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The formula is valid for all index values, and for any n (when n = 0 or n = 1, this is the empty product). However, computing the formula above naively has a time complexity of O( n 2 ) , whereas the sign can be computed from the parity of the permutation from its disjoint cycles in only O( n log( n )) cost.
In mathematics, a versor is a quaternion of norm one (a unit quaternion).Each versor has the form = = + , =, [,], where the r 2 = −1 condition means that r is a unit-length vector quaternion (or that the first component of r is zero, and the last three components of r are a unit vector in 3 dimensions).
The gradient of a function is obtained by raising the index of the differential , whose components are given by: =; =; =, = = The divergence of a vector field with components is
The metric g can take up to two vectors or vector fields X, Y as arguments. In the former case the output is a number, the (pseudo-) inner product of X and Y . In the latter case, the inner product of X p , Y p is taken at all points p on the manifold so that g ( X , Y ) defines a smooth function on M .
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable.It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings.
A quaternion of the form a + 0 i + 0 j + 0 k, where a is a real number, is called scalar, and a quaternion of the form 0 + b i + c j + d k, where b, c, and d are real numbers, and at least one of b, c, or d is nonzero, is called a vector quaternion. If a + b i + c j + d k is any quaternion, then a is called its scalar part and b i + c j + d k ...
A four-vector A is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations: [3] = (,,,) = + + + = + = where A α is the magnitude component and E α is the basis vector component; note that both are necessary to make a vector, and that when A α is seen alone, it refers strictly to the components of the vector.
A convenient way to work with the quaternion product is to write a quaternion as the sum of a scalar and a vector (strictly speaking a bivector), that is A = a 0 + A, where a 0 is a real number and A = A 1 i + A 2 j + A 3 k is a three dimensional vector. The vector dot and cross operations can now be used to define the quaternion product of A ...