gjm

Mathematician. Reader. Fiftysomething. Cruciverbalist. Not very good at thinking of witty things to put in short bios.

I wasn't on Twitter before that nice Mr Musk suggested that everyone go to Mastodon, and I don't expect to be very active here, but who knows?

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gjm_at_mathstodon.xyz@momostr.pink

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Last Notes

2026-10-24 13:16:43 UTC

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This is the kind of thing the robots are good at finding nowadays and, lo, I asked a robot and it found a reference.
(If you are sufficiently allergic to robot-derived information, you may choose to stop reading here. It looks to me as if the robot got it right, though.)
Luis A. Santaló, “Convexidad en el plano hiperbólico”, Matemática y Física Teórica XIX (1969), pp. 174–184, equation (2.6). https://ciaem-iacme.org/wp-content/uploads/2019/05/Convexidad-en-el-plano-hiperbolico.pdf
It's in Spanish, which I don't read very well, so I haven't actually checked that what Santaló is calculating is the same thing as you are, but the robot thinks it is.
The formula isn't exactly the same as yours but is equivalent. Santaló's formula is 4 sinh d/2 - 4 atan sinh d/2, which will be the same as yours if atan sinh d/2 = 2 atan tanh d/4, i.e., if sinh d/2 = 2 tanh d/4 / (1 - tanh^2 d/4), which is true.
There's a paper from Santaló two years earlier (in English) that might but probably doesn't have the same or an equivalent formula in, but it isn't freely available online.
If there's an earlier appearance of the formula, the robot doesn't know of it and couldn't find it. (Neither do I or could I, but I didn't find the Santaló paper either.)

2026-10-22 12:29:21 UTC

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All the graphs in the paper by Chao and Plymen are prettier than the one in Bays & Hudson.
Unfortunately, this is because there are no graphs in the paper by Chao and Plymen.
(Incidentally, Chao and Plymen do not appear to attempt to determine whether the curve goes below the x-axis at the point highlighted by Bays & Hudson.)

2026-09-21 17:29:45 UTC

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I think there's something odd about phrasings like "why candidates or orgs want to use celebrities to pitch their causes" and "why use this person for this pitch".
_Some_ celebrity politics-pushing is in (say) advertisements put out by political organizations, and for those that's a very reasonable question. But e.g. so far as I can tell, in the incident that prompted that piece in the _Guardian_ that wasn't what happened at all. A celebrity decided that they wanted to talk about a political issue, and they did. Unless you reckon some pro-Palestinian organization paid him to say what he did, no one was "using" him. And I assume he considered he was giving, or at least gesturing towards, good reasons for agreeing with him.
So in the primary case being discussed by that piece in the _Guardian_, it seems to me like your issues 3 and 4 don't arise here, and no one needs to be being cynical or underestimating anyone's intelligence or anything.
In cases where a person or organization _is_ "using" a celebrity, I would make a sizeable bet that while they might be underestimating _your_ intelligence, they are not underestimating the intelligence of the voting public at large, which for obvious reasons they care about more.
(And, if indeed it "works", the fact that they've done it doesn't really let us infer much about the person or organization beyond "sometimes they do something that works even though we might prefer it not to work".)

2026-09-09 16:44:47 UTC

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Yeah, "fill in the details" was more dismissively phrased than I really intended. Sorry about that.

2026-09-09 10:13:20 UTC

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The investors might care to whatever extent the bad press and scandal gives them the impression that the model isn't actually as capable as OpenAI say it is, and that the big ideas in its proof of Navier-Stokes were already in its training data and it was just _lucky_ that no human had finished putting them together yet.
To my mind, how much truth there is in that doesn't depend all that much on whether interactions with Buckmaster and Alpoge were in their training data. AIUI from Buckmaster's account, the biggest ideas were already in the literature; AIUI from Tao's account, the general feeling among experts before any of this came out was that the key ideas were all there and it was just a lot of work to fill in the details and prove that N-S can blow up. And it's clear that what OpenAI's model produced is a significant advance on Buckmaster & Alpoge. So the big picture is "it did something very impressive and finished solving a big problem that humans had done a lot of the conceptual work for but whose remaining details were really hard" regardless.
I agree that investors won't give a damn about e.g. OpenAI deliberately trying to scoop a competitor who was also one of their customers, or about the ethics of training on customers' data unless paid not to.
(They may also not care about the other stuff. Investors' level of sophistication varies a lot, and probably there are enough unsophisticated ones.)

