Talk:Mathematical theory of saturation

Origin of desaturating algorithm

I have been working on the topic of saturation off and on for months now, with the assistance of Dave Keenan. We cannot figure out how the algorithm that is used here — the one that uses the Smith Decomposition (named for Henry John Stephen Smith; no relation to Gene Ward Smith, as far as we know) — to desaturate matrices works. We've tried devising our own methods that feel more straightforward, but they fail on edge cases. At this point Gene's method here feels a bit like magic. Can anyone explain how it works, or maybe at least where it came from — Gene's own ingenuity, or perhaps from the Sage math software (where we suspect he draws his name for the concept, "saturation")? --Cmloegcmluin (talk) 22:53, 1 August 2021 (UTC)

If anyone comes to this later, Dave and I managed to answer our own question. Our findings are compiled here: Defactoring_algorithms#Precedent:_Smith_defactoring --Cmloegcmluin (talk) 06:15, 27 September 2021 (UTC)

how to relate this page to the new page for canonical form

I recently published a page on the xen wiki proposing a canonical form for RTT mappings (or comma-bases), a key component of which is "defactoring" — removing common factors from their rows (or columns).

This defactoring process is in fact the same mathematical thing as "saturation". And the opposite situation, being "enfactored", is identical to the situation of being "contorted", a term which redirects to this page on the wiki as well.

Of the several methods for eliminating enfactoring/contortion by achieving defactoring/saturation which are presented on my new page, one of them is the exact algorithm described here, and another of them was derived from it.

However, I chose not to add any new thoughts about this topic to this existing page. Instead I added only to my new page. My reasons were:

1. I have been asked not to significantly alter the works of Gene Ward Smith, and I respect the intent to protect his original work here. I have made a few changes to this page recently, but only changes that I thought should help preserve Gene's genius thinking as well as help it reach people even better: I cleaned up the mathematical formatting using recently developed Mediwiki templates and what I've learned about LaTeX, I added some parenthetical clarifications that my colleagues have found helpful when discussing this page, and I also included a code implementation of Gene's original algorithm that we developed.
2. I do think that it is a proper home for the topic of defactoring/saturation to be as a subtopic of canonical form. I think this especially because if the RTT community leverages this canonical form, then defactoring/saturation ceases to be a problem. At least, the average practitioner should barely need to understand it or be aware of it.
3. Gene's page has a different target audience than mine, at least insofar as the high level of mathematical rigor and technicality it uses. My page is written as a non-mathematician, for non-mathematicians. And my understanding is that most xen wiki users and contributors (including myself) are amenable to the idea of achieving whatever structure or organization is practical such that those who are interested in such mathematical theory and proof can find it, but those who aren't don't get overwhelming levels of exposure to it and get turned off from xen music-making. So, I think both pages have good reason to stick around.
4. I have serious terminological concerns about both the words "saturation" and "contorsion", which I go into in detail on my new page.

My question is, to what extent do people think is appropriate for me to recognize/associate the new canonical form page here on this page? It could be as slight as a "See also" at the bottom. It could be a new section at the bottom where I briefly summarize the relationship between this page and the new canonical form page. Or it could be some footnotes interspersed with the original text. Or something else entirely that I haven't thought of yet.

I don't want to be hasty here so I don't plan to take any action until I hear plenty of other opinions. Thanks for your attention to this. --Cmloegcmluin (talk) 17:39, 27 September 2021 (UTC)

Okay, I decided that at least a "See also" at the bottom seemed reasonable enough. --Cmloegcmluin (talk) 19:03, 29 September 2021 (UTC)

Alright, after discussing with Mike Battaglia, this page has been moved to be the mathematically-inclined page, and a new general audience page has been created here: Saturation,_torsion,_contorsion,_and_defactoring --Cmloegcmluin (talk) 19:13, 13 November 2021 (UTC)

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