User:Sintel/Dual Weil-Euclidean norm: Difference between revisions
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== Derivation == | |||
On some <math>p</math>-limit subgroup with <math>n</math> primes, define the <math>n \times n</math> Tenney weighting matrix <math>W</math>: | On some <math>p</math>-limit subgroup with <math>n</math> primes, define the <math>n \times n</math> Tenney weighting matrix <math>W</math>: | ||
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$$ | $$ | ||
< | == Cross-Weighted Metric == | ||
[[Graham Breed]] gives the following formula (adapted for the notation introduced here):<ref>See formula (16) in section 3.1 "Cross-Weighted Metrics" In Breed, G. (2008). RMS-Based Error and Complexity Measures Involving Composite Intervals http://x31eq.com/composite.pdf</ref> | |||
$$ | |||
\begin{aligned} | |||
G'(\lambda) &= W^{-2} - \lambda \frac{W^{-2}j^{\mathsf T}jW^{-2}}{jW^{-2}j^{\mathsf T}} \\ | |||
&= W^{-2} - \lambda \frac{l^{\mathsf T}l}{n} | |||
\end{aligned} | |||
$$ | |||
So this is equivalent to <math>G^{-1}</math> when we pick <math>\lambda = \frac{n}{n+1}</math>. | |||
==== References ==== | |||