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On the other hand, reflection groups are concrete, in the sense that each of its elements is the composite of finitely many geometric reflections about linear hyperplanes in some euclidean space. Technically, a reflection group is a subgroup of a linear group (or various generalizations) generated by orthogonal matrices of determinant -1.
If x is a reflection point (0, 5, 10, 15, 20, or 25), its stabilizer is the group of order two containing the identity and the reflection in x. In other cases the stabilizer is the trivial group. For a fixed x in X, consider the map from G to X given by g ↦ g · x. The image of this map is the orbit of x and the coimage is the set of all left ...
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis
For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x. The orthogonal group, consisting of all proper and improper rotations, is generated by reflections.
It follows rather readily (see orthogonal matrix) that any orthogonal matrix can be decomposed into a product of 2 by 2 rotations, called Givens Rotations, and Householder reflections. This is appealing intuitively since multiplication of a vector by an orthogonal matrix preserves the length of that vector, and rotations and reflections exhaust ...
The dilogarithm along the real axis. In mathematics, the dilogarithm (or Spence's function), denoted as Li 2 (z), is a particular case of the polylogarithm.Two related special functions are referred to as Spence's function, the dilogarithm itself:
In a given switching class of graphs of a regular two-graph, let Γ x be the unique graph having x as an isolated vertex (this always exists, just take any graph in the class and switch the open neighborhood of x) without the vertex x. That is, the two-graph is the extension of Γ x by x. In the first example above of a regular two-graph, Γ x ...