2026-08-27 11:52:37 UTC

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You could get a medal for having the mettle to meddle with a nettle.

2026-08-24 08:24:49 UTC

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Since one paper contains a claimed concrete construction and the other contains a claimed "calculational" proof that any given almost complex structure fails to be a complex structure, it seems like there's an obvious next step, and one that surely someone must already be getting their AI to do.
Or doing by hand, of course, like some sort of caveman :-).
If an AI does it, it seems like it should be able to provide either an explicit calculation showing that the alleged structure doesn't work, with all the details filled in, or a pointer to somewhere where the alleged impossibility proof breaks down when applied to the alleged structure, either of which should be human-checkable without too much pain.
Of course _both_ new results could turn out to be wrong, so neither of those outcomes would resolve things entirely.

2026-08-16 22:12:23 UTC

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Take a look at page 7.
The idea is to express the thing you're interested in in terms of integrals around the unit circle; the things in question are typically large near to points on the circle corresponding to angles that are simple rational multiples of pi; then you divide the circle into "major arcs" near to those points, and the "minor arcs" are everything else.
Then you want to prove something along the lines of "the integral over the major arcs is bigger than some upper bound on the size of integral over the minor arcs", and hence that the _whole_ integral is positive and hence nonzero.
(I am handwaving a lot because 1. the details are complicated and 2. I am far from expert on this and don't actually know the details. But the overall structure of this sort of "circle method" proof is pretty straightforward.)

2026-08-09 12:47:05 UTC

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My immediate thought: gotta be the naan because you can frisbee it. This turns out to match the observed result but it was closer than I expected.

2026-08-04 00:39:36 UTC

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A bit of pedantry: surely it's not true that the 2-adics "can go on infinitely in both directions". They can go on infinitely _to the left_ and any arbitrary finite amount to the right. Just as ordinary real numbers written in base 2 can go on infinitely to the right and any arbitrary finite amount to the left.

2026-07-25 16:55:36 UTC

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I want to see the followup from Roger Penrose and Julian Jaynes about how this means the Norse gods may have been conscious even though the ancient Greek gods weren't.
(Jaynes has been dead for nearly 30 years but of course that's far from the biggest reason why that will never happen.)

2026-07-21 14:03:27 UTC

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My reading of that thread is that this isn't a _nicer counterexample_ but a _nicer description of the same counterexample_. Have I misunderstood?
(It is very likely that I have. I am no algebraic geometer.)

2026-07-19 17:13:26 UTC

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Israel's treatment of the West Bank is shockingly evil, but I think calling human beings "brutish animals without hearts or souls" is rather how we got into this sort of situation in the first place. I bet that's exactly how a lot of Israelis talk about the Palestinians who are fighting them.
I'm all for using very harsh language for people's _actions_ but it makes my skin crawl a little when I see such things said about _people_, even if they are terrible people doing terrible things.

2026-07-18 20:13:55 UTC

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Superficial observation that might help somehow. (I suspect from Oscar's remark that he has made the same observation.)
Let A,B,C be the centres of the three half-size S. triangles making up the big S. triangle. For each 𝑛≥0, let 𝑋ₙ be the random variable (random one of A,B,C) - (random one of A,B,C), all 𝑋ₙ independent. Then the difference between any two points in the S. triangle is ∑2⁻ⁿ𝑋ₙ.
(So the expectation of the distance _squared_ is twice the expectation of |𝑋₀|² or twice (1/3)² so the RMS distance is √2/3, as per Oscar's remark.)
It feels like someone who knows more than I do about random vectors in euclidean space might find it pretty easy to use this to find the answer, or at least to find an efficient way to approximate it. Unfortunately, I know only exactly as much as I do, which isn't very much.

2026-07-18 17:42:06 UTC

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And, looking more carefully, my extrapolations do seem like they're closer to one another and to the value claimed on that page than they are to 41√3/168.
(Logs of differences: last Aitken / last Wynn -16.4, last Aitken / second-last Wynn -16.3, last two Wynn -19.9; last Aitken / claim -16.4, last Wynn / claim -24.0, second-last Wynn / claim -19.9.)
So I retract what I said before: it seems _very unlikely_ that 41√3/168 is the true value, and the available evidence suggests that the programming contest problem's claimed 0.4227021810348385578571 could well be correct.

