These properties are the reason why monotonic functions are useful in technical work in analysis. Some more facts about these functions are:. An important application of monotonic functions is in probability theory. A function is unimodal if it is monotonically increasing up to some point the mode and then monotonically decreasing.
In contrast, each constant function is monotonic, but not injective,  and hence cannot have an inverse. Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives. A monotone operator is said to be maximal monotone if its graph is a maximal monotone set.
Order theory deals with arbitrary partially ordered sets and preordered sets as a generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not total. The composite of two monotone mappings is also monotone.
The dual notion is often called antitoneanti-monotoneor order-reversing. Hence, an antitone function f satisfies the property. A constant function is both monotone and antitone; conversely, if f is both monotone and antitone, and if the domain of f is a latticethen f must be constant. Monotone functions are central in order theory.
They appear in most articles on the subject and examples from special applications are found in these places. In the context of search algorithms monotonicity also called consistency is a condition applied to heuristic functions.
A heuristic h n is monotonic if, for every node n and every successor n' of n generated by any action athe estimated cost of reaching the goal from n is no greater than Monotonic Wealthy Brook step cost of getting to n' plus the estimated cost of reaching the goal from n'.
This is a form of triangle inequalitywith nn'and the goal G n closest to n. Because every monotonic heuristic is also admissiblemonotonicity is a stricter requirement than admissibility.
In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that an n -ary Boolean function is monotonic when its representation as an n -cube labelled with truth values has no upward edge from true to false. The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs which may appear more than once using only the operators and and or in particular not is forbidden.
For instance "at least two of abc hold" is a monotonic function of abcsince it can be written for instance as a and b or a and c or b and c. Monotonic Wealthy Brook number of such functions on n variables is known as the Dedekind number of n. From Wikipedia, the free encyclopedia. Redirected from Monotonicity. For information on monotonicity as it pertains to voting systemssee monotonicity criterion.
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