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 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:

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 example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer can all be analysed using diffraction equations.

For example, array elements producing a 5 degree phase shift for each wavelength across the array face will produce a beam pointed 5 degree away from the centerline perpendicular to the array face.

For example, an autotrophic self-replicating machine could cover a moon or planet with solar cells, and beam the power to the Earth using microwaves.

For multimodal marginal distributions ( a beam profile with multiple peaks ), the 1 / e < sup > 2 </ sup > width usually does not yield a meaningful value and can grossly underestimate the inherent width of the beam.

For many applications, a lower-divergence beam is preferable.

For example, an ultraviolet laser that emits at a wavelength of 308 nm will have a lower divergence than an infrared laser at 808 nm, if both have the same minimum beam diameter.

For the optical regime, see beam divergence.

For example, the aviation glideslope vertical guidance clearance beam is sometimes described as a " capture effect " system, even though it operates using AM signals.

For an electromagnetic wave, the coherence time is the time over which a propagating wave ( especially a laser or maser beam ) may be considered coherent.

For a beam of wavelength λ at a distance z along the beam from the beam waist, the variation of the spot size is given by

For the end users K < sub > u </ sub > band is generally cheaper and enables smaller antennas ( both because of the higher frequency and a more focused beam ).

For a converging lens ( for example a convex lens ), the focal length is positive, and is the distance at which a beam of collimated light will be focused to a single spot.

For a diverging lens ( for example a concave lens ), the focal length is negative, and is the distance to the point from which a collimated beam appears to be diverging after passing through the lens.

For larger stones, the doctor may also use the extra tube in the ureteroscope to extend a flexible fiber that carries a laser beam to break the stone into smaller pieces that can then pass out of the body in the urine.

For the Deutsch-Jozsa oracle, P and BPP are also separated.

For BPP this is immediate, since Arthur can simply ignore Merlin and solve the problem directly ; for NP, Merlin need only send Arthur a certificate, which Arthur can validate deterministically in polynomial time.

For example, NP, co-NP, PLS, PPA all have known natural complete problems, while RP, ZPP, BPP and TFNP do not have any known complete problems ( although such a problem may be discovered in the future ).

For example, Sipser showed that there is a language M such that BPP < sup > M </ sup > ( BPP with oracle M ) has no complete problems.