Weighting by iteration: iterations of n variables means based on subdivisions of the standard $$\left( n-1\right) $$ n
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ORIGINAL RESEARCH PAPER
Weighting by iteration: iterations of n variables means based on subdivisions of the standard (n -- 1)-simplex Lucio R. Berrone1 Received: 20 July 2018 / Accepted: 13 October 2020 Ó Forum D’Analystes, Chennai 2020
Abstract The problem of weighting a general n variables mean is solved by using a class of algorithms which involve iterated subdivisions (triangulations) of the ðn 1Þ-simplex Dn1 . The instance based on the barycentric subdivision is treated at length, while a more sketchy presentation is given to the algorithm based on the Freudenthal triangulation. Generalizing the weightings of 2 variables means based on Acze´l iterations, the resulting weighting procedures turn out to be continuous and scale invariant, being geometric the rate of convergence of the algorithms on the class of scaled uniformly internal means. Keywords Means Weightings Triangulations
Mathematics Subject Classification 54C30 26E60 54C20 39B12 26A15
1 Introduction Given a real interval I and an integer number n 2 N, a function M : I n ! I is a (n variables) mean when it is internal; i.e., provided that the twofold inequality minfx1 ; . . .; xn g M ðx1 ; . . .; xn Þ maxfx1 ; . . .; xn g;
ð1Þ
is satisfied for every x1 ; . . .; xn 2 I. M is said to be a strict mean when the inequalities in (1) are both strict unless x1 ¼ ¼ xn (strict internality). The compact form min x M ð xÞ max x of the inequalities (1) illustrates the usage of
& Lucio R. Berrone [email protected] 1
Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas (CONICET), Laboratorio de Acu´stica y Electroacu´stica, Facultad de Cs. Exactas, Ing. y Agrim., Univ. Nac. de Rosario, Riobamba 245 bis, 2000 Rosario, Argentina
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L. R. Berrone
the notation made in this paper. M is said to be symmetric when, for every x1 ; . . .; xn 2 I, M xrð1Þ ; . . .; xrðnÞ ¼ M ðx1 ; . . .; xn Þ whichever be the permutation of indices r 2 Sn . Means enjoying properties other than internality and symmetry will often arise in this paper but, in order to quickly introduce its subject matter, it will be useful to conceive means from the standpoint of the aggregation functions theory; i.e., as stated in the preface of [31], as a singular way of ‘‘combining several numerical values into a single representative value’’. The arithmetic mean An ðxÞ ¼
n 1X xi n i¼1
ð2Þ
combines the real values x1 ; x2 ; . . .; xn in a uniform manner; in other terms, it does not assign a particular relevance to any one of them. Given a continuous and strictly monotonic function f : I ! R, the change of coordinates y ¼ f ðxÞ transforms the arithmetic mean (2) in the quasiarithmetic mean (defined on I) ! n X 1 1 An;f ðxÞ ¼ f f ðxi Þ ; ð3Þ n i¼1 so that the uniformity in assigning relevances to the data of the arithmetic mean is inherited by the quasiarithmetic ones. Now, the combination of the numerical values in such a way of assigning them different weights depending on their respective relevances is, in
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