where n < sup > c </ sup > denotes the charge conjugate state, i. e., the antiparticle.

where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E ( k ), and a < sub > k </ sub > denotes the corresponding annihilation operators.

where r is the position vector of the particle relative to the origin, p is the linear momentum of the particle, and × denotes the cross product.

where and denotes the portfolio value at time t.

respectively, where e < sub > x </ sub >, e < sub > y </ sub >, e < sub > z </ sub > denotes the cartesian basis vectors ( all are orthogonal unit vectors ) and A < sub > x </ sub >, A < sub > y </ sub >, A < sub > z </ sub > are the corresponding coordinates, in the x, y, z directions.

where denotes the complex conjugate of.

denotes the rank-one operator that maps the ket to the ket ( where is a scalar multiplying the vector ).

Just as kets and bras can be transformed into each other ( making into ) the element from the dual space corresponding with is where A < sup >†</ sup > denotes the Hermitian conjugate ( or adjoint ) of the operator A.

where Z denotes J, Y, H < sup >( 1 )</ sup >, or H < sup >( 2 )</ sup >.

where C < sub > α </ sub > denotes I < sub > α </ sub > or e < sup > απi </ sup > K < sub > α </ sub >.

where ε denotes the Levi-Civita symbol, the metric tensor is used to lower the index on F, and the Einstein summation convention implies that repeated indices are summed over.

where denotes the length of a string p.

where J denotes the Jacobian matrix of partial derivatives of ƒ and is the induced norm on the matrix.

for every automorphism φ of G ( where φ ( H ) denotes the image of H under φ ).

where denotes the vector cross product and square brackets denote evaluation in the rotating frame of reference.

This functional relationship is often denoted y = f ( x ), where f denotes the function.

The common notation for the divergence ∇· F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ ( see del ), apply them to the components of F, and sum the results.

where the star denotes complex conjugation.

A *( B *( C * D )) where A, B, C, D are partially transparent image layers and "*" denotes a compositing operator ( with the left layer on top of the right layer ).

With this view it turns out that the extractor property is equivalent to: for any source of randomness that gives bits with min-entropy, the distribution is-close to, where denotes the uniform distribution on.

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

where denotes the largest integer that

where ζ denotes the Riemann zeta function ( see Lehmer ; one approach to prove the inequality is to obtain the Fourier series for the polynomials B < sub > n </ sub >).

where denotes the largest integer that is not greater than x.

The first known instance of Newton's lines joined to music was in A Companion to the Countess of Huntingdon's Hymns ( London, 1808 ), where it is set to the tune " Hephzibah " by English composer John Jenkins Husband.

By starting with an inertial frame, where Newton's laws of motion hold, and keeping track of how the time derivatives of a position vector change when transforming to a rotating reference frame, the various fictitious forces and their forms can be identified.

It can also be generalized to situations where Newton's laws do not hold, for example in the theory of relativity and in electrodynamics.

By Newton's law of universal gravitation and laws of motion, a body of mass m a distance R from the center of a sphere of mass M feels a force equivalent to an acceleration, where:

Under an assumption of constant gravity, Newton's law of universal gravitation simplifies to F = mg, where m is the mass of the body and g is a constant vector with an average magnitude of 9. 81 m / s < sup > 2 </ sup >.

The opening sections of the ' Principia ' contain, in revised and extended form, nearly all of the content of Newton's 1684 tract ' De motu ...' ( see article De motu corporum in gyrum which summarises the topics and indicates where they reappear in the ' Principia ').

Section IX includes Newton's demonstration ( Propositions 43-45 ) that in an eccentric orbit under centripetal force where the apse may move, a steady nonmoving orientation of the line of apses is an indicator of an inverse-square law of force.

She included an analytical section where she applied the new mathematics of calculus to Newton's most controversial theories.

For example, there is no limit where Newton's mechanics reduces to Aristotle's mechanics because Aristotle's mechanics, although academically viable for 18 centuries, do not have any domain of validity.

Normally wave equations in physics can be derived from other physical laws-the wave equation for mechanical vibrations on strings and in matter can be derived from Newton's laws-where the analogue wavefunction is the displacement of matter, and electromagnetic waves from Maxwell's equations, where the wavefunctions are electric and magnetic fields.

In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used ( derived from Newton's laws of motion and Newton's law of universal gravitation ).

Mae took her to sing in Nashville, Tennessee, where she performed at the Ralph Emery Show, singing Juice Newton's " Queen of Hearts " and Tammy Wynette's " Your Good Girl's Gonna Go Bad ".

The writing in which Einstein found inspiration from Mach was " The Science of Mechanics ", where the philosopher criticized Newton's idea of absolute space, in particular the argument that Newton gave sustaining the existence of an advantaged reference system: what is commonly called " Newton's bucket argument ".

In the most common version, the trajectories of molecules and atoms are determined by numerically solving the Newton's equations of motion for a system of interacting particles, where forces between the particles and potential energy are defined by molecular mechanics force fields.

They achieve that by sometimes using the average of two differences where Newton's or Gauss's would use just one difference.

For a particle of constant invariant mass m > 0, where is the four-velocity, so we can relate the four-force with the four-acceleration as in Newton's second law:

The points ( α, β ) are plotted as with Newton's diagram method but the line α + β = n, where n is the degree of the curve, is added to form a triangle which contains the diagram.

Not concluding that Newton's law of universal gravitation was flawed, however, astronomers John Couch Adams as well as Urbain Le Verrier independently predicted a new planet, calculated its weight and orbit through Newton's theory, and so was discovered the planet Neptune where predicted.

A West Country native where many well-known English pirates hailed from, Robert Newton's portrayal of Long John Silver in 1950s films popularized the stereotypical West Country pirate accent.

where we have used Newton's third law F < sub > 12 </ sub > = − F < sub > 21 </ sub > and where