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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.
A benefit of isotonic regression is that it is not constrained by any functional form, such as the linearity imposed by linear regression, as long as the function is monotonic increasing. Another application is nonmetric multidimensional scaling , [ 1 ] where a low-dimensional embedding for data points is sought such that order of distances ...
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 .
A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
As explained in Riesz & Sz.-Nagy (1990), every non-decreasing non-negative function F can be decomposed uniquely as a sum of a jump function f and a continuous monotone function g: the jump function f is constructed by using the jump data of the original monotone function F and it is easy to check that g = F − f is continuous and monotone. [10]
Suppose (,) is a directed-complete partial order (dcpo) with a least element, and let : be a Scott-continuous (and therefore monotone) function. Then f {\displaystyle f} has a least fixed point , which is the supremum of the ascending Kleene chain of f . {\displaystyle f.}
Let (L, ≤) be a complete lattice and let f : L → L be an order-preserving (monotonic) function w.r.t. ≤ . Then the set of fixed points of f in L forms a complete lattice under ≤ . It was Tarski who stated the result in its most general form, [1] and so the theorem is often known as Tarski's fixed-point theorem.
This illustrates clearly demonstrates the effectiveness of the MUSCL approach to solving the Euler equations. The simulation was carried out on a mesh of 200 cells using Matlab code (Wesseling, 2001), adapted to use the KT algorithm and Ospre limiter. Time integration was performed by a 4th order SHK (equivalent performance to RK-4) integrator.