Defactoring: Difference between revisions
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In accordance with this research and reasoning, this article henceforth will eschew the terms saturation and contorsion in favor of defactored and enfactored. | In accordance with this research and reasoning, this article henceforth will eschew the terms saturation and contorsion in favor of defactored and enfactored. | ||
=== deeper dive on concerns re: contorsion concept | === deeper dive on concerns re: contorsion concept; enfactorization illustrated with lattices to demonstrate pathology of temperoids === | ||
== identifying enfactored mappings == | == identifying enfactored mappings == | ||
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So this implementation begins by transposing the matrix, so that when it then performs the Hermite Decomposition, it is doing a column decomposition. We then take the unimodular matrix from the decomposition using <code>First[]</code>, and <code>Transpose[]</code> it to in effect undo the transposition we did at the beginning. | So this implementation begins by transposing the matrix, so that when it then performs the Hermite Decomposition, it is doing a column decomposition. We then take the unimodular matrix from the decomposition using <code>First[]</code>, and <code>Transpose[]</code> it to in effect undo the transposition we did at the beginning. | ||
==== by hand ==== | ==== by hand ==== | ||
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This is pretty long so I've put it in a collapsible section. Just click "expand" if you're interested: | This is pretty long so I've put it in a collapsible section. Just click "expand" if you're interested: | ||
<div class="mw-collapsible mw-collapsed"> | <div class="mw-collapsible mw-collapsed"> | ||
</div> | </div> | ||
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== other stuff to report == | == other stuff to report == | ||
The rest of the material in this article is not strictly necessary to understand canonical form and defactoring, but for posterity the work Dave and Douglas did to attain their insights has been summarized here in case it may be helpful to anyone else who might want to iterate on this later. | |||
=== criteria === | === criteria === | ||
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* But now if we look at the RREF of this mapping with this HNF, we have to divide rows until all pivots are 1. So that second row would be changed to {{map|0 1 <span><math>\frac12</math></span>}}. So now RREF won't match IRREF, because it contains a noninteger. The only way to prevent this would be if a pivot of the HNF was not 1, but still every entry in that row was evenly divisible by that value, such as {{vector|{{map|1 1 0}} {{map|0 2 4}}}}. But now that row is enfactored by its pivot's value. | * But now if we look at the RREF of this mapping with this HNF, we have to divide rows until all pivots are 1. So that second row would be changed to {{map|0 1 <span><math>\frac12</math></span>}}. So now RREF won't match IRREF, because it contains a noninteger. The only way to prevent this would be if a pivot of the HNF was not 1, but still every entry in that row was evenly divisible by that value, such as {{vector|{{map|1 1 0}} {{map|0 2 4}}}}. But now that row is enfactored by its pivot's value. | ||
* If the mapping is enfactored, that is the case when the HNF ≠ DCF. | * If the mapping is enfactored, that is the case when the HNF ≠ DCF. | ||
=== sum-and-difference defactoring === | === sum-and-difference defactoring === | ||