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Gaussian and integers
* 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.
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 ).

Gaussian and are
In 1976 the International Astronomical Union ( IAU ) revised the definition of the AU for greater precision, defining it as that length for which the Gaussian gravitational constant ( k ) takes the value when the units of measurement are the astronomical units of length, mass and time.
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.
Furthermore, within CGS, there are several plausible choices of electromagnetic units, leading to different unit " sub-systems ", including Gaussian, " ESU ", " EMU ", and Heaviside – Lorentz.
Among these choices, Gaussian units are the most common today, and in fact the phrase " CGS units " is often used to refer specifically to CGS-Gaussian units.
** 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 )
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.
Gaussian elimination does not generalize in any simple way to higher order tensors ( matrices are order 2 tensors ); even computing the rank of a tensor of order greater than 2 is a difficult problem.
Many media transforms, such as Gaussian blur, are, like lossy compression, irreversible: the original signal cannot be reconstructed from the transformed signal.
Some terms associated with gravitational mass and its effects are the Gaussian gravitational constant, the standard gravitational parameter and the Schwarzschild radius.
Equivalent technical statements are that the sum total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.
Other units commonly used are Gaussian units ( based on the cgs system ), Lorentz – Heaviside units ( used mainly in particle physics ) and Planck units ( used in theoretical physics ).
Methods that employ a distance function, such as nearest neighbor methods and support vector machines with Gaussian kernels, are particularly sensitive to this.
This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications.
The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination.
Its prime elements are known as Gaussian primes.
Rational primes ( i. e. prime elements in Z ) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.
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.

Gaussian and special
Non-Gaussian beams also exhibit this effect, but a Gaussian beam is a special case where the product of width and divergence is the smallest possible.
* Gaussian quadrature, a special case of numerical integration
Even for the special case of the Gaussian scenario, the capacity region of the other 3 channels except the broadcast channel is unknown in general.
The period lattices are of a very special form, being proportional to the Gaussian integers.
Generalized hypergeometric functions include the ( Gaussian ) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.
In special cases the beam can be described as a single transverse mode and the spatial properties can be well described by the Gaussian beam, itself.
One can make assumptions about the joint distribution of the interest rates and then calculate VaR by Monte Carlo simulation or, in some special cases ( e. g., Gaussian distribution assuming a linear approximation ), even analytically.
Gram was the first mathematician to provide a systematic theory of the development of skew frequency curves, showing that the normal symmetric Gaussian error curve was but one special case of a more general class of frequency curves.
SNESAmp can even produce higher-quality output than the SPC700 itself by outputting the sound at a higher sampling rate and using more complex interpolation ( such as cubic interpolation ) than the original SPC700's Gaussian interpolation and using the built-in special " High Quality " enhancement feature.
* Gaussian process emulation, a special case of the Gaussian process in statistics
A special case is white Gaussian noise, in which the values at any pairs of times are statistically independent ( and uncorrelated ).
An important special case of a GRF is the Gaussian free field.
The process defined by a time-series model which represents values as a linear combination of past values and of present and past innovations ( see Autoregressive moving average model ) is, except for limited special cases, not time-reversible unless the innovations have a normal distribution ( in which case the model is a Gaussian process ).
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.

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