iSoul Time has three dimensions

# Means and operations

The power means are defined for a set of real numbers, a1, a2, …, an:

$\textup{Power&space;Mean}\;&space;M_{p}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;\frac{1}{n}\sum_{i=1}^{n}a_{i}^{p}&space;\right&space;)^{\frac{1}{p}}$

The best-known of these are the arithmetic, geometric, and harmonic means, with p = 1, p → 0, and p = –1:

$\textup{Arithmetic&space;Mean}\;&space;M_{1}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;\frac{1}{n}\sum_{i=1}^{n}a_{i}&space;\right&space;)$

$\textup{Geometric&space;Mean}\;&space;M_{0}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(\prod_{i=1}^{n}{a_{i}}&space;\right&space;)^{\frac{1}{n}}$

$\textup{Harmonic&space;Mean}\;&space;M_{-1}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;n\sum_{i=1}^{n}\frac{1}{a_{i}}&space;\right&space;)^{-1}$

The other well-known power mean is the root mean square with p = 2:

$\textup{Root&space;Mean&space;Square&space;(RMS)}\;&space;M_{2}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;\frac{1}{n}\sum_{i=1}^{n}a_{i}^{2}&space;\right&space;)^{\frac{1}{2}}$

These are all examples of quasi-arithmetic means:

$\textup{Quasi-Arithmetic&space;Mean}\;&space;QM_{f}(a_{1},a_{2},...,a_{n})\equiv&space;f^{-1}&space;\left&space;(\frac{f(a_{1},a_{2},...,a_{n})}{n}&space;\right&space;)$

The power means suggest power operations:

$\textup{Power&space;Operation}\;&space;O_{p}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;\sum_{i=1}^{n}a_{i}^{p}&space;\right&space;)^{\frac{1}{p}},&space;p\neq&space;0$

These include addition (p = 1), harmonic addition (p = –1), and the root sum squared (p = 2):

$\textup{Addition}\;&space;O_{1}(a_{1},a_{2},...,a_{n})\equiv&space;\sum_{i=1}^{n}a_{i}$

$\textup{Harmonic&space;addition}\;&space;O_{-1}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;\sum_{i=1}^{n}a_{i}^{-1}&space;\right&space;)^{-1}$

$\textup{Root&space;Sum&space;Squared&space;(RSS)}\;&space;O_{2}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;\sum_{i=1}^{n}a_{i}^{2}&space;\right&space;)^{\frac{1}{2}}$

In order to avoid roots of negative numbers, the absolute value is often taken: | ai |. If the ai represent differences such as | xiyi |, then the operations are p-norms:

$\textup{\textit{p}-norm}\;&space;N_{p}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;\sum_{i=1}^{n}&space;|a_{i}|^{p}&space;\right)^{\frac{1}{p}},&space;\,&space;p\neq&space;0$

The best-known norm is the Euclidean norm or distance (p = 2):

$\textup{Euclidean&space;norm}\;&space;N_{2}(a_{1},a_{2},...,a_{n})\equiv&space;\left&space;(&space;\sum_{i=1}^{n}&space;|a_{i}|^{2}&space;\right&space;)^{\frac{1}{2}}$