Applying Chow's analysis to the case of the unexpected hanging ( again with the week shortened to two days for simplicity ), we start with the observation that the judge's announcement seems to affirm three things:

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 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 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

Applying Faraday's law of induction to plane waves, we find, implying that an electrostatic wave must be purely longitudinal.

Applying the techniques developed at Harvard for generating a paradigm from a representative form and its classification, we can add all forms of a word to the dictionary at once.

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 carefully selected rules at the social level and encouraging appropriate motives at the personal level is, so it is argued, likely to lead to a better overall outcome even if on some individual occasions it leads to the wrong action when assessed according to act utilitarian standards.

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 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 a delta-v at the LEO of only 0. 78 km / s more ( 3. 20-2. 42 ) would give the rocket the escape speed, while at the geostationary orbit a delta-v of 1. 46 km / s is needed for reaching the escape speed of this circular orbit.

Applying the principles of comparative negligence, the jury found that McDonald's was 80 % responsible for the incident and Liebeck was 20 % at fault.

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 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 the formula at the top shows this to be about 386 cents.

* Applying a Doppler ultrasonic scanner, the surgeon identifies the axial pedicle of the tissue-flap ( composed of the supraorbital artery and the supratrochlear artery ), usually at the base, next to the medial brow ; the point usually is between the midline and the supraorbital notch.

* Free Advice On Applying To Graduate School ( PDF )-Written by a professor of psychology at Portland State University.

Applying Equation ( 1 ) and KCL at the OA's non-inverting input v < sub >+</ sub > gives

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 many of the PLATO concepts used in the late 1970s, E. Clarke Porter led the Drake Training and Technologies testing business ( today Thomson Prometric ) in partnership with Novell, Inc. away from the mainframe model to a LAN-based client server architecture and changed the business model to deploy proctored testing at thousands of independent training organizations on a global scale.

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 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 to him the term " absolute " would, however, give a false impression: he is not free to issue decrees at whim.

Applying symmetry reduced the computational and memory requirements associated with the problem, allowing the researchers to obtain results on the supercomputers available at the time.

Applying the physics of high pressure jets exiting circular apertures, a doubling of velocity and gas concentration at a fixed point in the whistle mouth would require a quadrupling of either aperture area or absolute pressure.

Applying the exterior connection to both sides gives the transformation law for ω:

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 an annihilation operator followed by its corresponding creation operator returns the number of particles in the k < sup > th </ sup > single-particle eigenstate:

Applying this estimate to any even power V < sup > k </ sup > of V and using the Künneth formula shows that the eigenvalues of Frobenius on the middle cohomology of a variety V of any dimension d satisfy

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 duality, this means that a morphism in some category C is a monomorphism if and only if the reverse morphism in the opposite category C < sup > op </ sup > is an epimorphism.

Applying Ohm's Law, the relationship between the input voltage, V < sub > in </ sub >, and the output voltage, V < sub > out </ sub >, can be found:

Applying the Tannakian formalism, one concludes that M < sub > num </ sub > is equivalent to the category of representations of an algebraic group G, which is called motivic Galois group.

Applying any additive functor to a split sequence results in a split sequence, so in particular R < sup > 1 </ sup > F ( A )

Applying this to the function defined by f () = log ( a < sub > 1 </ sub >), we obtain that

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 Ampère-Maxwell equation to surface S < sub > 1 </ sub > we find:

Applying these ladder operators to the eigenstates of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator C < sub > 1 </ sub > are n < sup > 2 </ sup > − 1 ; importantly, they are independent of the l and m quantum numbers, making the energy levels degenerate.

Applying the Nernst Equation above, one may account for these differences by changes in relative K < sup >+</ sup > concentration or differences in temperature.

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 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 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 >.