* Gaussian integers: those complex numbers where both and are integers are also quadratic integers.

* Z, the ring of Gaussian integers.

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.

This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers.

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

An example of such a domain is the Gaussian integers Z, that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers.

* Z: the ring of Gaussian integers

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

The Gaussian integers are a special case of the quadratic integers.

Gaussian integers as lattice point s in the complex plane

Formally, Gaussian integers are the set

Note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice.

The Gaussian integers form a principal ideal domain with units 1, − 1, i, and − i. If x is a Gaussian integer, the four numbers x, ix, − x, and − ix are called the associates of x.

As for every principal ideal domain, the Gaussian integers form also a unique factorization domain.

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 ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity ( 1832 ) ( see ).

Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about " whole complex numbers " ( i. e. the Gaussian integers ) than they are as statements about ordinary whole numbers ( i. e. the integers ).

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.

" 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.

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

** 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 )

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.

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 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.

If the torus carries the ordinary Riemannian metric from its embedding in R < sup > 3 </ sup >, then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature is indeed 0.

In the case of the plane ( where the Gaussian curvature is 0 and geodesics are straight lines ), we recover the familiar formula for the sum of angles in an ordinary triangle.

Here " whole real numbers " are ordinary integers and " whole complex numbers " are Gaussian integers ; one should similarly interpret " real and complex prime numbers ".

Helmut Wolf ( 1910 – 1994 ) published his direct semianalytic solution based on ordinary Gaussian elimination in matrix form in his paper " The Helmert block method, its origin and development ", Proceedings of the Second International Symposium on Problems Related to the Redefinition of North American Geodetic Networks, Arlington, Va. April 24 – April 28, 1978, pages 319 – 326.

Thus, Stein's example can be simply stated as follows: The ordinary decision rule for estimating the mean of a multivariate Gaussian distribution is inadmissible under mean squared error risk.

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.