2026-07-18 17:32:11 UTC

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A commenter on math.stackexchange.com at https://math.stackexchange.com/questions/4047421/what-is-the-average-distance-between-two-randomly-chosen-points-on-a-sierpi%C5%84ski links to a programming-contest site and claims that it mentions someone who's found an efficient way to compute this value. For me the link goes somewhere uninformative but archive.org has what I think the link was to. The page is in Korean so I am relying on machine translation. I think the mathematician mentioned on that page may possibly be fictional.
The page does give what it claims to be the correct answer to 22dp and it _doesn't_ match the conjecture: 0.4227021810348385578571 versus 0.4227028756566902918727.

2026-07-18 17:22:47 UTC

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Some numerology:
Write 𝐷ₙ for the average distance between points in the 𝑛th-order approximation to the S. triangle, where the 1st order is just the vertices and the (𝑛+1)th order consists of three scaled copies of the 𝑛th order. (Note in particular that some points appear multiple times. I think this is OK.)
Then, aside from mere accumulated floating-point calculation errors, the first ten values of 𝐷ₙ are:
0.6666666666666667
0.4986704301902867
0.44520689777765476
0.4289567387204627
0.4243722628662094
0.42313749521839766
0.4228139755506694
0.4227306238555088
0.42270937415127596
Writing 𝑔 for the conjectured 41√3/168, the logs of the differences 𝐷ₙ−𝑔 are: -1.41, -2.58, -3.79, -5.07, -6.40, -7.74, -9.11, -10.49, -11.94 which seem to be decreasing at a pretty consistent rate.
It seems plausible that the 𝐷ₙ might converge in a clean enough way to try standard convergence-acceleration techniques. Aitken Δ² extrapolation gives things for which the logs of the differences are: -6.01, -7.08, -8.93, -10.79, -12.49, -13.65, -14.07 which are more-negative than the originals and seem to be decreasing nicely. Wynn ε extrapolation, which is a bit like an iterated version of Aitken, produces broadly similar results.
All of this seems _very consistent_ with your proposed answer being correct. None of it, of course, comes within a light-year of proving anything.

2026-04-27 22:23:18 UTC

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This may well be an important and very correct and necessary book, but I really really dislike the title.
By and large, the actual nerds of Silicon Valley -- the engineers and programmers and whatnot -- aren't fascist or authoritarian at all. (Even, I _think_, the ones who work for the most blatantly evil tech companies: even a lot of employees at bloody _Palantir_ turn out to have been worried by its embracing of fascism.)
The people who are disproportionately fascist (or fascist-adjacent, or out past fascism and on the other side, or otherwise anti-democratic and generally awful) are not the nerds but the venture capitalists and big-tech-company CEOs. Some of them _started out_ as nerdy engineering types, but that's not what they are any more. (And most of them seem to have been idealistic liberals back when they were nerds. They became fascists when they _stopped_ being nerds and started being Masters of the Universe.)
It's a conspiracy of billionaires and their hangers-on, not of nerds. Nerds get enough hate and contempt without encouraging people to blur the lines between them and the bastards who own the companies some of them work for.
For the avoidance of doubt: yes, some individual nerds (e.g., Curtis Yarvin) are appalling authoritarians; yes, lots of nerds are working for companies that do evil and you can make a good case that that makes them complicit. But the people who deserve that word "Reich" attached to them are not generally the nerds.
(Declaration of interest: I am myself by any reasonable definition a nerd, I have never worked in Silicon Valley, I despise authoritarianism, and part of what bothers me is seeing People Like Me unfairly associated with fascism.)

2026-04-08 22:25:23 UTC

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Au contraire, it's definitely rouge.

2026-02-10 22:38:19 UTC

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Are you sure about your Juliuses and your Scaligers?
It looks to me as if:
Julius Caesar (the emperor) introduced the Julian calendar.
Julius Caesar Scaliger was an erudite chap but had nothing much to do with calendars.
Joseph Justus Scaliger, Julius Scaliger's son, introduced the "Julian period" and (kinda) the "Julian day number".
The "Julian" there is a reference to the Julian calendar, _not_ to his dad.
... But I am not an expert, I'm just reporting what I find in Wikipedia, and maybe the above is all wrong?

