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 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 the linearly parameterized Hamiltonian, simply stands for the generalized force operator.

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 fireflies, it is more difficult to find a microscopic explanation for the polarization, because the left and right lanterns of the larvae were found to emit polarized light of opposite senses.

For example, if the electric field vector is vertical ( alternately up and down as the wave travels ) the radiation is said to be vertically polarized.

For example, this is the case with macroscopic crystals of calcite, which present the viewer with two offset, orthogonally polarized images of whatever is viewed through them.

For left circularly polarized light ( LCP ) with propagation towards the observer, the electric vector rotates counterclockwise.

For right circularly polarized light ( RCP ), the electric vector rotates clockwise.

For the first 50 years of its existence, Rockwood was polarized by the temperance issue.

For effective CP, the potential of the steel surface is polarized ( pushed ) more negative until the metal surface has a uniform potential.

For polarized light the end of the vector lies on the surface of the sphere, whereas the vector has zero length for unpolarized light.

For effective CP, the potential of the steel surface is polarized ( pushed ) more negative until the surface has a uniform potential.

For example, large molecules ( e. g. proteins ) in solution, which rotate relatively slowly because of their size, will emit polarized light when excited with polarized light.

For example, the two groups polarized the population of one village, and each group eventually had its own mosque and separate religious organization.

For comparison, the Jones vector only spans the space of fully polarized light, but is more useful for problems involving coherent light.

For a vertically polarized antenna, the H-plane usually coincides with the horizontal / azimuth plane.

For example, the radar scattering of radio waves by a light styrofoam column can be approximated by assuming that each part of the plastic is polarized by the same electric field that would be present at that point without the column, and then calculating the scattering as a radiation integral over that polarization distribution.

For an ideal polarizing beamsplitter these would be fully polarized, with orthogonal polarizations.

For many common beam-splitting polarizers, however, only one of the two output beams is fully polarized.

For the young child this may be no more than providing food, light or movement.

For those who put their trust in Him He still says every day again: `` Let there be light ''!!

For if I knew the secret of letting this power in my body change directly into electricity, I could rent myself out to the electric companies and with just the power in my body I could light all the lights and run all the factories in the entire United States for some days.

For instance, a local society in the middle of a large city may have regular meetings with speakers, focusing less on observing the night sky if the membership is less able to observe due to factors such as light pollution.

For example, changes in signal strength may be used to specify the sounds to be reproduced by a loudspeaker, or the light intensity of television pixels.

For instance, 202 obsolete Panzer I light tanks were modified by removing the turret and were rebuilt as the Panzerjäger I self-propelled Skoda anti-tank gun.

For example, PAC expression in certain neurons has been shown to alter the grooming behavior in fruit flies exposed to blue light.

For the simple case where the direction of the light beam along the line SE is perpendicular to the motion of the observer then only light emitted at an angle equal to the arc cosine of the ratio of the observer's speed to the speed of light will reach the telescope.

For its time, the game was considered " light years " ahead of the Original Dungeons & Dragons.

For Satan himself transforms himself into an angel of light.

* For some reactions, the presence of electromagnetic radiation, most notably ultraviolet light, is needed to promote the breaking of bonds to start the reaction.

For example, the Diels-Alder reactions can be assisted by heat whereas the cycloaddition is selectively induced by light.

For example, the thermite reaction is used to generate light and heat in pyrotechnics and welding.

For identification by light microscopy, a scraping or swab of the affected area is placed on a microscope slide.

For example, diodes are used to regulate voltage ( Zener diodes ), to protect circuits from high voltage surges ( avalanche diodes ), to electronically tune radio and TV receivers ( varactor diodes ), to generate radio frequency oscillations ( tunnel diodes, Gunn diodes, IMPATT diodes ), and to produce light ( light emitting diodes ).

For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength.

For waves which do not require a medium, such as light or gravity in general relativity, only the relative difference in velocity between the observer and the source needs to be considered.

For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity.

For lower frequencies of EMR up to those of visible light ( i. e., radio, microwave, infrared ), the damage done to cells and also to many ordinary materials under such conditions is determined mainly by heating effects, and thus by the radiation power.

For instance, dark bands in the light emitted by a distant star are due to the atoms in the star's atmosphere.

For low-frequency radiation ( radio waves to visible light ) the best-understood effects are those due to radiation power alone, acting through the effect of simple heating when the radiation is absorbed by the cell.

For example, the sentence " The coat is red " has no observer, the sentence " We see the coat as red " ( where " we " indicates observers ) appears more specific in context as regards light waves and colour as determined by modern science, that is, colour results from a reaction in the human brain.