A survey of the m-M calculus

A survey of the m-M calculus

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Article ID: iaor19982407
Country: Serbia
Volume: 8
Issue: 1
Start Page Number: 137
End Page Number: 168
Publication Date: Jan 1998
Journal: Yugoslav Journal of Operations Research
Authors:
Abstract:

This paper is a brief survey of the m-M calculus. The m-M calculus deals with the so-called m-M functions, i.e. functions of the form f:DR(D=[a1,b1]×…×[an,bn], where n>0 is any integer and ai,biR) subject to the following supposition: For each n-dimensional segment Δ = [α11]×…×[αnn]⊂D a pair of real numbers, denoted by m(f)(Δ), M(f)(Δ), satisfying the conditions m(f)(Δ)≤f(X)≤M(f)(Δ) (for all Δ⊂D, X∈Δ) and lim(M(f)(Δ)−m(f)(Δ))=0 (where diamΔ:(Σ(βi−αi)2)1/2) is effectively given. Such an ordered pair ⟨m(f),M(f)⟩ of mappings m(f),M(f) (both mapping the set of all Δ⊂D into R) is called an m-M pair of the function f. We also say that m(f),M(f) are generalized minimum and maximum for f respectively. For instance, with only few exceptions all elementary functions are m-M functions (Lemma 1.2).

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