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In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution.Also known as the Pareto–Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf, who suggested a simpler distribution called Zipf's law, and the mathematician Benoit Mandelbrot, who subsequently generalized it.
A generalization of Zipf's law is the Zipf–Mandelbrot law, proposed by Benoit Mandelbrot, whose frequencies are: (;,,) = (+) . [clarification needed] The constant C is the Hurwitz zeta function evaluated at s.
By even portioning, Mandelbrot meant that the addends were of same order of magnitude, otherwise he considered the portioning to be concentrated. Given the moment of order q of a random variable, Mandelbrot called the root of degree q of such moment the scale factor (of order q). The seven states are:
Misiurewicz points in the context of the Mandelbrot set can be classified based on several criteria. One such criterion is the number of external rays that converge on such a point. [4] Branch points, which can divide the Mandelbrot set into two or more sub-regions, have three or more external arguments (or angles). Non-branch points have ...
The Mandelbrot set within a continuously colored environment. The Mandelbrot set (/ ˈ m æ n d əl b r oʊ t,-b r ɒ t /) [1] [2] is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified.
Self-similarity in the Mandelbrot set shown by zooming in on the Feigenbaum point at (−1.401155189..., 0) An image of the Barnsley fern which exhibits affine self-similarity The Mandelbrot set is also self-similar around Misiurewicz points .
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." [ 1 ] Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension.
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