Height: Difference between revisions
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Wilson height |
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| | <math>2^{\large{\|q\|_{T1}}}</math> | | | <math>2^{\large{\|q\|_{T1}}}</math> | ||
| | <math>\|q\|_{T1}</math> | | | <math>\|q\|_{T1}</math> | ||
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| | Wilson Height | |||
| | Height | |||
| | <math>\text{sopf}(n d)</math><br><small>* | |||
| | <math>2^{\large{\text{sopf}(n d)}}</math> | |||
| | <math>sopf(q)</math> | |||
|- | |- | ||
| | Weil Height | | | Weil Height | ||
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| | <math>\|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |</math> | | | <math>\|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |</math> | ||
|} | |} | ||
Where ||q||<span style="font-size: 80%; vertical-align: sub;">T1</span> is the [[Generalized_Tenney_Norms_and_Tp_Interval_Space#The Tenney Norm (T1 norm)|tenney norm]] of q in monzo form, and v<span style="vertical-align: sub;">p</span>(x) is the [http://en.wikipedia.org/wiki/P-adic_order p-adic valuation] of x. | Where ||q||<span style="font-size: 80%; vertical-align: sub;">T1</span> is the [[Generalized_Tenney_Norms_and_Tp_Interval_Space#The Tenney Norm (T1 norm)|tenney norm]] of q in monzo form, and v<span style="vertical-align: sub;">p</span>(x) is the [http://en.wikipedia.org/wiki/P-adic_order p-adic valuation] of x. | ||
The function <math>sopf(nd)</math> is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors"] of n*d. Equivalently, this is the L1 norm on monzos, but where each prime is weighted by "p" rather than "log(p)". This is called "Wilson Complexity" in John Chalmers "Division of the Tetrachord." | |||
Some useful identities: | Some useful identities: | ||