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[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.
Three examples of Turing patterns Six stable states from Turing equations, the last one forms Turing patterns. The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and spots, can arise naturally and autonomously from a homogeneous, uniform state.
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.
A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [1] [2] [3] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
Reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics (neutron diffusion theory) and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential ...
The class of firmly nonexpansive operators is equal to the set of resolvents of maximally monotone operators. [6] Surprisingly, while iterating non-expansive maps has no guarantee to find a fixed point (e.g. multiplication by -1), firm non-expansiveness is sufficient to guarantee global convergence to a fixed point, provided a fixed point exists.
Dynamical properties of reaction networks were studied in chemistry and physics after the invention of the law of mass action.The essential steps in this study were introduction of detailed balance for the complex chemical reactions by Rudolf Wegscheider (1901), [1] development of the quantitative theory of chemical chain reactions by Nikolay Semyonov (1934), [2] development of kinetics of ...
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 ...