On MathOverflow the relevant vendor of the Computer Algebra System said something like "Borwein playfully called it a curiosity rather than claiming it was a bug, but it took me 3 days to figure out what was going on."
Happy to see the new video/s I got slightly divergent as well from the channel and branched off a bit into the Iwahori-Hecke and Steenrod Algebras. I'll be finishing a couple texts on Optic Modal Analysis and Regression Analysis where the Fourier transform is used somewhat heavily so the reference will not go to waste on me. I'd rather spare myself the N-teenth tangential textbook for the moment.
Btw, when I finish cracking the Riemann Hypothesis, I'll make sure to break you off 90% of the prize money. I've already come to the conclusion that a genuine proof of it would require at least 9/10s plagiarism from a great mind than I. 😂
3blue1brown covered this very well. I cannot add links for some reason, including links to TH-cam itself. So "3blue1brown borwein" will help you find it yourself.
3b1b released a valuable elementary proof of sin(x)=x(1-(x/π)²)(1-(x/2π))²··· He also coded popular animation software, and with his reach, he inspires a lot of people.
Gentleman Borcherds is always better if you have some background in math. His own explanation is super terse, covers all interesting details. Honestly 3b1b is a joke compared to this gentleman because 3b1b only deals with only *relatively* basic stuff with just fancy animations. It is not even comparable in terms of content.
I'm confused as to how the last integral diverges. Can't you use the double angle formula to show that it's just an integral of sinc? I'm pretty certain it's just pi/(2*a^2).
I liked the explanation. It is a bit exaggerated to call the change in the integral result a "sudden" change. A delta in the 10th decimal digit is not "sudden", although perhaps a mathematician might consider it so.
What was sudden was the fact that there was any change at all from the constant values for the first six integrals. (As in, after 6 days of not showing up, the student suddenly showed up on the 7th day of class.) I don't think it was intended to be a comment on the magnitude of the change in any way.
Hi Dr Borcherds, i just wanted to ask you about what resources i should expose myself to in order to delve into the mathematics of the langlands program. Hope you consider my request.
Is there a satisfying explanation for why the sums and integrals of sin(x)/x and sin^2(x)/x^2 are all equal to pi? I haven't really seen a justification for why it "should" be true like with these integrals.
I’ve spent some time with that as well and haven’t found a satisfying answer. One approach I’ve tried is using the fact that the inverse Fourier transform “should” actually be an inverse, and funny enough my argument just shows it “should” be a constant multiple of the inverse. That constant? The integral of sinc(x) again!
Interesting: naïveté suggested to me that 17 would be the infemal oddity to toss a monkey into that wrench. I wonder what happens when you break the compactitude of support by restricting scope to semi-infinitude & only summing inversions of oddities… where’s the kernel encoded about how much these sincs surpass unity? Is anything salient about the initial half term describing how three and five appearing as duplicated divisors is what knocks an error into the totality’s deviation from unity? If only there were some salience if additive connection between them predicting divergence in the 10th…
In general, when speaking about math, large words with no symbols will be extremely difficult for most people to parse. I honestly am almost completely unable to extract any meaning from this comment, other than the fact that you guessed the pattern would break down at 17. Just pointing that out to hopefully be helpful! Often times I find the concepts are fiddly enough that the best way to explain something is with very simple, precise language.
@@huzzzzzzahh the thought was about summing over every other term rather than every natural number , omitting the lowest order 1/2 term… Having just watched the first video on elliptic integrals, it looks like that was Weirstrass’s strategy for getting the curly P and P’ to converge. Thank you Richard Borcherds, for bringing an intelligible explanation to such tricky content.
It is understood that his lectures are usually aimed at a graduate student level, so he does assume background knowledge. But if you have the right background, he is excellent. Much better than many of the lecturers I have experienced. Different channels cater to different audiences.
On MathOverflow the relevant vendor of the Computer Algebra System said something like "Borwein playfully called it a curiosity rather than claiming it was a bug, but it took me 3 days to figure out what was going on."
Just noticed you started uploading videos again, welcome back! What you're doing is extremely valuable, thank you!!
Yes absolutely!!
He doesn't know who you are nor will he read this message
Happy to see the new video/s
I got slightly divergent as well from the channel and branched off a bit into the Iwahori-Hecke and Steenrod Algebras. I'll be finishing a couple texts on Optic Modal Analysis and Regression Analysis where the Fourier transform is used somewhat heavily so the reference will not go to waste on me. I'd rather spare myself the N-teenth tangential textbook for the moment.
