Morneau-Guérin, Frédéric 
ORCID: https://orcid.org/0000-0001-7610-4648
(sous presse).
Means along the hyperoperation ladder: up the rungs and in between. Elemente der Mathematik.
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- elemente_means_10pp_revised.pdf
Contenu du fichier : Manuscrit soumis (avant évaluation) Accès restreint jusqu'à fin- décembre 2028. |
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| Catégorie de document : | Articles de revues |
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| Évaluation par un comité de lecture : | Oui |
| Étape de publication : | Accepté (sous presse) |
| Résumé : | The arithmetic mean of two numbers is the number that can replace both of them without changing their sum. The geometric mean is the number that can replace both without changing their product. Oscar Chisini noticed in 1929 that this idea works for any operation: the mean of m and n with respect to an operation is the number x for which x combined with x gives the same result as m combined with n. Now, addition and multiplication have a natural successor. Multiplication is repeated addition, and exponentiation is repeated multiplication. So what is the mean, in the sense of Chisini, of exponentiation? Which number can stand in for both the base and the exponent of a power without changing its value? This simple question appears to have received remarkably little attention. The answer turns out to be a genuine mean: it always lies between the two numbers it averages. But it is a mean of an unfamiliar kind -- not symmetric, not homogeneous, although it still has a surprisingly rigid structure -- with a closed form involving the Lambert W function. And the question does not stop there. Addition, multiplication and exponentiation are the first rungs of a ladder of hyperoperations that continues with tetration (towers of exponents) and beyond, and the ladder can even be filled in between the rungs with operations of non-integer rank. Each rung and each step in between has its Chisini mean, and together they form a single continuous curve of means, whose first stretch is the familiar family of power means. |
| Déposant: | Morneau-Guérin, Frédéric |
| Responsable : | Frédéric Morneau-Guérin |
| Dépôt : | 08 oct. 2026 12:41 |
| Dernière modification : | 08 oct. 2026 12:41 |
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