Height: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 447781960 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 447782290 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2013-08-31 06:49 | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2013-08-31 06:53:49 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>447782290</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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2^{-v_2 \left( {d} \right)} d} \right) | 2^{-v_2 \left( {d} \right)} d} \right) | ||
[[math]] || [[math]] | [[math]] || [[math]] | ||
2^{\large{\left(\frac{1}{2}\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(2) - v_2(q) \mid \right)}} | 2^{\large{\left(\frac{1}{2}\left(\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(2) - v_2(q) \mid \right)\right)}} | ||
[[math]] || [[math]] | [[math]] || [[math]] | ||
\|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) | | \|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) | | ||
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Some useful identities: | Some useful identities: | ||
[[math]] | [[math]] | ||
n = \ | n = 2^{\large{\frac{1}{2}\|q\|_{T1} + \log_2(q)}} | ||
[[math]] | [[math]] | ||
[[math]] | [[math]] | ||
d = \ | d = 2^{\large{\frac{1}{2}(\|q\|_{T1} - \log_2(q))}} | ||
[[math]] | [[math]] | ||
[[math]] | [[math]] | ||
n d = 2^{ | n d = 2^{\|q\|_{T1}} | ||
[[math]] | [[math]] | ||
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<td><!-- ws:start:WikiTextMathRule:17: | <td><!-- ws:start:WikiTextMathRule:17: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
2^{\large{\left(\frac{1}{2}\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(2) - v_2(q) \mid \right)}}&lt;br/&gt;[[math]] | 2^{\large{\left(\frac{1}{2}\left(\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(2) - v_2(q) \mid \right)\right)}}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">2^{\large{\left(\frac{1}{2}\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(2) - v_2(q) \mid \right)}}</script><!-- ws:end:WikiTextMathRule:17 --><br /> | --><script type="math/tex">2^{\large{\left(\frac{1}{2}\left(\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(2) - v_2(q) \mid \right)\right)}}</script><!-- ws:end:WikiTextMathRule:17 --><br /> | ||
</td> | </td> | ||
<td><!-- ws:start:WikiTextMathRule:18: | <td><!-- ws:start:WikiTextMathRule:18: | ||
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<!-- ws:start:WikiTextMathRule:19: | <!-- ws:start:WikiTextMathRule:19: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
n = \ | n = 2^{\large{\frac{1}{2}\|q\|_{T1} + \log_2(q)}}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">n = \ | --><script type="math/tex">n = 2^{\large{\frac{1}{2}\|q\|_{T1} + \log_2(q)}}</script><!-- ws:end:WikiTextMathRule:19 --><br /> | ||
<!-- ws:start:WikiTextMathRule:20: | <!-- ws:start:WikiTextMathRule:20: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
d = \ | d = 2^{\large{\frac{1}{2}(\|q\|_{T1} - \log_2(q))}}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">d = \ | --><script type="math/tex">d = 2^{\large{\frac{1}{2}(\|q\|_{T1} - \log_2(q))}}</script><!-- ws:end:WikiTextMathRule:20 --><br /> | ||
<!-- ws:start:WikiTextMathRule:21: | <!-- ws:start:WikiTextMathRule:21: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
n d = 2^{ | n d = 2^{\|q\|_{T1}}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">n d = 2^{ | --><script type="math/tex">n d = 2^{\|q\|_{T1}}</script><!-- ws:end:WikiTextMathRule:21 --><br /> | ||
<br /> | <br /> | ||
Height functions can also be put on the points of <a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html" rel="nofollow">projective varieties</a>. Since <a class="wiki_link" href="/Abstract%20regular%20temperament">abstract regular temperaments</a> can be identified with rational points on <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow">Grassmann varieties</a>, complexity measures of regular temperaments are also height functions.</body></html></pre></div> | Height functions can also be put on the points of <a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html" rel="nofollow">projective varieties</a>. Since <a class="wiki_link" href="/Abstract%20regular%20temperament">abstract regular temperaments</a> can be identified with rational points on <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow">Grassmann varieties</a>, complexity measures of regular temperaments are also height functions.</body></html></pre></div> | ||