Convex scale: 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>2011-10-19 12:47:38 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-19 12:57:35 UTC</tt>.<br>

: The original revision id was <tt>~~266450892~~</tt>.<br>

: The original revision id was <tt>266455180</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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[[math]]

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 n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by

[[math]]

$c = c_1 + c_2 + \dots + c_k$

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$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$

[[math]]

where d_i = c_i/c.

where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.

===Convex set===

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--><script type="math/tex">$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$</script><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 n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by <br />

<!-- ws:start:WikiTextMathRule:1:

[[math]]&lt;br/&gt;

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$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$&lt;br/&gt;[[math]]

--><script type="math/tex">$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$</script><br />

where d_i = c_i/c.<br />

where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.<br />

<br />

<h3 id="toc2"><a name="x-Formal definition-Convex set"></a>Convex set</h3>

@@ 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>2011-10-19 12:47:38 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-19 12:57:35 UTC</tt>.<br>
-: The original revision id was <tt>266450892</tt>.<br>
+: The original revision id was <tt>266455180</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 21: / Line 21: @@
 [[math]]
-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 n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by
 [[math]]
 $c = c_1 + c_2 + \dots + c_k$
@@ Line 29: / Line 29: @@
 $b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$
 [[math]]
-where d_i = c_i/c.
+where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.
 ===Convex set===
@@ Line 57: / Line 57: @@
   --&gt;&lt;script type="math/tex"&gt;$(c_1 + c_2 + \dots + c_k) b = c_1 a_1 + c_2 a_2 + \dots + c_k a_k$&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow"&gt;injective hull&lt;/a&gt; of the Z-module, which extends monzos to fractional monzos and allows for the c_i to be rational numbers. By dividing through by &lt;br /&gt;
+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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow"&gt;injective hull&lt;/a&gt; of the Z-module, which extends n-tuples of integers to n-tuples of rational numbers (a vector space over the rational numbers Q) and allows for the c_i to be rational numbers. By dividing through by &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:1:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 67: / Line 67: @@
 $b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;$b = d_1 a_1 + d_2 a_2 + \dots + d_k a_k$&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
-where d_i = c_i/c.&lt;br /&gt;
+where d_i = c_i/c. Note that while the coefficients d_i are allowed to be positive rational numbers (now summing to 1), b is still an integral vector, ie an n-tuple of integers.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Formal definition-Convex set"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Convex set&lt;/h3&gt;