Tp tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 354257326 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 354326218 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-22 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-22 19:57:35 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>354326218</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G. | The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G. | ||
Note that while the Hahn-Banach theorem is usually proven using | Note that while the Hahn-Banach theorem is usually proven using Zorn's lemma and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique. | ||
=L2 tuning= | =L2 tuning= | ||
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The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.<br /> | The norm of the full p-limit Ƹ, ||Ƹ||, must also be minimal among all valid error maps for S*, or the restriction of Ƹ to G would improve on Ɛ. Hence, as ||Ƹ|| is minimal, J* + Ƹ, where J* is the full p-limit JIP, must equal the Lp tuning for S*. Thus to find the Lp tuning of S for the group G, we may first find the Lp tuning T* for S*, and then apply it to the normal interval list giving the standard form of generators for G.<br /> | ||
<br /> | <br /> | ||
Note that while the Hahn-Banach theorem is usually proven using | Note that while the Hahn-Banach theorem is usually proven using Zorn's lemma and does not guarantee any kind of uniqueness, in most cases there is only one Lp tuning and the extension of Ɛ to Ƹ is in that case unique.<br /> | ||
<br /> | <br /> | ||
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