* Normal ( Gaussian ) distribution, whose density function is a Gaussian function

** Normal dynamics, is a stochastic motion having a Gaussian probability density function in position with variance MSD that follows, MSD ~ t, where MSD is the mean squared displacement of the process, and t is the time the process is seen ( normal dynamics and Brownian dynamics are very similar ; the term used depends on the field )

Dyson recognized the formula as the pair correlation function of the Gaussian unitary ensemble, which has been extensively studied by physicists.

which can be found by setting z = 1 / 2 in the reflection or duplication formulas, by using the relation to the beta function given below with x = y = 1 / 2, or simply by making the substitution u = √ t in the integral definition of the gamma function, resulting in a Gaussian integral.

He pioneered the development of more sophisticated computational methods, called ab initio quantum chemistry methods, that use basis sets of either Slater type orbitals or Gaussian orbitals to model the wave function.

Methods that employ a distance function, such as nearest neighbor methods and support vector machines with Gaussian kernels, are particularly sensitive to this.

This has a Gaussian shape and reduces the peak strength of the line shape function.

* Gaussian filter – minimum group delay ; gives no overshoot to a step function.

In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet.

In applied mathematics, the delta function is often manipulated as a kind of limit ( a weak limit ) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.

The mathematical function that describes the Gaussian beam is a solution to the paraxial form of the Helmholtz equation.

The solution, in the form of a Gaussian function, represents the complex amplitude of the beam's electric field.

Gaussian beam width w ( z ) as a function of the axial distance z. w < sub > 0 </ sub >: beam waist ; b: depth of focus ; z < sub > R </ sub >: Rayleigh range ;: total angular spread

will give the probability that a single sample taken from a random process with zero-mean and unit-variance Gaussian probability density function will be greater or equal to.

It is a scaled form of the complementary Gaussian error function:

Gaussian quadrature as above will only produce accurate results if the function f ( x ) is well approximated by a polynomial function within the range.

Its general form is a Gaussian function ).

Many other authors have identified specific problems in financial engineering that caused catastrophes: Aaron Brown named confusion between quants and regulators over the meaning of “ capital ”, Felix Salmon fingered the Gaussian copula, Ian Stewart criticized the Black-Scholes formula, Pablo Triana < ref name = Triana >< Pablo Triana, The Number That Killed Us: A Story of Modern Banking, Flawed Mathematics, and a Big Financial Crisis, Wiley ( December 6, 2011 ) 978-0470529737 </ ref > dislikes Value-at-Risk and Scott Patterson

Quantum dots are valued for displays, because they emit light in very specific Gaussian distributions.

Hence, one may say that the primary way to generate a scale space is by the diffusion equation, and that the Gaussian kernel arises as the Green's function of this specific partial differential equation.

When looking at a specific behaviour, such as the frequency of lying, a researcher may use a Gaussian bell curve to plot all reactions, and a normal reaction would be within one standard deviation, or the most average 68. 3 %.

These perturbations are thought to have a very specific character: they form a Gaussian random field whose covariance function is diagonal and nearly scale-invariant.

The term " curve " refers to the bell curve, the graphical representation of the probability density of the normal distribution ( also called the Gaussian distribution ), but this method of grading does not necessarily make use of any specific frequency distribution such as the bell-shaped normal distribution.

In mathematics, the Gaussian or ordinary hypergeometric function < sub > 2 </ sub > F < sub > 1 </ sub >( a, b ; c ; z ) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.

In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity.

In other cases, some kind of numerical integration method is needed, either a general method such as Gaussian integration or a Monte Carlo method, or a method specialized to statistical problems such as the Laplace approximation, Gibbs sampling or the EM algorithm.

In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity ( irradiance ) distributions are well approximated by Gaussian functions.

In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers.

If is a Gaussian integer whose norm is a prime number, then is a Gaussian prime, because the norm is multiplicative.

The ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q ( i ) consisting of the complex numbers whose real and imaginary part are both rational.

The Gaussian functions are thus those functions whose logarithm is a quadratic function.

In probability theory and statistics, a Gaussian process is a stochastic process whose realizations consist of random values associated with every point in a range of times ( or of space ) such that each such random variable has a normal distribution.

The Brownian bridge is a Gaussian process whose increments are not independent.

The fractional Brownian motion is a Gaussian process whose covariance function is a generalisation of Wiener process.

Given any set of points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian.

A standard way to compute the Gaussian integral, whose idea goes back to Poisson, is

These splats are rendered as disks whose properties ( color and transparency ) vary diametrically in normal ( Gaussian ) manner.

In symmetric unimodal distributions, such as the normal ( or Gaussian ) distribution ( the distribution whose density function, when graphed, gives the famous " bell curve "), the mean ( if defined ), median and mode all coincide.

When applied in two dimensions, this formula produces a surface whose contours are concentric circles with a Gaussian distribution from the center point.

The graph of a Gaussian is a characteristic symmetric " bell curve " shape that quickly falls off towards plus / minus infinity.

* For a surface described as graph of a function, Gaussian curvature is: