Interior product: Difference between revisions
Wikispaces>genewardsmith **Imported revision 315715304 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 509654768 - 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 13:54:32 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>509654768</tt>.<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= | ||
The //interior product// is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or [[Wedgies and Multivals|n-map]], a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn). | The //interior product// is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or [[Wedgies and Multivals|n-map]], a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn). | ||
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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Interior product</title></head><body><!-- ws:start:WikiTextTocRule:4:&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:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: --> | <a href="#Applications">Applications</a><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --> | <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>Interior product</title></head><body><!-- ws:start:WikiTextTocRule:4:&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:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: --> | <a href="#Applications">Applications</a><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --> | ||
<!-- ws:end:WikiTextTocRule:7 --><br /> | <!-- ws:end:WikiTextTocRule:7 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | <!-- ws:start:WikiTextLocalImageRule:8:&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:8 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | ||
The <em>interior product</em> is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or <a class="wiki_link" href="/Wedgies%20and%20Multivals">n-map</a>, a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).<br /> | The <em>interior product</em> is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or <a class="wiki_link" href="/Wedgies%20and%20Multivals">n-map</a>, a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).<br /> | ||
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