Alternative and complementary algorithms include evolution strategies, evolutionary programming, simulated annealing, Gaussian adaptation, hill climbing, and swarm intelligence ( e. g.: ant colony optimization, particle swarm optimization ) and methods based on integer linear programming.

NA is also good at climbing sharp crests by adaptation of the moment matrix, because NA may maximise the disorder ( average information ) of the Gaussian simultaneously keeping the mean fitness constant.

Specifically, invariance ( or more appropriately covariance ) to local geometric transformations, such as rotations or local affine transformations, can be obtained by considering differential invariants under the appropriate class of transformations or alternatively by normalizing the Gaussian derivative operators to a locally determined coordinate frame determined from e. g. a preferred orientation in the image domain or by applying a preferred local affine transformation to a local image patch ( see the article on affine shape adaptation for further details ).

" Four normal distribution | Gaussian distributions in statisticsThis unproved statement put a strain on his relationship with János Bolyai ( who thought that Gauss was " stealing " his idea ), but it is now generally taken at face value.

** 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 )

In reality the pattern is closer to a Gaussian, or normal distribution, with a higher density in the center that tapers off at the edges.

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.

In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution, is a generalization of the one-dimensional ( univariate ) normal distribution to higher dimensions.

This definition of Gaussian curvature is extrinsic in that it uses the surface's embedding in R < sup > 3 </ sup >, normal vectors, external planes etc.

As described above, a Rayleigh fading channel itself can be modelled by generating the real and imaginary parts of a complex number according to independent normal Gaussian variables.

An integer is a prime for the Gaussian integers if it is a prime number ( in the normal sense ) that is congruent to 3 modulo 4.

If n is sufficiently large, the above binomial distribution may be approximated by a Gaussian ( normal ) distribution and thus the Pearson test statistic approximates a chi-squared distribution,

Gaussian functions are widely used in statistics where they describe the normal distributions, in signal processing where they serve to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics where they are used to solve heat equations and diffusion equations and to define the Weierstrass transform.

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 concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the normal distribution which is often called the Gaussian distribution.

Gaussian processes are important in statistical modelling because of properties inherited from the normal.

Note that although this model is termed a " Gaussian chain ", the distribution function is not a gaussian ( normal ) distribution.

The normal, exponential, log-normal, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, geometric, inverse Gaussian, von Mises and von Mises-Fisher distributions are all exponential families.

Examples are typical Gaussian mixture models as well as many heavy-tailed distributions that result from compounding ( i. e. infinitely mixing ) a distribution with a prior distribution over one of its parameters, e. g. the Student's t-distribution ( compounding a normal distribution over a gamma-distributed precision prior ), and the beta-binomial and Dirichlet-multinomial distributions.

Some, like height for a given sex, vary in close to a " normal " or Gaussian distribution.

Out of all distributions with a given variance, the normal or Gaussian distribution is the one with the highest entropy.

** normal ( or Gaussian ) and multivariate normal,

The original algorithm was described only for natural numbers and geometric lengths ( real numbers ), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials in one variable.

The norm of a Gaussian integer is the natural number defined as

Wideband Gaussian noise comes from many natural sources, such as the thermal vibrations of atoms in conductors ( referred to as thermal noise or Johnson-Nyquist noise ), shot noise, black body radiation from the earth and other warm objects, and from celestial sources such as the Sun.

In analytical chemistry, a solution of any substance which contains one equivalent per litre is known as a normal solution ( abbreviated N ), so the example sodium hydroxide solution would be 0. 0893 N. The relative uncertainty ( u < sub > r </ sub >) in the measured concentration can be estimated by assuming a Gaussian distribution of the measurement uncertainties:

The NA of a Gaussian laser beam is then related to its minimum spot size by

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

The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis ( the symmetrical axis of the system ) are infinitely small, i. e. with infinitesimal objects, images and lenses ; in practice these conditions are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits.

If the angle u1 is very small, O ' 1 is the Gaussian image ; and O ' 1 O ' 2 is termed the longitudinal aberration, and O ' 1R the lateral aberration of the pencils with aperture u2.

This ray, named by Abbe a principal ray ( not to be confused with the principal rays of the Gaussian theory ), passes through the center of the entrance pupil before the first refraction, and the center of the exit pupil after the last refraction.

Among other things he came up with the notion of Gaussian curvature.

The resulting linear circuit matrix can be solved with Gaussian elimination.

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.

The Selective Gaussian Blur tool works in a similar way, except it blurs areas of an image with little detail.

The three elementary row operations used in the Gaussian elimination ( multiplying rows, switching rows, and adding multiples of rows to other rows ) amount to multiplying the original matrix with invertible matrices from the left.

Therefore, the Gaussian Elimination algorithm applied to the augmented matrix begins with:

It is also concerned with the physics of laser beam propagation, particularly the physics of Gaussian beams, with laser applications, and with associated fields such as nonlinear optics and quantum optics.

Some terms associated with gravitational mass and its effects are the Gaussian gravitational constant, the standard gravitational parameter and the Schwarzschild radius.

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

If each of the features makes an independent contribution to the output, then algorithms based on linear functions ( e. g., linear regression, logistic regression, Support Vector Machines, naive Bayes ) and distance functions ( e. g., nearest neighbor methods, support vector machines with Gaussian kernels ) generally perform well.

An effective alternative is the singular value decomposition ( SVD ), but there are other less expensive choices, such as QR decomposition with pivoting ( so-called rank-revealing QR factorization ), which are still more numerically robust than Gaussian elimination.

For example, a mixture of Gaussians with one Gaussian at each data point is dense is the space of distributions.

Hence, bending a surface will not alter the Gaussian curvature and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it.

All these other surfaces would have boundaries and the sphere is the only surface without boundary with constant positive Gaussian curvature.

The pseudosphere is an example of a surface with constant negative Gaussian curvature.