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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.
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four [a] linearly independent vector fields called a tetrad or vierbein. [1]
In numerical analysis, the Weierstrass method or Durand–Kerner method, discovered by Karl Weierstrass in 1891 and rediscovered independently by Durand in 1960 and Kerner in 1966, is a root-finding algorithm for solving polynomial equations. [1]
The prime examples of such four-vectors are the four-position and four-momentum of a particle, and for fields the electromagnetic tensor and stress–energy tensor. The fact that these objects transform according to the Lorentz transformation is what mathematically defines them as vectors and tensors; see tensor for a definition.
The root system E 7 is the set of vectors in E 8 that are perpendicular to a fixed root in E 8. The root system E 7 has 126 roots. The root system E 6 is not the set of vectors in E 7 that are perpendicular to a fixed root in E 7, indeed, one obtains D 6 that way. However, E 6 is the subsystem of E 8 perpendicular to two suitably chosen roots ...
The rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane. Both are vectors. The first is parallel to the plane, the second is orthogonal. For a given vector and plane, the sum of projection and rejection is equal to the original vector.
In the complex case, the line bundles or their first characteristic classes are called Chern roots. The fact that p ∗ : H ∗ ( X ) → H ∗ ( Y ) {\displaystyle p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)} is injective means that any equation which holds in H ∗ ( Y ) {\displaystyle H^{*}(Y)} (say between various Chern classes) also holds in ...
Minkowski addition and convex hulls. The sixteen dark-red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left plus-signs.