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In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems.
Choropleth map showing estimated percent of the population below 150% poverty in the Contiguous United States by county, 2020 that uses the Jenks natural breaks classification Jenks’ goal in developing this method was to create a map that was absolutely accurate, in terms of the representation of data's spatial attributes.
The term monotonic transformation (or monotone transformation) may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences ). [ 5 ]
These maps are quasisymmetric, although they are a much narrower subclass of quasisymmetric maps. For example, while a general quasisymmetric map in the complex plane could map the real line to a set of Hausdorff dimension strictly greater than one, a δ-monotone will always map the real line to a rotated graph of a Lipschitz function L:ℝ → ...
Kneading theory provides an effective calculus for describing the qualitative behavior of the iterates of a piecewise monotone mapping f of a closed interval I of the real line into itself. Some quantitative invariants of this discrete dynamical system , such as the lap numbers of the iterates and the Artin–Mazur zeta function of f are ...
[4] [5] Another related concept is that of a completely/absolutely monotonic sequence. This notion was introduced by Hausdorff in 1921. This notion was introduced by Hausdorff in 1921. The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics.
In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated. Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation .
In statistics and numerical analysis, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing (or non-increasing) everywhere, and lies as close to the observations as possible.