* 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, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z.

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

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 ).

Gauss's circle problem does not deal with the Gaussian integers per se, but instead asks for the number of lattice points inside a circle of a given radius centered at the origin.

The neighborhood function Θ ( u, v, s ) depends on the lattice distance between the BMU ( neuron u ) and neuron v. In the simplest form it is 1 for all neurons close enough to BMU and 0 for others, but a Gaussian function is a common choice, too.

The E < sub > 6 </ sub > lattice, E < sub > 8 </ sub > lattice and Coxeter – Todd lattice also have constructions as complex lattices, over either the Eisenstein or Gaussian integers.

Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

Its double cover acts on a 28-dimensional lattice over the Gaussian integers.

The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers which form a square lattice in the complex plane.

Another point of view, which turns out to be very useful to analyze the algorithm, is that Gaussian elimination computes a matrix decomposition.

When applied to floating point computations on computers, basic Gaussian elimination ( LU decomposition ) can be unreliable, and a rank revealing decomposition should be used instead.

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

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. She runs around P while the thread is completely stretched and measures the length C ( r ) of one complete trip around P. If the surface were flat, she would find C ( r ) = 2πr.

On curved surfaces, the formula for C ( r ) will be different, and the Gaussian curvature K at the point P can be computed by the Bertrand – Diquet – Puiseux theorem as

The averaging is often done by convolution with a Gaussian filter, which, at every spatial point, weights neighboring voxels by their distance, with the weights falling exponentially following the bell curve.

The exponential map is also useful in relating the abstract definition of curvature to the more concrete realization of it originally conceived by Riemann himself — the sectional curvature is intuitively defined as the Gaussian curvature of some surface ( i. e., a slicing of the manifold by a 2-dimensional submanifold ) through the point p in consideration.

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ < sub > 1 </ sub > and κ < sub > 2 </ sub >, of the given point.

* If both principal curvatures are the same sign: κ < sub > 1 </ sub > κ < sub > 2 </ sub >> 0, then the Gaussian curvature is positive and the surface is said to have an elliptic point.

* If the principal curvatures have different signs: κ < sub > 1 </ sub > κ < sub > 2 </ sub >< 0, then the Gaussian curvature is negative and the surface is said to have a hyperbolic point.

* If one of the principal curvature is zero: κ < sub > 1 </ sub > κ < sub > 2 </ sub >= 0, the Gaussian curvature is zero and the surface is said to have a parabolic point.

At a point p on a regular surface in R < sup > 3 </ sup >, the Gaussian curvature is also given by

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 principal reason for the use of Gaussian basis functions in molecular quantum chemical calculations is the ' Gaussian Product Theorem ', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them.

As a result, usually reference beams are Gaussian beams or spherical wave beams ( beams that radiate from a single point ) which are fairly easy to reproduce.

a Gaussian density function is placed at each data point,

For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw – Curtis quadrature are generally far more accurate ; Clenshaw – Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.

In the language of the renormalization group, this theory is known as the Gaussian fixed point.

Even though the quantized massless φ < sup > 4 </ sup > is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point.

The monkey saddle has an isolated umbilical point with zero Gaussian curvature at the origin, while the curvature is strictly negative at all other points.

There will be two asymptotic directions through every point with negative Gaussian curvature, these directions are bisected by the principal directions.

For one thing, in Gaussian units, all of the following quantities have the same dimension: E, D, P, B, H, and M. Another important point is that the electric and magnetic susceptibility of a material is dimensionless in both Gaussian and SI units, but a given material will have a different numerical susceptibility in the two systems.