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Applying Van Kampen's theorem gives the result.
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Applying and Van
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 and theorem
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 the rational root theorem thus yields the following possible roots for t:
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 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 and gives
Another way of computing this derivative is to view the composite function as the composite of and h. Applying the chain rule to this situation gives:
Applying the definition of the derivative gives:
Applying this technique to the Pleiades gives an age of about 115 million years.
Applying this recursively gives an algorithm with a multiplicative cost of.
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 Itō's lemma with f ( Y ) = log ( Y ) gives
Applying this technique to the boundary layer equations gives the full turbulent boundary layer equations not often given in literature:
Applying a sinusoidal stress to a Maxwell model gives: where is the Maxwell relaxtion time.
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 these simplifying assumptions to the spatial components of the geodesic equation gives
Applying Equation ( 1 ) and KCL at the OA's non-inverting input v < sub >+</ sub > gives
Applying the quantum oracle gives
Applying the inverse system to gives the following.
Applying the exterior connection to both sides gives the transformation law for ω:
Applying these distinctions to the categories in German wine classification gives the following overview.
Applying the same operators to the right hand side gives
Applying the e < nowiki >'</ nowiki > th iterate of F to a ring which contains a field K of p < sup > e </ sup > elements gives us a fixed point set equal to K, similar to the example above.
Applying a smear of pine tar on the wound gives the attacking hens something else to do.
Applying the system operator first gives
Applying and result
Applying both sides to e < sub > j </ sub >, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula (*).
Applying heat to the inner cable for soldering may weaken the steel, and although soft soldering is less strong than silver solder, a lower temperature is required to form the joint, and there is less likelihood of the inner cable being damaged as a result.
Applying perspective ( pharmacoeconomic ) to analyze cost reduction, it has been shown that savings made by individual hospitals result in actual cost increases to the Provinces.
Applying a result of linear algebra to both orthogonality relations (| C < sub > i </ sub >| is always positive ), we find that the number of conjugacy classes is greater than or equal to the number of inequivalent irreducible representations ; and also at the same time less than or equal to.
Applying this result repeatedly shows that there is a basis for V such that all elements of the Lie algebra are represented by upper triangular matrices.
Applying excessive force to the cube when twisting it may result in broken pieces.
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 the result of the previous subsection to the displacement, the velocities in the two reference frames are related by the equation
Applying this method instead of Euler's method gives a much better result ( blue ).
Applying finding from affective forecasting research to happiness also raises methodological issues: should happiness measure the outcome of an experience, or the choice that is satisfied as a result of a forecast?
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