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The Euclidean rhythm in music was discovered by Godfried Toussaint in 2004 and is described in a 2005 paper "The Euclidean Algorithm Generates Traditional Musical Rhythms". [1] The greatest common divisor of two numbers is used rhythmically giving the number of beats and silences, generating almost all of the most important world music rhythms ...
A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. A particular Euclidean function f is not part of the definition of a Euclidean domain, as, in general, a Euclidean domain may admit many different Euclidean functions. In this context, q and r are called respectively a quotient and a remainder of ...
A modular multiplicative inverse of a modulo m can be found by using the extended Euclidean algorithm. The Euclidean algorithm determines the greatest common divisor (gcd) of two integers, say a and m. If a has a multiplicative inverse modulo m, this gcd must be 1. The last of several equations produced by the algorithm may be solved for this gcd.
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.
The rotation group SO(3) can be described as a subgroup of E + (3), the Euclidean group of direct isometries of Euclidean . This larger group is the group of all motions of a rigid body : each of these is a combination of a rotation about an arbitrary axis and a translation, or put differently, a combination of an element of SO(3) and an ...
A radial function is a function : [,).When paired with a norm on a vector space ‖ ‖: [,), a function of the form = (‖ ‖) is said to be a radial kernel centered at .A radial function and the associated radial kernels are said to be radial basis functions if, for any finite set of nodes {} =, all of the following conditions are true:
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (projective space) and a selective set of basic geometric concepts.
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