On mean values

Abstract

Introduction. Historical. In 1930 Kolmogoroff and Nagumo 1 proved simultaneously a fundamental theorem on mean values. In their definition a mean value is an infinite sequence of functions; M 1 (xi)=*Xi 1 M2(x u X2),M z (xi,X2,x z) f • • • ,M n (xu •••,*»),•••. Each function of this sequence has to satisfy the following conditions: M nix, • • • , x)~x> M n (xi, • • • , x n) must be a continuous, (strictly) increasing (cf. §2) and symmetric function. The terms of this sequence are connected by the "associative property": (j|f* = Af*(*i, • • • , Xk)

Square roots and continuity in strictly linearly ordered semigroups on real intervals

“…See e.g. [2], [14] and section XI.8 in [9] for the usage of roots and dyadic numbers in some approaches to abstract mean values. Moreover, roots are involved in many of the above mentioned embedding results in order to construct some kind of exponential-functions which serve as a semigroup-homomorphisms.…”

The limit of inconsistency reduction in pairwise comparisons

“…In other words, the sample mean is also the best predictor in the sense of having the lowest root mean squared error. Means were analyzed by Aczel (1948) as well as Aczel and Saaty (1983). It is difficult to state when exactly the logarithmic mapping was used for PC matrices as an alternative method for scaling priorities in hierarchical structures.…”

Expectation Values and Variance Based on Lp-Norms

“…Aczél [23] suggested an axiomatic characterization of means, settled by five fundamental properties of the ordinary arithmetic means, and succeeded to reproduce the φ-means. In particular, any univalued, bivariable function M (y 1 , y 2 ), y 1 , y 2 ∈ D y constitutes a general mean of y 1 , y 2 , if the following preconditions are fulfilled: (i) Continuity; (ii) Strict monotonicity: if y 1 < y 1 (>), then M (y 1 , y 2 ) < M (y 1 , y 2 ) (>), and the same holds for y 2 < y 2 ; (iii) Bisymmetry:…”

“…Aczél also pointed [23] that the "internness" property, i.e., M in(y 1 , y 2 ) ≤ M (y 1 , y 2 ) ≤ M ax(y 1 , y 2 ), follows from preconditions (i), (ii), (iv). However, it is evident that the internness, together with the continuity (i), leads to the reflexivity (iv).…”