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 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.

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:

When Fosdick showed the letter to Baker his negative response was: `` For God's sake, Raymond, don't show this to the President or he'll stop the war ''.

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

For example the M-T2 protein of myxoma viruses can bind TNF preventing it from binding the TNF receptor and inducing a response.

) For the casuist, the circumstances of a case are essential for evaluating the proper response.

For instance, Microsoft Word files are normally created and modified by the Microsoft Word program in response to user commands, but the user can also move, rename, or delete these files directly by using a file manager program such as Windows Explorer ( on Windows computers ) or by command lines ( CLI ).

For example, an externalist response to the Gettier problem is to say that, in order for a justified true belief to count as knowledge, it must be caused, in the right sort of way, by relevant facts.

For example, it is not the feeling of fear that produces an increase in heart beat, both are symptomatic of a common physiological origin, possibly in response to a legitimate external threat.

For the same reason, filter functions whose critical response is at lower frequencies ( compared to the sampling frequency 1 / T ) require a higher order, more computationally intensive FIR filter.

For instance, if all of the coefficients were equal to unity, a so-called boxcar function, then it would implement a low-pass filter with a low frequency gain of N + 1 and a frequency response given by the sinc function.

For practical filters, a custom design is sometimes desirable, that can offer the best tradeoff between different design criteria, which may include component count and cost, as well as filter response characteristics.

For example, the experience of fear usually occurs in response to a threat.

For instance, they interpret the exhortation to defend one ’ s faith “ with gentleness and reverence ” in 3: 15-16 as a response to Pliny executing Christians for the obstinate manner in which they professed to be Christians.

For users with many or large messages, this IMAP4 usage pattern can result in faster response times.

For this reason, knitting was initially developed for garments that must be elastic or stretch in response to the wearer's motions, such as socks and hosiery.

For a number of years French officials had been unsuccessful in dealing with the squalor of the growing Parisian slums, and Le Corbusier sought efficient ways to house large numbers of people in response to the urban housing crisis.

For example, if the suspect nodded his head up and down in response to the question " did you kill the victim " the conduct is testimonial, it is the same as saying " yes I did " and Miranda would apply.

For Heidegger, however, communication in the first place is not among human beings, but language itself shapes up in response to questioning ( the inexhaustible meaning of ) being.

For example, there are certain formulaic greetings such as “ hello ” and “ how are you ?” that elicit a certain formulaic response such as a returning hello, or an indication of one's state of being.

For domestic dogs, the response is to observe, standing in place where they were when the alarm was sounded, again with the underground prairie dogs emerging to watch.

For example, a Safety PLC might be used to control access to a robot cell with trapped-key access, or perhaps to manage the shutdown response to an emergency stop on a conveyor production line.

For some schools, these plans have included the use of door numbering to aid public safety response.

For example, cell membrane integrins on circulating leukocytes are maintained in an inactive state to avoid epithelial cell attachment ; they are only activated in response to stimuli such as those received at the site of an inflammatory response.

For example, in some areas, workers ' compensation laws arose as a legislative response to court rulings restricting the extent to which employees could sue their employers in respect of injuries sustained during employment.