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Generalized Tenney norms and Tp interval space: Difference between revisions

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Wikispaces>genewardsmith
**Imported revision 356588326 - Original comment: **
Wikispaces>mbattaglia1
**Imported revision 356643826 - 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>2012-08-06 18:52:56 UTC</tt>.<br>
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-07 03:02:52 UTC</tt>.<br>
: The original revision id was <tt>356588326</tt>.<br>
: The original revision id was <tt>356643826</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.\frac{9}{7}.\frac{5}{3}} = \left \|
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.9/7.5/3} = \left \|
\begin{bmatrix}
\begin{bmatrix}
\log_2(2) &amp; 0 &amp; 0 &amp; 0\\
\log_2(2) &amp; 0 &amp; 0 &amp; 0\\
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[[math]]
[[math]]
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| |0 \;-7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = 15.861
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.9/7.5/3} = \left \| |0 \;-7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = 15.861
[[math]]
[[math]]


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&lt;!-- ws:start:WikiTextMathRule:7:
&lt;!-- ws:start:WikiTextMathRule:7:
[[math]]&amp;lt;br/&amp;gt;
[[math]]&amp;lt;br/&amp;gt;
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \|&amp;lt;br /&amp;gt;
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.9/7.5/3} = \left \|&amp;lt;br /&amp;gt;
\begin{bmatrix}&amp;lt;br /&amp;gt;
\begin{bmatrix}&amp;lt;br /&amp;gt;
\log_2(2) &amp;amp; 0 &amp;amp; 0 &amp;amp; 0\\&amp;lt;br /&amp;gt;
\log_2(2) &amp;amp; 0 &amp;amp; 0 &amp;amp; 0\\&amp;lt;br /&amp;gt;
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\end{array} \right]&amp;lt;br /&amp;gt;
\end{array} \right]&amp;lt;br /&amp;gt;
\right \|_\mathbf{1}&amp;lt;br/&amp;gt;[[math]]
\right \|_\mathbf{1}&amp;lt;br/&amp;gt;[[math]]
  --&gt;&lt;script type="math/tex"&gt;\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \|
  --&gt;&lt;script type="math/tex"&gt;\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.9/7.5/3} = \left \|
\begin{bmatrix}
\begin{bmatrix}
\log_2(2) &amp; 0 &amp; 0 &amp; 0\\
\log_2(2) &amp; 0 &amp; 0 &amp; 0\\
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&lt;!-- ws:start:WikiTextMathRule:8:
&lt;!-- ws:start:WikiTextMathRule:8:
[[math]]&amp;lt;br/&amp;gt;
[[math]]&amp;lt;br/&amp;gt;
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| |0 \;-7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = 15.861&amp;lt;br/&amp;gt;[[math]]
\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.9/7.5/3} = \left \| |0 \;-7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = 15.861&amp;lt;br/&amp;gt;[[math]]
  --&gt;&lt;script type="math/tex"&gt;\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.\frac{9}{7}.\frac{5}{3}} = \left \| |0 \;-7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = 15.861&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:8 --&gt;&lt;br /&gt;
  --&gt;&lt;script type="math/tex"&gt;\left \| \vec{v} \right \|_\mathbf{T1}^\mathbf{2.9/7.5/3} = \left \| |0 \;-7.925 \;\; 2.322 \;\; 5.615 \rangle \right \|_\mathbf{1} = 15.861&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:8 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Note that the value 15.861 is obtained by |0| + |-7.925| + |2.322| + |5.615|, which is the L1 norm of the vector.&lt;br /&gt;
Note that the value 15.861 is obtained by |0| + |-7.925| + |2.322| + |5.615|, which is the L1 norm of the vector.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
To confirm this, we can put smonzo |0 -2 1&amp;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.&lt;/body&gt;&lt;/html&gt;</pre></div>
To confirm this, we can put smonzo |0 -2 1&amp;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.&lt;/body&gt;&lt;/html&gt;</pre></div>
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