\(n\)-variable bisection.
type: Item
instance of: scholarly article
title: \(n\)-variable bisection.
zbMATH Open document ID: 1073.91639
author: Ákos Münnich
author: Gyula Maksa
published in: Journal of Mathematical Psychology
publication date: 2000-01-01T00:00:00Z
review text: Study of the equation \[ B[B(x_{11},\dots,x_{1n}),\dots,B(x_{n1},\dots,x_{nn})]=B[B(x_{11},...,x_{n1}),. ,\dots,B(x_{1n},\dots,x_{nn})] \] for real functions is considered to have started with papers by \textit{J. Aczél} [[1] Norske Vid. Selsk. Forh. (Trondheim) 19, No. 23, 83--86 (1946; Zbl 0030.02601); [2] Bull. Am. Math. Soc. 54, 392--400 (1948; Zbl 0030.02702)], who called it bisymmetry, even though for \(n=2\) it was subject of some algebraic studies earlier and also later under different names, like ``abelian'' (e.g. \textit{D.C. Murdoch} [Trans. Am. Math. Soc. 49, 392--409 (1941; Zbl 0025.10003; JFM 67.0082.01)], ``alternation'' (\textit{M. Sholander} [Bull. Am. Math. Soc. 55, 746--755 (1949; Zbl 0034.29802)], even ``entropic'' (\textit{I.M.H. Etherington}, Proc. Edinb. Math. Soc. (2) 13, 219--222 (1963; Zbl 0115.02201)]; ``etymologically `entropic' means `inner turning'\dots'' wrote the author), and ``medial'' (\textit{S. K. Stein}, Trans. Amer. Math. Soc. 85, 228--256 (1957; Zbl 0079.02402)]. [The reviewer tried later to switch the ``bisymmetric, bisymmetry'' terminology to ``medial, mediality'', see e.g. \textit{J. Aczél} and \textit{J. Dhombres}, Functional equations in several variables with applications to mathematics, information theory, and to the natural and social sciences, Chapter 17, Cambridge University Press (1989; Zbl 0685.39006) - to no avail]. In [1] it has been proved that, \(I\) being a nontrivial real interval, the function \(B: I^n\to I\) is a mean value (\(B(x,\dots,x)=x\) for all \(x\in I\)), bisymmetric, continuous, strictly increasing in each variable and symmetric if, and only if, there exists a continuous strictly monotonic function \(f: I \to \mathbb{R}\) such that \(B(x_1,\dots,x_n)=f^{-1}[\sum_{k=1}^n f(x_k)/n]\) for all \(x_k\in I\;(k=1,\dots,n)\). It has been stated in [2] that, if we drop the symmetry from among the above assumptions, then the general form of \(B\) will be \(B(x_1,\dots,x_n)=f^{-1}[\sum_{k=1}^n q_k f(x_k)],\) …
reviewed by: János Aczél
Mathematics Subject Classification ID: 91E45
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 1750031
zbMATH Keywords: functional equation
zbMATH Keywords: means
zbMATH Keywords: symmetric and nonsymmetric ontinuous
zbMATH Keywords: strictly monotonic mean values
zbMATH Keywords: Bisection
zbMATH Keywords: Measurement
Wikidata QID: Q52069714
MaRDI profile type: Publication
full work available at URL: jmps.1999.1262
OpenAlex ID: W2021492304
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DOI: 10.1006/JMPS.1999.1262
description: scientific article; zbMATH DE number 1750031