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For a Gaussian intensity ( i. e. power density, W / m < sup > 2 </ sup >) distribution in a single-mode optical fiber, the mode field diameter is that at which the electric and magnetic field strengths are reduced to 1 / e of their maximum values, i. e., the diameter at which power density is reduced to 1 / e < sup > 2 </ sup > of the maximum power density, because the power density is proportional to the square of the field strength.
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For and Gaussian
For a Gaussian response system ( or a simple RC roll off ), the rise time is approximated by:
For general matrices, Gaussian elimination is usually considered to be stable in practice if you use partial pivoting as described below, even though there are examples for which it is unstable.
For example, if one assumes that data arise from a univariate Gaussian distribution, then one has assumed a Gaussian model:.
For example, a mixture of Gaussians with one Gaussian at each data point is dense is the space of distributions.
For example, a common choice for A < sub > o </ sub > is a Gaussian wave packet:
For example, a Gaussian wavefunction ψ might take the form:
For an ideal single-mode Gaussian beam, the D4σ, D86 and 1 / e < sup > 2 </ sup > width measurements would give the same value.
For both families, the lowest-order solution describes a Gaussian beam, while higher-order solutions describe higher-order transverse modes in an optical resonator.
For a Gaussian beam propagating in free space, the spot size ( radius ) w ( z ) will be at a minimum value w < sub > 0 </ sub > at one place along the beam axis, known as the beam waist.
For a Gaussian beam, the BPP is the product of the beam's divergence and waist size.
For large numbers the Poisson distribution approaches a normal distribution, typically making shot noise in actual observations indistinguishable from true Gaussian noise except when the elementary events ( photons, electrons, etc.
For example, the inverse of cumulative Gaussian distribution
For sufficiently nice prior probabilities, the Bernstein-von Mises theorem gives that in the limit of infinite trials and the posterior converges to a Gaussian distribution independent of the initial prior under some conditions firstly outlined and rigorously proven by Joseph Leo Doob in 1948, namely if the random variable in consideration has a finite probability space.
For any affine transformation of the Gaussian plane, z mapping to a z + b, a ≠ 0, a triangle is transformed but does not change its shape.
For a Gaussian distribution this is the best unbiased estimator ( that is, it has the lowest MSE among all unbiased estimators ), but not, say, for a uniform distribution.
For a Gaussian distribution, where, this means the MSE is minimized when dividing the sum by, whereas for a Bernoulli distribution with p = 1 / 2 ( a coin flip ),, the MSE is minimized for.
For example, for a laser producing pulses with a Gaussian temporal shape, the minimum possible pulse duration Δt is given by
For the HeNe laser with a 1. 5 GHz spectral width, the shortest Gaussian pulse consistent with this spectral width would be around 300 picoseconds ; for the 128 THz bandwidth Ti: sapphire laser, this spectral width would be only 3. 4 femtoseconds.
For large α, the shape of the Kaiser window ( in both time and frequency domain ) tends to a Gaussian curve.
For the Gaussian random approximation, the structure function of Tatarski ( 1961 ) can be described in terms
For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the " unrolled " tube ( which is flat ).
* For an orthogonal parametrization ( i. e., ), Gaussian curvature is:
* For a surface described as graph of a function, Gaussian curvature is:
For and intensity
For those affiliated with it, the A.L.A.M. pool was a haven from the infringement actions involving detail patents that beset the industry with mounting intensity after 1900.
For a serious young man who plays golf with a serious intensity, Palmer has such an inherent sense of humor that it relieves the strain and keeps his nerves from jangling like banjo strings.
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 fluorophores in solution this means that the intensity and polarization of the emitted light is dependent on rotational diffusion.
For platformist anarcho-communist Wayne Price " Today ’ s proposals for Parecon, in which workers are rewarded for the intensity and duration of their labor in a cooperative economy, would fit into Bakunin ’ s or Marx ’ s concept of a transitory, beginning, phase, of a free society ... Kropotkin rejected the two-phase approach of the Marxists and the anarchist-collectivists.
For example, on 19 May 2011 an earthquake of magnitude 0. 7 in Southern California, USA 4 km deep was classified as of intensity III by the United States Geological Survey ( USGS ), while a 4. 5 magnitude quake in Salta, Argentina 164 km deep was of intensity I.
For a given metal and frequency of incident radiation, the rate at which photoelectrons are ejected is directly proportional to the intensity of the incident light.
For an incident intensity, this implies the following equality
For a transmitted ray we could do something similar but in that case the distance traveled through the object would cause even faster intensity decrease.
For a known albedo, it is possible to estimate the surface temperature, and correspondingly the intensity of heat radiation.
For every reflection corresponding to a point q in the reciprocal space, there is another reflection of the same intensity at the opposite point-q.
For example, the sampling process in any standard image sensor ( CCD or CMOS camera ) is relatively far from the ideal sampling which would measure the image intensity at a single point.
For example, the intensity of radiation from the Sun is 9126 watts per square meter at the distance of Mercury ( 0. 387 AU ); but only 1367 watts per square meter at the distance of Earth ( 1 AU )— an approximate threefold increase in distance results in an approximate ninefold decrease in intensity of radiation.
For a lightly damped linear oscillator with a resonance frequency Ω, the intensity of oscillations I when the system is driven with a driving frequency ω is typically approximated by a formula that is symmetric about the resonance frequency:
For example, " the intensity of pressure " is meaningless, since the parameters of those variables do not match.
For an ideal point source the intensity of the Arago spot equals that of the undisturbed wave front.
" For the climactic battle, Eidelman starts the music quietly, building the intensity as the battle progresses.
For many services there is labor intensity as services usually involve considerable human activity, rather than a precisely determined process ; exceptions include utilities.
For example, measurements for the intensity of usage, such as incidence and frequency, vary by study.
For confined housing where light control is possible, lowering light intensity so that birds cannot see each other as easily reduces antagonistic encounters and aggressive behaviour.
For example, the Solar storm of 1859 struck the Earth with such intensity that the northern lights could be seen as far south as Rome, approximately 42 ° north of the equator.
For a detailed discussion of the intensity and spectrum ( color ) of incandescence, see the article: thermal radiation.
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