Btw, when I finish cracking the Riemann Hypothesis, I'll make sure to break you off 90% of the prize money. I've already come to the conclusion that a genuine proof of it would require at least 9/10s plagiarism from a great mind than I. 😂
Nothing ever beats picking a single textbook and doing as many exercises as possible
Welcome back, Sir.
3blue1brown covered this very well. I cannot add links for some reason, including links to TH-cam itself. So "3blue1brown borwein" will help you find it yourself.
Great explanation! Also check out the 3b1b series on this where he goes into more detail about convolutions
Comparing with his video, prof borcherds is using reverse click bait
3b1b is a joke
3b1b released a valuable elementary proof of
sin(x)=x(1-(x/π)²)(1-(x/2π))²···
He also coded popular animation software, and with his reach, he inspires a lot of people.
Gentleman Borcherds is always better if you have some background in math. His own explanation is super terse, covers all interesting details. Honestly 3b1b is a joke compared to this gentleman because 3b1b only deals with only *relatively* basic stuff with just fancy animations. It is not even comparable in terms of content.
very nice
Plot twist: the bug report was for the universe itself.
Keep up the good work
Great video as always
I'm confused as to how the last integral diverges. Can't you use the double angle formula to show that it's just an integral of sinc? I'm pretty certain it's just pi/(2*a^2).
I liked the explanation. It is a bit exaggerated to call the change in the integral result a "sudden" change. A delta in the 10th decimal digit is not "sudden", although perhaps a mathematician might consider it so.
What was sudden was the fact that there was any change at all from the constant values for the first six integrals. (As in, after 6 days of not showing up, the student suddenly showed up on the 7th day of class.) I don't think it was intended to be a comment on the magnitude of the change in any way.
Hi Dr Borcherds, i just wanted to ask you about what resources i should expose myself to in order to delve into the mathematics of the langlands program. Hope you consider my request.
Finally professor is back!
welcome back
hi guys do uk anyone who teaches physics in similar fashion of this level?
@@EdwinSteiner Thank you
"via science" their Quantum videos build up from basic thermodynamics in a very intuitive way www.youtube.com/@viascience
yeeeee
🤯🤯🤯
Is there a satisfying explanation for why the sums and integrals of sin(x)/x and sin^2(x)/x^2 are all equal to pi? I haven't really seen a justification for why it "should" be true like with these integrals.
I’ve spent some time with that as well and haven’t found a satisfying answer. One approach I’ve tried is using the fact that the inverse Fourier transform “should” actually be an inverse, and funny enough my argument just shows it “should” be a constant multiple of the inverse. That constant? The integral of sinc(x) again!
It has a really simple power series, so that might help
@@huzzzzzzahh Thanks for the suggestion. I thought of trying something like that too, maybe something with the Poisson summation formula?
@@MathFromAlphaToOmega I believe the content around 11:30 in the video has already addressed your question.
Interesting: naïveté suggested to me that 17 would be the infemal oddity to toss a monkey into that wrench. I wonder what happens when you break the compactitude of support by restricting scope to semi-infinitude & only summing inversions of oddities… where’s the kernel encoded about how much these sincs surpass unity? Is anything salient about the initial half term describing how three and five appearing as duplicated divisors is what knocks an error into the totality’s deviation from unity? If only there were some salience if additive connection between them predicting divergence in the 10th…
In general, when speaking about math, large words with no symbols will be extremely difficult for most people to parse. I honestly am almost completely unable to extract any meaning from this comment, other than the fact that you guessed the pattern would break down at 17. Just pointing that out to hopefully be helpful! Often times I find the concepts are fiddly enough that the best way to explain something is with very simple, precise language.
@@huzzzzzzahh the thought was about summing over every other term rather than every natural number , omitting the lowest order 1/2 term…
Having just watched the first video on elliptic integrals, it looks like that was Weirstrass’s strategy for getting the curly P and P’ to converge. Thank you Richard Borcherds, for bringing an intelligible explanation to such tricky content.
This guy may be a good and successful mathematician. But he's a horrible and lazy teacher. Always assumes we know everything before hand.
It is understood that his lectures are usually aimed at a graduate student level, so he does assume background knowledge. But if you have the right background, he is excellent. Much better than many of the lecturers I have experienced.
Different channels cater to different audiences.