Structure metric: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2015-11-22 14:10:14 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-22 20:36:25 UTC</tt>.<br>

: The original revision id was <tt>~~567379759~~</tt>.<br>

: The original revision id was <tt>567411253</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>

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==Sparcity==

If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the //average distance// in the space is S/(2(n-1)(n-2)), and the //sparcity// of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less. </pre></div>

If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the //average distance// in the space is S/(2(n-1)(n-2)), and the //sparcity// of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less.

Tempering will often shrink distances and so increase density. For example, the duodene has a sparcity of 0.3686. Tempering by [[srutal]], where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparcity becomes 0.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Structure metric</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

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<br />

<h2 id="toc6"><a name="Invariants-Sparcity"></a>Sparcity</h2>

If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the <em>average distance</em> in the space is S/(2(n-1)(n-2)), and the <em>sparcity</em> of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less.</body></html></pre></div>

If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the <em>average distance</em> in the space is S/(2(n-1)(n-2)), and the <em>sparcity</em> of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less. <br />

<br />

Tempering will often shrink distances and so increase density. For example, the duodene has a sparcity of 0.3686. Tempering by <a class="wiki_link" href="/srutal">srutal</a>, where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparcity becomes 0.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2015-11-22 14:10:14 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-11-22 20:36:25 UTC</tt>.<br>
-: The original revision id was <tt>567379759</tt>.<br>
+: The original revision id was <tt>567411253</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>
@@ Line 73: / Line 73: @@
 ==Sparcity==
-If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the //average distance// in the space is S/(2(n-1)(n-2)), and the //sparcity// of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less. </pre></div>
+If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the //average distance// in the space is S/(2(n-1)(n-2)), and the //sparcity// of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less.
+Tempering will often shrink distances and so increase density. For example, the duodene has a sparcity of 0.3686. Tempering by [[srutal]], where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparcity becomes 0.</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Structure metric&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;/div&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Invariants-Sparcity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Sparcity&lt;/h2&gt;
-If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the &lt;em&gt;average distance&lt;/em&gt; in the space is S/(2(n-1)(n-2)), and the &lt;em&gt;sparcity&lt;/em&gt; of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less.&lt;/body&gt;&lt;/html&gt;</pre></div>
+If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the &lt;em&gt;average distance&lt;/em&gt; in the space is S/(2(n-1)(n-2)), and the &lt;em&gt;sparcity&lt;/em&gt; of the space is S/(2(n-1)^2(n-2)). The sparcity is 1 when all points are at the same distance, but otherwise less. &lt;br /&gt;
+&lt;br /&gt;
+Tempering will often shrink distances and so increase density. For example, the duodene has a sparcity of 0.3686. Tempering by &lt;a class="wiki_link" href="/srutal"&gt;srutal&lt;/a&gt;, where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparcity becomes 0.&lt;/body&gt;&lt;/html&gt;</pre></div>