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Wikispaces>genewardsmith **Imported revision 278342188 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 509655488 - 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> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 13:59:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>509655488</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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<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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | <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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | ||
[[image:mathhazard.jpg align="center"]] | |||
=Definition= | =Definition= | ||
Given a reduced list of [[Harmonic limit|p-limit]] vals V, we may define a set of //transversal generators// for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By //reduced// is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r. | Given a reduced list of [[Harmonic limit|p-limit]] vals V, we may define a set of //transversal generators// for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By //reduced// is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r. | ||
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* Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.) | * Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.) | ||
* Consider the rest to be a monzo, which may be converted to a rational number if you prefer | * Consider the rest to be a monzo, which may be converted to a rational number if you prefer | ||
* This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney height]] by multiplying by the commas of V | * This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney height]] by multiplying by the commas of V</pre></div> | ||
</pre></div> | |||
<h4>Original HTML content:</h4> | <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>Transversal generators</title></head><body><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Finding the transversal generators">Finding the transversal generators</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <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>Transversal generators</title></head><body><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Finding the transversal generators">Finding the transversal generators</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | ||
<!-- ws:end:WikiTextTocRule:10 --><br /> | <!-- ws:end:WikiTextTocRule:10 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | <!-- ws:start:WikiTextLocalImageRule:31:&lt;div style=&quot;text-align: center&quot;&gt;&lt;img src=&quot;/file/view/mathhazard.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/div&gt; --><div style="text-align: center"><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /></div><!-- ws:end:WikiTextLocalImageRule:31 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | ||
Given a reduced list of <a class="wiki_link" href="/Harmonic%20limit">p-limit</a> vals V, we may define a set of <em>transversal generators</em> for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By <em>reduced</em> is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r.<br /> | Given a reduced list of <a class="wiki_link" href="/Harmonic%20limit">p-limit</a> vals V, we may define a set of <em>transversal generators</em> for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By <em>reduced</em> is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r.<br /> | ||
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