Generalized Tenney norms and Tp interval space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 356542386 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 356543432 - 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>2012-08-06 | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 13:01:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>356543432</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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[[math]] | [[math]] | ||
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\ | \left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| | ||
\begin{bmatrix} | \begin{bmatrix} | ||
\log_2(2) & 0 & 0 & 0\\ | \log_2(2) & 0 & 0 & 0\\ | ||
| Line 103: | Line 103: | ||
0 & 0 & \log_2(5) & 0\\ | 0 & 0 & \log_2(5) & 0\\ | ||
0 & 0 & 0 & \log_2(7) | 0 & 0 & 0 & \log_2(7) | ||
\end{bmatrix} | \end{bmatrix} \cdot \left[ \begin{array}{rrrrrl} | ||
| & 1 & 0 & 0 & 0 & \rangle\\ | |||
| & 0 & 2 & 0 & -1 & \rangle\\ | |||
| & 0 & -1 & 1 & 0 & \rangle | |||
\end{array} \right] \cdot \left[ \begin{array}{rrrrl} | |||
| & 0 & -2 & 1 & \rangle | |||
\end{array} \right] | |||
\right \|_\mathbf{1} | |||
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| |0 \;\text{-}7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = |0| + |-7.925| + |2.322| + |5.615| = 15.861 | |||
[[math]] | [[math]] | ||
To confirm this, we can put smonzo |0 -2 1> back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(n·d) for any interval n/d, we can confirm that we have the right answer above by noting that log(245·243) is indeed equal to 15.861.</pre></div> | To confirm this, we can put smonzo |0 -2 1> back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(n·d) for any interval n/d, we can confirm that we have the right answer above by noting that log(245·243) is indeed equal to 15.861.</pre></div> | ||
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<!-- ws:start:WikiTextMathRule:7: | <!-- ws:start:WikiTextMathRule:7: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\ | \left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \|&lt;br /&gt; | ||
\begin{bmatrix}&lt;br /&gt; | \begin{bmatrix}&lt;br /&gt; | ||
\log_2(2) &amp; 0 &amp; 0 &amp; 0\\&lt;br /&gt; | \log_2(2) &amp; 0 &amp; 0 &amp; 0\\&lt;br /&gt; | ||
| Line 227: | Line 233: | ||
0 &amp; 0 &amp; \log_2(5) &amp; 0\\&lt;br /&gt; | 0 &amp; 0 &amp; \log_2(5) &amp; 0\\&lt;br /&gt; | ||
0 &amp; 0 &amp; 0 &amp; \log_2(7)&lt;br /&gt; | 0 &amp; 0 &amp; 0 &amp; \log_2(7)&lt;br /&gt; | ||
\end{bmatrix}&lt;br/&gt;[[math]] | \end{bmatrix} \cdot \left[ \begin{array}{rrrrrl}&lt;br /&gt; | ||
--><script type="math/tex">\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\ | | &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; \rangle\\&lt;br /&gt; | ||
| &amp; 0 &amp; 2 &amp; 0 &amp; -1 &amp; \rangle\\&lt;br /&gt; | |||
| &amp; 0 &amp; -1 &amp; 1 &amp; 0 &amp; \rangle&lt;br /&gt; | |||
\end{array} \right] \cdot \left[ \begin{array}{rrrrl}&lt;br /&gt; | |||
| &amp; 0 &amp; -2 &amp; 1 &amp; \rangle&lt;br /&gt; | |||
\end{array} \right]&lt;br /&gt; | |||
\right \|_\mathbf{1}&lt;br /&gt; | |||
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| |0 \;\text{-}7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = |0| + |-7.925| + |2.322| + |5.615| = 15.861&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| | |||
\begin{bmatrix} | \begin{bmatrix} | ||
\log_2(2) & 0 & 0 & 0\\ | \log_2(2) & 0 & 0 & 0\\ | ||
| Line 234: | Line 248: | ||
0 & 0 & \log_2(5) & 0\\ | 0 & 0 & \log_2(5) & 0\\ | ||
0 & 0 & 0 & \log_2(7) | 0 & 0 & 0 & \log_2(7) | ||
\end{bmatrix}& | \end{bmatrix} \cdot \left[ \begin{array}{rrrrrl} | ||
& | | & 1 & 0 & 0 & 0 & \rangle\\ | ||
| & 0 & 2 & 0 & -1 & \rangle\\ | |||
| & 0 & -1 & 1 & 0 & \rangle | |||
\end{array} \right] \cdot \left[ \begin{array}{rrrrl} | |||
| & 0 & -2 & 1 & \rangle | |||
\end{array} \right] | |||
\right \|_\mathbf{1} | |||
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| |0 \;\text{-}7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = |0| + |-7.925| + |2.322| + |5.615| = 15.861</script><!-- ws:end:WikiTextMathRule:7 --><br /> | |||
<br /> | <br /> | ||
To confirm this, we can put smonzo |0 -2 1&gt; back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(n·d) for any interval n/d, we can confirm that we have the right answer above by noting that log(245·243) is indeed equal to 15.861.</body></html></pre></div> | To confirm this, we can put smonzo |0 -2 1&gt; back into rational form to see that it represents the interval 245/243. As the L1 norm is supposed to give log(n·d) for any interval n/d, we can confirm that we have the right answer above by noting that log(245·243) is indeed equal to 15.861.</body></html></pre></div> | ||