2026-01-20 20:40:07 UTC

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I suspect DLL's comment is a deliberate reference to this, but in case (1) it isn't or (2) others didn't notice, Raymond Smullyan wrote a whole book with this premise: https://en.wikipedia.org/wiki/To_Mock_a_Mockingbird

2025-10-09 14:47:54 UTC

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By "fairly small" I meant on the order of 10^-10, which is the sort of thing I would expect to be fairly easily achievable in multiple ways by things looking like linear combinations of spherical shells.

2025-10-09 12:49:30 UTC

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Note that the gravitational field of _any_ spherically-symmetrical thing is the same (up to a scaling of the total mass) _everywhere_ outside the sphere containing it. So it doesn't particularly matter whether the sample points are all on a single sphere.
Of course, that's what's true for an _actually literally spherically symmetrical_ mass density, and what you've got here is just an approximation to that. I don't think I'd have predicted ahead of time that the determinant would be so very small ... though maybe what's going on isn't that there are _incredibly small eigenvectors_ but that there are _quite a lot of fairly small eigenvectors_.
Further conjecture: if you move your sample points further out, the determinant will get smaller; if you move them further in, it will get larger. (On the grounds that I expect the spherically-symmetrical approximation to get better at larger distances.)

2025-10-09 12:22:50 UTC

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Have you looked to see what the almost-nullspace looks like? That is, what not-very-short vectors are mapped almost-to-zero? Or, to put it differently: your empirical finding shows that some distributions of mass have almost the same effect on your 83 red dots as others; what are they?
My best guess, which I don't trust much: your 83 red dots are all at a single distance, so any distribution of mass that looks "enough" like a spherical shell of any size smaller than that will look (up to scaling the masses) exactly like any other, and perhaps your nearly-nullspace vectors look like linear combinations of spherical shells.

2025-09-02 11:48:27 UTC

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It's too bad it can't be A000001. Though it wouldn't by any means be the first known-to-be-finite OEIS sequence; for instance A100000 is another.

2025-08-22 12:48:25 UTC

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Some years ago, in the UK there was an election which the Liberal Democrat candidate was expected to win, with the Conservative candidate a close second. So the Conservatives got someone to run as a "Literal Democrat", he picked up a bunch of votes from people who didn't read carefully, and the Conservatives won. (The election regulations got changed after that so we didn't get the otherwise-inevitable deluge of "Conversatives" and "Literal Democrats" fighting to see who could confuse more voters.)
Are these people hoping to attract parents who think they're getting a Montessori school? It seems kinda unlikely, so I guess they either don't know or don't care about the similarity in name. Neither is very encouraging.

2025-08-06 14:15:16 UTC

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By a curious coincidence, RFK Jr. _is_ an evil troll whose actions are likely to make some children sneeze non-stop until they drop dead, or at least something in the same conceptual vicinity.

2025-08-02 18:38:45 UTC

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I hope they made it not too difficult for people to leave early. I'm going to guess you weren't the only ones.

2025-08-02 12:26:50 UTC

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Good luck! (I like Tavener but I don't think I would cope well with 8 hours of one Tavener piece at a stretch.)

2025-08-02 10:30:57 UTC

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Pedantic note: I think you mean not "The Protecting Veil", a famous Tavener piece but only ~45 minutes long, but "The Veil of the Temple", a newer piece by the same composer that is, indeed, 8 hours long.

2025-07-22 00:04:07 UTC

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This is a slightly atypical case because usually when something's named after X it's because someone else discovered the thing first and then X wrote about it later[1], whereas in this case IIUC Moessner really did get there first but didn't _prove_ it.
[1] Though my actual favourite example is Pell's equation, which so far as I can tell Pell never worked on _at all_.

2025-06-15 13:21:05 UTC

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That paper also contains the line "the d-ism of Leibniz overtook the dotage of Newton" which is rather magnificent. Knuth puts it in quotes but doesn't say who it's from; it seems to be Charles Babbage, though his wording was a little different.

2024-12-02 21:18:59 UTC

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Whether or not _that_ was a joke, there's something very appropriate about your photo showing Aristote's Fragme Sele.

2024-10-20 15:05:25 UTC

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You don't have to expend time or money getting to and from the movie theatre. You have comfier seating. If you have friends or family, you can all watch at once for the same price. You are much less likely to catch whatever infectious diseases may be going around.
How all of that weighs against the larger screen, the maybe-better sound system, the appeal of the traditional movie-theatre atmosphere, etc., will vary from person to person. But I'm betting they wouldn't be setting their prices at 150% of cinema ticket prices if there weren't plenty of people for whom watching at home is a significantly better experience.