Hahn distance: Difference between revisions
Wikispaces>genewardsmith **Imported revision 357283890 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 357283950 - 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>2012-08-11 02: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-11 02:27:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>357283950</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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= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|) | = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|) | ||
[[math]] | [[math]] | ||
We may take this formula | We may take this formula and apply it to any triple of real numbers ||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|) | ||
If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by | If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by | ||
[[math]] | [[math]] | ||
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= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)&lt;br/&gt;[[math]] | = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)&lt;br/&gt;[[math]] | ||
--><script type="math/tex">= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)</script><!-- ws:end:WikiTextMathRule:1 --><br /> | --><script type="math/tex">= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
We may take this formula | We may take this formula and apply it to any triple of real numbers ||(a, b, c)||_hahn = max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)<br /> | ||
If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by<br /> | If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by<br /> | ||
<!-- ws:start:WikiTextMathRule:2: | <!-- ws:start:WikiTextMathRule:2: | ||