Convex scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 266449450 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 266450248 - 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>2011-10-19 12: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-19 12:46:16 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>266450248</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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Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent defintion can be given in terms of the [[http://en.wikipedia.org/wiki/Injective_hull|injective hull]] of the Z-module, which extends monzos to fractional monzos and allows for the c_i to be rational numbers. By dividing through by | Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent defintion can be given in terms of the [[http://en.wikipedia.org/wiki/Injective_hull|injective hull]] of the Z-module, which extends monzos to fractional monzos and allows for the c_i to be rational numbers. By dividing through by | ||
[[math]] | [[math]] | ||
$c = c_1 + c_2 + \dots + c_k$ | $c = c_1 + c_2 + \dots + c_k$ | ||
[[math]] | [[math]] | ||
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<br /> | <br /> | ||
Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent defintion can be given in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hull</a> of the Z-module, which extends monzos to fractional monzos and allows for the c_i to be rational numbers. By dividing through by <br /> | Note that in this definition, a1, a2.. and b are elements of the Z-module, and c1, c2... are integers, so the only operations used are those defined for every Z-module. An equivalent defintion can be given in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hull</a> of the Z-module, which extends monzos to fractional monzos and allows for the c_i to be rational numbers. By dividing through by <br /> | ||
<!-- ws:start:WikiTextMathRule:1: | <!-- ws:start:WikiTextMathRule:1: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
$c = c_1 + c_2 + \dots + c_k$&lt;br/&gt;[[math]] | |||
--><script type="math/tex"> | --><script type="math/tex">$c = c_1 + c_2 + \dots + c_k$</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
we obtain<br /> | |||
<!-- ws:start:WikiTextMathRule:2: | <!-- ws:start:WikiTextMathRule:2: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
[[math]]&lt;br /&gt; | |||
$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$&lt;br/&gt;[[math]] | $b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$&lt;br/&gt;[[math]] | ||
--><script type="math/tex">$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</script><!-- ws:end:WikiTextMathRule:2 --><br /> | --><script type="math/tex">[[math]] | ||
$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</script><!-- ws:end:WikiTextMathRule:2 --><br /> | |||
where d_i = c_i/c.<br /> | where d_i = c_i/c.<br /> | ||
<br /> | <br /> | ||