For non-integer α, the functions J < sub > α </ sub >( x ) and J < sub >− α </ sub >( x ) are linearly independent, and are therefore the two solutions of the differential equation.

For fluids which are sufficiently dense to be a continuum, do not contain ionized species, and have velocities small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier-Stokes equations, which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on velocity gradients and pressure.

: For both of these processes, the energy loss per cycle of alternating current is constant, so core losses increase linearly with frequency.

For a better appreciation of the nature of circularly polarized light one may find it useful to read how circularly polarized light is converted to and from linearly polarized light in the circular polarizer article.

For these mechanisms the maximum expected CP fraction is, where is the fraction of linearly polarized ( LP ) light.

For example, a linearly encoded 16-bit PCM signal can be converted to an 8-bit WAV or AU file while maintaining a decent SNR by compressing before the transition to 8-bit and expanding after a conversion back to 16-bit.

For a system that responds linearly to its input, there is a unique responsivity.

For a given spectral distribution, the photocurrent is linearly proportional to the illuminance ( and to the irradiance ).

For example, consider a market for nails where the cost of each nail is 10 cents and the demand will decrease linearly from a high demand for free nails to zero demand for nails at $ 1. 10.

For linearly polarized light, this is equivalent to saying that the effect of the half-wave plate is to rotate the polarization vector through an angle 2θ ; however, for elliptically polarized light the half-wave plate also has the effect of inverting the light's handedness.

For amplitudes up to about 60 °, the velocity of a saccade linearly depends on the amplitude ( the so called " saccadic main sequence ").

For 1 < p, q < ∞ and f ∈ L < sup > p </ sup >( μ ) and g ∈ L < sup > q </ sup >( μ ), Hölder's inequality becomes an equality if and only if | f |< sup > p </ sup > and | g |< sup > q </ sup > are linearly dependent in L < sup > 1 </ sup >( μ ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α | f |< sup > p </ sup > = β | g |< sup > q </ sup > μ-almost everywhere.

For a given capacitor charging current, the amplitude of the output waveform will decrease linearly with frequency.

For example, a linearly ordered group that is Archimedean is an Archimedean group.

For example, if the manifold has dimension 6, there are three linearly independent Chern numbers, given by c < sub > 1 </ sub > ³, c < sub > 1 </ sub > c < sub > 2 </ sub >, and c < sub > 3 </ sub >.

For n functions of several variables, a generalized Wronskian is the determinant of an n by n matrix with entries D < sub > i </ sub >( f < sub > j </ sub >) ( with 0 ≤ i < n ), where each D < sub > i </ sub > is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish.

For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent.

For any rocket propulsion, since the kinetic energy of exhaust goes up with velocity squared ( kinetic energy = ½ mv < sup > 2 </ sup >), whereas the momentum and thrust goes up with velocity linearly ( momentum = mv ), obtaining a particular level of thrust ( as in a number of g acceleration ) requires far more power each time that exhaust velocity and specific impulse ( Isp ) is much increased in a design goal.

For every element a of a linearly ordered group G either a ∈ G < sub >+</ sub >, or − a ∈ G < sub >+</ sub >, or a = 0.

For each event, a first place gives 60 points, a 2nd place 54 pts, a 3rd place 48 pts, a 4th place 43 pts, a fifth place 40 pts, a 6th place 38 pts, 7th 36 pts 8th 34 points, 9th 32 points, 10th 31 points, then linearly decreasing by one point down to the 40th place.

For small amplitudes, sound pressure and particle velocity are linearly related and their ratio is the acoustic impedance.

For example, buyers of an oil ETF such as USO might think that as long as oil goes up, they will profit roughly linearly.

For example, it has been shown that the widely available atomic conditional primitives, CAS and LL / SC, cannot provide starvation-free implementations of many common data structures without memory costs growing linearly in the number of threads.

For a linear medium, only the first term of this equation is significant and the polarization varies linearly with the electric field.

For example, given the four points the pencil of conics through them can be parameterized as yielding the following pencil ; in all cases the center is at the origin :< ref group =" note "> A simpler parametrization is given by which are the affine combinations of the equations and corresponding the parallel vertical lines and horizontal lines, and results in the degenerate conics falling at the standard points of

For such an observer, the incremental () form of the proper time equation is needed, along with a parameterized description of the path being taken, as shown below.

For example, given the four points the pencil of conics through them can be parameterized as which are the affine combinations of the equations and corresponding to the parallel vertical lines and horizontal lines ; this yields degenerate conics at the standard points of A less elegant but more symmetric parametrization is given by in which case inverting a () interchanges x and y, yielding the following pencil ; in all cases the center is at the origin:

For a parametric family of distributions one compares a code with the best code based on one of the distributions in the parameterized family.

For different situations and / or number of particles, the Hamiltonian is different since it includes the sum of kinetic energies of the particles, and the potential energy function corresponding to the situation.

For this reason cross terms for kinetic energy may appear in the Hamiltonian ; a mix of the gradients for two particles:

For this reason, the Hamiltonian for the observables is called " free Hamiltonian " and the Hamiltonian for the states is called " interaction Hamiltonian ".

For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation.

For graphs of maximum degree three, a careful backtracking search can find a Hamiltonian cycle ( if one exists ) in time O ( 1. 251 < sup > n </ sup >).

For example, ordinary differential equations and symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics.

For, the Hamiltonian can be solved analytically, resulting in the Breit-Rabi formula.

For instance, often it is possible to choose the Hamiltonian itself H = H ( q, p ; t ) as one of the new canonical momentum coordinates.

For the first-order perturbation we need to solve the perturbed Hamiltonian restricted to the degenerate subspace D

For an elementary derivation, we will take Hamiltonian to commute with itself at different times, and further, be independent of time, in which case it simplifies to:

For this reason finding Hamiltonian cycles and longest cycles in outerplanar graphs may be solved in linear time, in contrast to the NP-completeness of these problems for arbitrary graphs.

For this reason, Heisenberg investigated the anharmonic oscillator, with Hamiltonian

For example, if Peggy knew ahead of time that Victor would ask to see the Hamiltonian Cycle in H then she could generate a Hamiltonian cycle for an unrelated graph.

For a nuclear magnetic dipole moment, μ < sub > I </ sub >, placed in a magnetic field, B, the relevant term in the Hamiltonian is given by:

For particles in a time-independent potential ( see Schrödinger equation ), it also labels the nth eigenvalue of Hamiltonian ( H ), i. e. the energy, E with the contribution due to angular momentum ( the term involving J < sup > 2 </ sup >) left out.

For the most common case that the target Hamiltonian contains only pairwise interactions, i. e.,

For a detector coupled to modes with a definite frequency in, the boost operator is then the Hamiltonian.

For a truly adiabatic process we require ; in this case the final state will be an eigenstate of the final Hamiltonian, with a modified configuration:

For example, the time t can be separated if the Hamiltonian does not depend on time explicitly.

For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic in the generalized momenta, S will be completely separable if the potential energy is additively separable in each coordinate, where the potential energy term for each coordinate is multiplied by the coordinate-dependent factor in the corresponding momentum term of the Hamiltonian ( the Staeckel conditions ).