For the near term, however, it must be realized that the industrial and commercial market is somewhat more sensitive to general business conditions than is the military market, and for this reason I would expect that any gain in 1961 may be somewhat smaller than those of recent years ; ;

For Euclid ’ s method to succeed, the starting lengths must satisfy two requirements: ( i ) the lengths must not be 0, AND ( ii ) the subtraction must be “ proper ”, a test must guarantee that the smaller of the two numbers is subtracted from the larger ( alternately, the two can be equal so their subtraction yields 0 ).

For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

For the first time, armored cars or limousines were put into service for safer transport, with modern versions virtually invulnerable to small arms fire, smaller bombs and mines.

For instance, in hydrogenation operations, large planes of metal surface function as sites of hydrogenolysis catalysis while sites catalyzing hydrogenation of unsaturates are smaller.

For example, periods of high oxygen concentrations determined from ice core samples have been associated with fauna of a larger scale in the fossil record, while periods of low oxygen concentrations have been associated with fauna of a smaller scale in the fossil record.

For CRTs, the viewable size is typically smaller than the tube itself.

For example, if a diagonal line with a width smaller than a full pixel must be rendered, then this can be done by lighting only the subpixels that the line actually touches.

For the west of the country, the major airport is Billund ( 2. 7m passengers in 2011 ) although both Aalborg ( 1. 4m passengers in 2011 ) and Aarhus ( 591. 000 passengers in 2011 ) have smaller airports with regular connections to Copenhagen.

For bassists with smaller hands, the large spaces between pitches may present a significant challenge, especially in the lowest range, where the spaces between notes are largest.

For example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.

For example, when electromagnetic radiation is absorbed by matter, particle-like properties will be more obvious when the average number of photons in the cube of the relevant wavelength is much smaller than 1.

For spherical bodies, when the occulting object is smaller than the star, the length ( L ) of the umbra's cone-shaped shadow is given by:

For polynomials of degree four and smaller there are explicit formulae for the roots ( the formulae are elementary functions ).

For example, Ireland currently has 4 licenced operators with their own networks despite a smaller and sparser population than France's making improved coverage less economically rewarding.

The red dot overtakes two green dots when moving from the left to the right of the figure. New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front. For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.

For much of its history, the Empire consisted of hundreds of smaller sub-units, principalities, duchies, counties, Free Imperial Cities and other domains.

: For smaller units of time, see Measurement of hours below.

For example, at Cannae, the Romans deployed 80, 000 men, and generally could put tens of thousands more into smaller combat actions ).

For smaller prey up to the size of a goat, their loosely articulated jaws, flexible skull, and expandable stomach allow it to swallow its prey whole.

For the cooler stars, dwarfs such as red dwarfs, orange dwarfs, and yellow dwarfs are indeed much smaller and dimmer than other stars of those colors.

For example, white dwarfs are a different kind of star that is much smaller than main-sequence stars — being roughly the size of the Earth.

For example, a road map may not show railroads, smaller waterways or other prominent non-road objects, and even if it does, it may show them less clearly ( e. g. dashed or dotted lines / outlines ) than the main roads.

For the same reason, larger nuclei ( more than about eight nucleons in diameter ) are less tightly bound per unit mass than are smaller nuclei ; breaking a large nucleus into two or more intermediate-sized nuclei, releases energy.

For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative ( since it decreases from zero ) for smaller finite distances.

For example, if a single photon strikes the electrons, only a single electron changes states in response to the photon.

For instance, each photon that travels through intergalactic space loses energy due to the redshift effect.

For commonly used fluorescent compounds, typical excited state decay times for photon emissions with energies from the UV to near infrared are within the range of 0. 5 to 20 nanoseconds.

For high energy gamma, this relies on interaction of the photon radiation with the tube wall material, usually 1-2mm of chrome steel to produce electrons which can enter and ionise the fill gas.

For example, a single photon may be refracted by a lens or exhibit wave interference with itself, but also act as a particle giving a definite result when its position is measured.

For example, QED has one electron field and one photon field ; QCD has one field for each type of quark, etc.

For example, in most string theories one of the closed string modes is the graviton, and one of the open string modes is the photon.

For electromagnetic radiation, the amplitude of a photon corresponds to the changes in the electric field of the wave.

For example, for the probability that a photon will reflect off of a mirror and be observed at a given point a given amount of time later, one sets the photon's probability amplitude spinning as it leaves the source, follows it to the mirror, and then to its final point, even for paths that do not involve bouncing off of the mirror at equal angles.

For a photon, its momentum, and thus can be substituted for for all photon momentum terms which arise in course of the derivation below.

For the spontaneous Raman effect, which is a form of light scattering, a photon excites the molecule from the ground state to a virtual energy state.

For example, if a photon in a plus spin state has a 0. 1 amplitude to be absorbed and take an atom to the second energy level, and if the photon in a minus spin state has a − 0. 1 amplitude to do the same, a photon which has an equal amplitude to be plus or minus would have zero amplitude to take the atom to the second excited state and the atom will not be excited.

For example, the pattern ( probability distribution ) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method.

For example, the APOLLO Collaboration photon pulse return graph, shown here, has a pattern consistent with a retroreflector array near a known landing site.

For the final space battle, Bill George redesigned the photon torpedoes to have a hotter core and larger flare, because he felt that the weapons in earlier films looked " too pretty ".

For electromagnetic radiation in vacuum, wavenumber is proportional to frequency and to photon energy.

For instance, if the polarization of two of the photons are measured and determined to be rotated + 45 ° from horizontal, then local hidden variable theory predicts that the polarization of the third photon will also be + 45 ° from horizontal.

For an approximate estimate of indirect illumination, the photon map is used to calculate the radiance contribution.

For soft indirect illumination, radiance is calculated using the photon map directly.

# For each photon, divide the amount of flux ( real photons ) that the photon represents by the area of S and multiply by the BRDF applied to that photon.