[HN Gopher] A Curious Integral ___________________________________________________________________ A Curious Integral Author : todsacerdoti Score : 173 points Date : 2023-01-06 17:32 UTC (5 hours ago) (HTM) web link (golem.ph.utexas.edu) (TXT) w3m dump (golem.ph.utexas.edu) | tomrod wrote: | I have a social group of folks that work out hard integrals via | social media -- friends and friends of friends over the years in | quant jobs and graduate school where we work on fun things. | | This author obviously knows this at a deeper level, especially | the academic literature, where our group is more raw about it. So | fun. Thanks to the author and the poster for a pleasant Friday | afternoon read. | dxbydt wrote: | > I have a social group of folks that work out hard integrals | via social media | | Heh heh! Glad am not the only one. I used to think this was | purely an Asian habit, borne out of excessive focus on math | problems during the 11-12 grades in order to pass the grueling | entrance exams. I mostly share pdfs of math problems with | former classmates thru linkedin. | chrisshroba wrote: | Any chance I could get in on this? Sounds like a lot of fun | and I love chatting about interesting math problems and | meeting like-minded folks! | dxbydt wrote: | Well, I live in the midwest, a small republican town in | flyover country, with no fancypants private schools nearby | :) The kids here go to public school. I thought I can help | them out. So I volunteered my "math services" to the public | school teachers & set up a side project website, weekly 1-1 | math sessions. Long story short - this year three 6th | graders I work with ended up acing the AMC 10, which is a | contest that American kids take in the 10th grade. One of | them even made the AIME cutoff, which is the top 1% i.e. | top 3000 of the 300,000 contestants. So I'm thinking of | increasing my efforts in that direction. Maybe host a math | contest forum for working adults where we can work on | interesting integrals & suchlike :) Lemme know what you | think, perhaps we can collab. | hgsgm wrote: | EDIT: apparently the dictionary supports "acing" to mean | "doing very well" even if there are clearly higher | measures of performance. Still, "acing" generally means | "the highest rank" which would at least be the | Distinguished Honor Roll (approx 125-130) if you don't | want to go all the way to exact 100% score. | | Original post: | | > three 6th graders I work with ended up acing the AMC | 10, which is a contest that American kids take in the | 10th grade. One of them even made the AIME cutoff, | | This is 100% impossible. How could you "ace" a contest | and not qualify for the lext level? No one did better | than "acing" and there is no lottery. | | Do you mean "100" on the AMC 10, not "acing" ? That's | impressive and plausible, but "acing" is 150, which is | achieved by only about 10 people in the world each year, | and takes years of intense study, and all of them qualify | for the AIME. | | Also, AIME is currently top 2.5% of AMC 10, not top 1%. | | Score reports at: http://amc- | reg.maa.org/reports/GeneralReports.aspx | dxbydt wrote: | > "acing" is 150 | | Sorry, I meant acing in the "scored way above what I'd | expect 6th graders to do" sense. One of them made the | AIME cutoff, two missed that cutoff but quite narrowly. | | > "acing" is 150, which is achieved by only about 10 | people in the world each year | | Agreed, that is pretty impressive. | c7b wrote: | Do you people have any more open social channels? I used to be | active a bit on CrossValidated (my favorite math problems are | in probability and stats), but something more social could be | nice. | tom-thistime wrote: | A lot of good points here. | | Another possible lesson from these integrals: if someone has | deliberately prepared the problem you're studying, and made it | misleading on purpose, then solving the problem may be a | different game from studying a problem that arose for some other | reason. | heliophobicdude wrote: | Very similar to Cunningham's Law. | | https://meta.wikimedia.org/wiki/Cunningham%27s_Law | w10-1 wrote: | It seems like an exercise in mathematical self-absorption, but | it's really a fundamental question for science: assuming you have | equations and even a model of the world that is really, really | close, is that any kind of proof? | | And even if math/logic is granted its perfect world, it's never | even self-complete. | | Welcome back to the world of practical wisdom, where the rest of | us live and work :) | [deleted] | [deleted] | [deleted] | hintymad wrote: | It's always heartwarming to see people passionately geek out, | seeking to advance our civilization. It's also curious that a | land of geeks like the US would have a culture of looking down | upon geeks, to the point that Paul Graham would write essays like | Why Nerds Are Unpopular[1]. I couldn't even understand that essay | when reading it for the first time, as it was a novel concept in | my country that people didn't appreciate hardworking and passion | in hard subjects. It's even stranger that Americans thought it | was a great virtual to toil in sports, like shooting hoops | thousands of times a day, but it was a sin to toil in STEM, like | solving maths problems for fun. But well, it's a topic for | another day. | | [1] http://www.paulgraham.com/nerds.html. Quote: "I know a lot of | people who were nerds in school, and they all tell the same | story: there is a strong correlation between being smart and | being a nerd, and an even stronger inverse correlation between | being a nerd and being popular. Being smart seems to make you | unpopular." | [deleted] | hk__2 wrote: | As a non-math person I didn't understand anything, but I enjoyed | reading just for the enthusiasm of the author. | pipingdog wrote: | If you want to get a flavor of the article without being too | much of a math person, you can watch the 3Blue1Brown video | linked in the article, which captures a similar surprise that | for certain values a formula maps exactly to a "round" | quantity, then starts to diverge after a while. | | https://www.youtube.com/watch?v=851U557j6HE | schlauerfox wrote: | My ignorant intuition is since it's the area between sin(x)/x | and the x axis, but as you get further into infinity the 1/x | still keeps getting smaller slowly but sin(x) always is the | same magnitude, the integral gets slightly further away the | closer you get to infinity and that slight difference adds | up? | hgsgm wrote: | Yes but the question is why the value is _exact_ for small | N. | singularity2001 wrote: | It reminds me of those 'surprising' rational approximations of p | where xyz/abc[?]p with 7 digits precision and abcd/efgh[?]p with | 9 digits precision. It's not that surprising when you take into | account that abcd/efgh does not save you many digits compared to | just writing p out. | | Somewhat similarly the +-23 chars of | | "[?] 