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The compositions GF : A → A and FG : B → B are the associated closure operators; they are monotone idempotent maps with the property a ≤ GF(a) for all a in A and b ≤ FG(b) for all b in B. The implications of the two definitions of Galois connections are very similar, since an antitone Galois connection between A and B is just a monotone ...
The following is a selection from the large body of literature on absolutely/completely monotonic functions/sequences. René L. Schilling, Renming Song and Zoran Vondraček (2010). Bernstein Functions Theory and Applications. De Gruyter. pp. 1– 10. ISBN 978-3-11-021530-4. (Chapter 1 Laplace transforms and completely monotone functions)
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 ...
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 ]
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.
Structure of cisplatin, an example of a molecule with the square planar coordination geometry. In chemistry, the square planar molecular geometry describes the stereochemistry (spatial arrangement of atoms) that is adopted by certain chemical compounds. As the name suggests, molecules of this geometry have their atoms positioned at the corners.
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.
Chemical similarity (or molecular similarity) refers to the similarity of chemical elements, molecules or chemical compounds with respect to either structural or functional qualities, i.e. the effect that the chemical compound has on reaction partners in inorganic or biological settings.