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Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed out of statistics and mathematical physics.
In mathematics and its applications, the signed distance function or signed distance field (SDF) is the orthogonal distance of a given point x to the boundary of a set Ω in a metric space (such as the surface of a geometric shape), with the sign determined by whether or not x is in the interior of Ω.
The Fourier transform of a bump function is a (real) analytic function, and it can be extended to the whole complex plane: hence it cannot be compactly supported unless it is zero, since the only entire analytic bump function is the zero function (see Paley–Wiener theorem and Liouville's theorem).
In category theory, "map" is often used as a synonym for "morphism" or "arrow", which is a structure-respecting function and thus may imply more structure than "function" does. [9] For example, a morphism : in a concrete category (i.e. a morphism that can be viewed as a function) carries with it the information of its domain (the source of the ...
More formally, one describes it in terms of functions on closed sets of points. If we let d A denote the dilation by a factor of 1 / 2 about a point A, then the Sierpiński triangle with corners A, B, and C is the fixed set of the transformation d A ∪ d B ∪ d C {\displaystyle d_{\mathrm {A} }\cup d_{\mathrm {B} }\cup d_{\mathrm ...
For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology feature and a group feature, such that these two features are related in a certain way, then the structure becomes a topological group.
The sector contour used to calculate the limits of the Fresnel integrals. This can be derived with any one of several methods. One of them [5] uses a contour integral of the function around the boundary of the sector-shaped region in the complex plane formed by the positive x-axis, the bisector of the first quadrant y = x with x ≥ 0, and a circular arc of radius R centered at the origin.