so the prize and its cargo were " condemned ", to be sold at auction with the proceeds divided among the privateer's owner and crew.

Applying power to one cathode surrounds it with an orange glow discharge.

Applying a pulse with specified width and voltages to the spades made the sheet advance to the next anode, where it stayed until the next advance pulse.

* Greening The Desert: Applying natural farming techniques in Africa, interview with Masanobu Fukuoka ( retrieved 30 November 2010 )

Applying this recursively gives an algorithm with a multiplicative cost of.

The Cholesky decomposition is commonly used in the Monte Carlo method for simulating systems with multiple correlated variables: The correlation matrix is decomposed, to give the lower-triangular L. Applying this to a vector of uncorrelated samples, u, produces a sample vector Lu with the covariance properties of the system being modeled.

Applying Itō's lemma with f ( S ) =

Applying Itō's lemma with f ( Y ) = log ( Y ) gives

Applying to the French classics the rigorous method used with regard to the texts of the Middle Ages, he published the Pensées of Pascal, revised with the original manuscript ( 1887 – 1889 ), and the Provinciales ( 1891 ), edited with notes.

Applying this with, then and

Applying procedural fairness to an entire society therefore seems certain to create recognizable inefficiencies, and therefore be unfair, and ( by the equivalence of justice with fairness ) unjust.

Applying the rule of description, one occupies oneself with describing instead of explaining.

Applying these methods in the workplace, such as Csikszentmihalyi did with Swedish police officers, can improve morale by fostering a sense of greater happiness and accomplishment, and in correlated to increased performance.

Applying the chaos theory to the second law of thermodynamics, the paradox of irreversibility can be explained in the errors associated with scaling from microstates to macrostates and the degrees of freedom used when making experimental observations.

Applying the two sine waves at frequencies ω < sub > 1 </ sub >

Applying a rotation means here rotating one of the two rods 360 degrees.

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

Applying the XOR gate's output to a low-pass filter results in an analog voltage that is proportional to the phase difference between the two signals.

Applying one of these two processes, short-term redundancy ( positive correlation of nearby values ) of the signal is eliminated ; compression ratios on the order of 2 to 4 can be achieved if differences are subsequently entropy coded, because the entropy of the difference signal is much smaller than that of the original discrete signal treated as independent samples.

Applying the result of the previous subsection to the displacement, the velocities in the two reference frames are related by the equation

Applying the same approach of exhaustive research and writing narrative based on objective fact, Freeman completed the first two volumes, titled Young Washington, in 1948.

Applying the same techniques ( which involved gradual immunization of horses and sheep by administering small doses of venoms, and then extracting, purifying and freeze-drying the antibody portion from the blood of injected animals ), Vital Brazil and his coworkers were able to discover the first sera against two species of scorpions ' ( 1908 ) and spiders ' ( 1925 ) venoms.

Applying these two formulas to a Savonius constructed with the two halves of an oil barrel of approx.

Applying the Liénard transformation, where the dot indicates the time derivative, the Van der Pol oscillator can be written in its two dimensional form:

Applying the state-transition function to a state may yield two possible results: either an element of together with the next state of the stream, or the element of the singleton set 1 as a separate " final state " indicating that there are no more values in the stream.

Applying the democratic principle, we can calculate the influence of these two rotations on the satellite by calculating the relative contributions of these two collections of massenergy to the background gravitational field strength at the satellite's location, and then weighting their contributions on the satellite's " sense of rotation " accordingly.

Applying the Fourier transform to the two spatial variables gives