0 [?] cos(2x)[?] n 1 [?] cos(xn)dx" | | giving 43 digits of another number is not _that_ surprising and | in the realm of what you can expect with some likelihood. | | I expect that one could theoretically find some double integral | with less then 25 signs which approximates e^p or ANY number to | 50 digits. | | Now if you find some 20 char expression which approximates | another 20 char expression up to 100000 digits and THEN suddenly | takes a different turn, that would be really curious. Like those | properties of natural numbers which are true for some orders of | magnitude before someone found a counter example. | version_five wrote: | From the article: Jaded nonmathematicians told | us it's just a coincidence, so what is there to explain? | | The actual reason they get into is interesting and much deeper | and orthogonal to the information content of the representation | Jabbles wrote: | (1+9^-4(6*7))^3^2^85 is an expression that uses all digits 1-9 | once and approximates e to 10^25 digits. | | https://math.stackexchange.com/questions/1945026/an-amazing-... | wyager wrote: | This one isn't so interesting, as it's super easy to generate | `e` in lots of different ways (such as the limit the | expression is approximating). `e` is a very low-kolmogorov- | complexity constant. The error term in the integral, on the | other hand, has no apparent reason to be. | cozzyd wrote: | yeah but that's silly since | | lim n->\infty (1 + \frac{1}{n})^ n | | is even easier to remember | mananaysiempre wrote: | Having good "small" rational approximations is actually a | characterization of transcendental numbers: there are upper | bounds for how good rational approximations to an algebraic | number can be as the maximum allowed denominator grows, so by | proving that this bound is violated for your number you can | prove it's not algebraic; that's how people initially went | about constructing transcendental numbers and--later--proving p | and _e_ were such. | | (Nowadays, it's easy to construct a transcendental number | because it's easy to construct a noncomputable one; | constructing a transcendental _computable_ number still | requires additional ideas such as those bounds--I don't really | know of a simple way to do it.) | wyager wrote: | The kolmogorov complexity of the provided integral is vastly | lower than the >140 bits needed to naively represent the error | term. Something else is going on here. | hgsgm wrote: | What are 140bits of the error term? The _approximation_ has | 140 bits. | kloch wrote: | Coincidences are _everywhere_ in math and physics, but our monkey | brains just can 't accept that they are meaningless. | olddustytrail wrote: | Your post is meaningless. Where else does meaning exist other | than within a brain? | marshray wrote: | I counted the number of symbols in the equation after "As far as | I can tell, the known proofs that"... | | It contained 41 letters and other miscellaneous math symbols. | | Therefore, I find it unsurprising that it generates a specific | constant having a magnitude of 10^-43. | | https://en.wikipedia.org/wiki/Kolmogorov_complexity | civilized wrote: | Well I hope it wouldn't be too different in French then! | abetusk wrote: | I only skimmed the article and skipped to the 3Brown1Blue [0] | video which I found very enlightening. | | Basically, the tldr version is that product under the integral | can be considered a sum, of sorts, in the Fourier domain | (convolution <-> product and products turn into sums under some | transformation of exponentiation) and when the coefficients of | that sum cross a constant, then the original integral becomes | less than pi. | | That is, when $\sum_{i=0}^n \frac{1}{2 i + 1} >= 1$, that's the | transition point. 15 in the denominator is where that sum is | greater than one. | | Awesome stuff. | | [0] https://youtu.be/851U557j6HE | deepspace wrote: | I have been fascinated by these integrals for a long time and am | happy to see them getting more attention. 3Blue1Brown recently | made a video on the topic: | https://www.youtube.com/watch?v=851U557j6HE | | What strikes me is the reminder that it is never possible to | "prove" something by pointing out that it is true for all known | cases. (See also Black Swan events). In Greg Egan's example, if | you stopped testing at 10^43 iterations, you would be _very | tempted_ to conclude that the identity holds for all n, for | example. | _nalply wrote: | To prove something for all _n_ you need to do mathematical | induction: | | First prove that something is true for some _n_ , usually _n_ = | 1. | | Then prove that if it's true for _n_ it 's for _n_ + 1, too. | | Boom. It's true for all _n_. | | But the second step is sometimes very hard or even perhaps | impossible. | Chinjut wrote: | Induction is one way to prove something for all n but hardly | the only way. | dysoco wrote: | I wondered if the Greg Egan he named provided a comment on the | integral was THE Greg Egan (author of Permutation City) and | following the comment indeed he was! | spindle wrote: | IIRC, THE Greg Egan has a (genuinely excellent, peer-reviewed) | maths paper with John Baez, and a number of smaller or | unpublished contributions to professional maths. | anthk wrote: | I was about to say this. | wyager wrote: | Yes, Greg is an active math guy. You can often find comments | from Greg, John Baez, and Scott on each other's blogs. They are | some of my favorite bloggers/writers! ___________________________________________________________________ (page generated 2023-01-06 23:00 UTC)