Hodge dual: 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>2012-01-02 01:36:44 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-02 01:37:38 UTC</tt>.<br>

: The original revision id was <tt>~~289020983~~</tt>.<br>

: The original revision id was <tt>289021025</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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=The dual=

Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that <U|Vº> = U∧V for every k-multival U.

Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that <U|Vº> = U∧V for every k-multival U.

=Computing the dual=

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<br />

<h1 id="toc1"><a name="The dual"></a>The dual</h1>

Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that &lt;U|Vº&gt; = U∧V for every k-multival U.<br />

Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that &lt;U|Vº&gt; = U∧V for every k-multival U.<br />

<br />

<h1 id="toc2"><a name="Computing the dual"></a>Computing the dual</h1>

@@ 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>2012-01-02 01:36:44 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-02 01:37:38 UTC</tt>.<br>
-: The original revision id was <tt>289020983</tt>.<br>
+: The original revision id was <tt>289021025</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 14: / Line 14: @@
 =The dual=
-Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that &lt;U|Vº&gt; = U∧V for every k-multival U.
+Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that &lt;U|Vº&gt; = U∧V for every k-multival U.
 =Computing the dual=
@@ Line 30: / Line 30: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="The dual"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;The dual&lt;/h1&gt;
-Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that &amp;lt;U|Vº&amp;gt; = U∧V for every k-multival U.&lt;br /&gt;
+Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that &amp;lt;U|Vº&amp;gt; = U∧V for every k-multival U.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Computing the dual"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Computing the dual&lt;/h1&gt;