[HN Gopher] A Curious Integral
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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!
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