Search results
Results From The WOW.Com Content Network
A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite.The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime.
Similarly, the product of the factors a − r 2 b is a square in Z[r 2], with a "square root" which also can be computed. It should be remarked that the use of Gaussian elimination does not give the optimal run time of the algorithm. Instead, sparse matrix solving algorithms such as Block Lanczos or Block Wiedemann are used.
The conjugate of a Gaussian prime is also a Gaussian prime (this implies that Gaussian primes are symmetric about the real and imaginary axes). A positive integer is a Gaussian prime if and only if it is a prime number that is congruent to 3 modulo 4 (that is, it may be written 4 n + 3 , with n a nonnegative integer) (sequence A002145 in the ...
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation [1] [2] is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting ...
"The table might have been further reduced to the integers in the first octant of the complex plane using the symmetry y + ix =i (x − iy)." That symmetry maps between the 1st and 4th quadrants. It rotates by +/- 90 degrees so maps between the 1st and 7th octant and between the 2nd and 8th octant. e.g. 2+i == i(1-2i) is a first octant term ...
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes.. A Gaussian integer is a complex number + such that a and b are integers. The norm (+) = + of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer.
Dixon's method is based on finding a congruence of squares modulo the integer N which is intended to factor. Fermat's factorization method finds such a congruence by selecting random or pseudo-random x values and hoping that the integer x 2 mod N is a perfect square (in the integers):
Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory. Gaussian beams are used in optical systems, microwave systems and lasers. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing.