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Andrea S (Mar 04 2026 at 05:59):I put a paper on the arXiv on some of the research I've been up to in grad school: https://arxiv.org/abs/2602.20125 . It's building on prior work by @John Baez, Adam Yassine, and my PhD advisor and collaborator in this project: https://arxiv.org/abs/1710.11392 . Some applied results appear to reconstruct classical results on the non-existence of certain joints as lower kinematic pairs, like universal joints and sliding hinges, from first principles under minimal assumptions of how physical systems can be structured and composed.
My advisor is more of an analyst than a category theorist, which deeply informs the approach of this paper, so I would be especially grateful for folks in this community to read and give their perspective on it.
I hope to recast some of the results of this paper in a more standard categorical language and unify it with the prior work on dynamics in the near future.
John Baez (Mar 04 2026 at 21:57):Interesting! You're reminding me that I recently got an email from your advisor (David Weisbart) and I should reply to it.
What's a "lower kinematic pair"? And who uses that term? I've never heard of it.
Andrea S (Mar 04 2026 at 21:59):It’s a kind of archaic term for nonholonomic constraint between two rigid bodies—Reuleaux used the term in his “The Kinematics of Machinery” in… 1875 I think? But some contemporary authors still use the term, albeit a little differently than how we do in the present work
Andrea S (Mar 04 2026 at 22:00):Mostly in mechanical engineering though, is my impression
John Baez (Mar 04 2026 at 22:46):Okay. It's generally not a great idea to use a little-known term in an abstract, where you don't have time to explain it. If you said "nonholonomic constraint", everyone serious about classical mechanics would understand it, which is good even if it's less precise than what you ultimately want.
Anyway, that's just a minor point. I'll look at the paper in a while!
John Baez (Mar 07 2026 at 00:36):The introduction says:
with F–limits over ACM-diagrams
without saying what an F-limit or an ACM-diagram is: these are not standard terms that mathematical physicists or category theorists know. In an introduction you want to give some idea of what the paper is about, so even if you don't give formal definitions of new terms, you should still give some rough idea of what they mean: otherwise people tend to lose interest. At the very least: what is "ACM" an acronym for?
Later in the introduction:
The ACM framework provides a minimal structure needed to formulate and prove classical non–existence results for kinematic pairs [...].
Now we wonder even more what "ACM" means... and I'm also wondering what a "kinematic pair" is.
Andrea S (Mar 07 2026 at 00:40):This is very helpful. I was mostly thinking of outlining the paper’s contents at that stage, but that would make it much more readable
Andrea S (Mar 07 2026 at 00:40):Thank you for going over this so carefully!
John Baez (Mar 07 2026 at 00:52):Sure. You can just outline the contents - but do it in a way that explains any new terminology you introduce, so people can understand why your paper is worth reading.
From my paper Why mathematics is boring:
I like to put it this way. Of the people who see your math paper, 90% will read only the title. Of those who continue, 90% will read only the abstract. Of those who go still further, 90% will read only the introduction, and then quit. Thus, it pays to put a huge amount of energy into making the front end of your paper clear and enticing. This can reduce those 90% figures (which I made up) to about 80%, leading ultimately to an eightfold increase in the number of people who read beyond your paper’s introduction.
Andrea S (Mar 07 2026 at 01:05):Would it suffice to link the reader to the introduction of these definitions / just the formal definitions or is more informal elaboration here necessary? The paper is so damn long already that we tried to cut out saying anything twice, but I also recognize that’s part of the point of an introduction
John Baez (Mar 07 2026 at 01:25):It's best to say in one sentence what's the basic idea of an ACM diagram - in a rough intuitive way that people who know classical mechanics and category theory can understand. It may take several hours to write this sentence, but it's worthwhile, because apparently it's one of the main ideas in your paper, and you need to convince readers that they want to understand it. A link to the definition won't do that.
John Baez (Mar 07 2026 at 01:26):I find it's often good to pretend I'm riding in an elevator with someone and I need to explain a concept before they get off the elevator. The elevator does not have a blackboard.
Kevin Carlson (Mar 08 2026 at 00:15):I think the incentive to cut down good material just for length should be mostly neglected these days, if you’re not submitting the paper to a venue with a page limit. Especially considering John’s heuristic that very few people will read past the intro anyway, and that few papers are read on physical media these days, the length of the body seems mostly irrelevant.
Andrea S (Mar 08 2026 at 01:04):I see. Does paper length not significantly affect reviewer decisions? My concern would be that a reviewer might be happier to reject a longer paper than a shorter one because it means less reading for them
John Baez (Mar 08 2026 at 21:21):Paper length does affect reviewer decisions. In my experience, referees become significantly tougher on papers over 35 pages in length. Perhaps that length is going up now that papers are electronic - but I don't see why, because the new technology doesn't make it any easier for referees to read long papers, or enjoy that task.
But if you want a referee to accept your paper, it helps tremendously to clearly explain what the paper is doing in the introduction, without relying on jargon that's only defined later in the paper. E.g. if you say "with F–limits over ACM-diagrams" early in the introduction, a referee is likely to think "oh, this paper is only for people who know what those things are - not me", and that's not good for getting the paper accepted. So the desire to save space does not justify writing an unclear introduction.
There could be other places in the paper where you could save space.
Andrea S (Mar 08 2026 at 21:24):Sweet! Thank you for the advice
John Baez (Mar 08 2026 at 21:29):Sure! I think this is the paragraph you need to work on most:
The results of this paper show that, under natural locality assumptions, a configuration space exists if the constraint data decomposes, and in that case it appears as a rigid universal object determined entirely by the interactions. Theorem 4.1 identifies configuration spaces with F–limits over ACM-diagrams and establishes rigidity, and Theorem 4.4 gives an existence criterion by reducing general systems to decomposing diagrams. When compatibility fails, locally specified interaction data need not assemble into a global configuration space; explicit examples exhibit this phenomenon (Examples 1 and 10). The ACM framework provides a minimal structure needed to formulate and prove classical non–existence results for kinematic pairs (Theorems 5.3, 5.4, and 5.5). The framework also serves as a foundation for subsequent work on dynamics.
This is the key place where you are describing your work for the first time. Since at this stage we don't know what "ACM-diagram", "F-limits", "ACM framework", "decomposing diagram" or "kinematic pair" means, it reads more like this:
The results of this paper show that, under natural locality assumptions, a configuration space exists if the constraint data decomposes, and in that case it appears as a rigid universal object determined entirely by the interactions. Theorem 4.1 identifies configuration spaces with SOMETHING and establishes rigidity, and Theorem 4.4 gives an existence criterion by reducing general systems to SOMETHING. When compatibility fails, locally specified interaction data need not assemble into a global configuration space; explicit examples exhibit this phenomenon (Examples 1 and 10). The SOMETHING framework provides a minimal structure needed to formulate and prove classical non–existence results for SOMETHING (Theorems 5.3, 5.4, and 5.5). The framework also serves as a foundation for subsequent work on dynamics.
The easiest way to improve this is to indicate with scare quotes that the reader isn't expected to understand the jargon - it's not the standard jargon in some field they don't know yet!
Like this:
The results of this paper show that, under natural locality assumptions, a configuration space exists if the constraint data decomposes, and in that case it appears as a rigid universal object determined entirely by the interactions. Theorem 4.1 identifies configuration spaces with something we call 'F-limits of ACM-diagrams'. and establishes rigidity, and Theorem 4.4 gives an existence criterion by reducing general systems to 'decomposing diagrams' [or better, if true: 'certain ACM-diagrams']. When compatibility fails, locally specified interaction data need not assemble into a global configuration space; explicit examples exhibit this phenomenon (Examples 1 and 10). The framework of ACM-diagrams provides a minimal structure needed to formulate and prove classical non–existence results for 'kinematic pairs', which are.... SOME INTUITIVE EXPLANATION (Theorems 5.3, 5.4, and 5.5). The framework also serves as a foundation for subsequent work on dynamics.