6 ms·

If we normalize the Fourier transform {{\mathcal F} f(\xi)} of a (Schwartz) function {f(x)} as {{\mathcal F} f(\xi) := \int_{\bf R} f(x) e^{-2\pi i \xi x}\ dx},

by kleiba 9y ago

If we normalize the Fourier transform {{\mathcal F} f(\xi)} of a (Schwartz) function {f(x)} as {{\mathcal F} f(\xi) := \int_{\bf R} f(x) e^{-2\pi i \xi x}\ dx}, it is well known that the Gaussian {x \mapsto e^{-\pi x^2}} is its own Fourier transform. Funny that he should mention it: just the other day, my mom and I were having a good laugh because as it turned out mom's neighbor, a Mrs. Cringleton, apparently had no idea that the Gaussian is its own Fourier transform under these circumstances. Apparently it came up over a cup of tea and bicuits they were having, and Mrs. Cringleton felt quite a bit embarrassed when my mom remarked: "But my dear, that is well-known!" Poor Mrs. Cringleton...

photon-torpedo 9y ago
Better not tell Mrs Cringleton about the hyperbolic secant...
- waqf 9y ago
  Most likely Mrs. Cringleton knew about the hyperbolic secant and just assumed it was unique up to linear factors.
jordigh 9y ago
All jokes aside, Terry is writing to an audience where this is utterly elementary. The phrase "it is well known" is seldom misused. Yes, anyone who knows anything about the Fourier transform will know the Gaussian is a fixed point, or can easily verify it for themselves. It would actually look very amateurish if Terry provided a citation for that. In fact, it would look borderline cranky. Most of the cranky "proofs" of the RH begin by dedicating a whole section or two to elementary results, such as analytic continuation, followed by three or four sections of misunderstandings or utter gibberish.
- meuk 9y ago
  Agreed. "It is well-known" should be read as "It is well-known among specialists". On the other hand, I have some professors who abused the word "trivial" to mean "it can be proven by me". One professor spent the better half of an hour thinking about a question, only to declare that the answer was trivial. His explanation followed and filled the rest of the hour.
  - WhitneyLand 9y ago
    That’s funny. I wonder if the origin of the concept of something being trivial only in hindsight can be dated, or has it always been a truism?
  - vostok 9y ago
    I think trivial means that the proof requires no serious amount of creativity.
    - zeth__ 9y ago
      >Takes less than a week to complete.
  - alphydan 9y ago
    Remember that problems in high school typically can take 1/2h to solve, in college a couple of hours ... as a PhD typically a couple of years and as a professional researcher ... many years. So for him, a problem where he was confused 20 minutes and solved it in 10 minutes is probably "trivial".
  - contravariant 9y ago
    There's a reason there's a somewhat famous joke about a professor who, when asked to explain an step he claimed to be trivial, furrowed his brow deeply, spent the best part of an hour looking at notes and scribbling on the blackboard only to declare afterwards "Yes! It is trivial!'.
    - SAI_Peregrinus 9y ago
      Or the numerous results which are obvious in hindsight.
  - ajkjk 9y ago
    (Not even specialists -- 2nd-3rd year undergraduates in engineering/math/physics!) In that case the word 'trivial' means it doesn't require complex machinery to solve. A simple consequence of arithmetic manipulations, say, may be extremely difficult to think of yourself, but once you know it it's trivial to see the validity of. I believe that counts as 'trivial' in mathematical terminology.
- munificent 9y ago
  > All jokes aside, Terry is writing to an audience where this is utterly elementary. One of the things I find fascinating about the Internet is the effect it has on writing when you have so little control over who your audience is. Every now and then, a blogger or Twitterer used to writing for some niche audience catches the larger eye of the Internet zeitgeist and all of the sudden momentarily finds themselves with a much larger, very different audience, who perceive their work entirely differently because they lack the context the author presumed. I wonder how much of the modern trends of political correctness, trigger warnings, etc. come from this experience where you can no longer safely make any assumptions about your audience. At the same time, it's not possible to write effectively when you assume the audience could be anyone, even a potentially malicious aggressor.
  - msla 9y ago
    It's the result of opening all the echo chambers to the world, including parts of the world who, for whatever reason or none at all, simply don't like you, who you are, what you are, or what you do. It also opens them to people who are just not going to get it, but who still feel compelled to comment and get a reaction, whatever kind of reaction they can elicit, and will not stop until they get a reaction or are forced to leave/stop participating. So, can we recreate the good echo chambers, such as academia, but do it right this time, such that irrelevant characteristics no longer matter but the people inside of them no longer have to deal with nutballs and assholes coming in and disrupting anything?
eesmith 9y ago
Did Mrs. Cringleton at least know about !! in the shell? https://news.ycombinator.com/item?id=2983466 https://news.ycombinator.com/item?id=2983466 ;)
- meuk 9y ago
  That is actually very good to know!
waqf 9y ago
> mom > tea and bicuits Where are you claiming to be from?
- adw 9y ago
  The same place as Dick van Dyke, one suspects.
  - munificent 9y ago
    That tiny little island in the middle of the Atlantic which, for several decades, was astonishingly highly populated. https://en.wikipedia.org/wiki/Mid-Atlantic_accent https://en.wikipedia.org/wiki/Mid-Atlantic_accent
dxbydt 9y ago
Back when I worked at Twitter, celebs used to show up randomly at the HQ at lunchtime. I was just your average Indian immigrant data scientist, not clued into American culture. The team I was on was trying to hire a lady for a front end eng role. I didn’t know what the lady’s name was, but I was told she was a very promising candidate, and we were all looking forward to the day when she joined because we had a ton of frontend work. Ok, so at lunch, my colleague said out aloud - Hey Katy Perry is in the lunchroom! Everyone got excited. So I said, Oh is that the new front end engineer ? The whole room exploded in laughter. I thought I must have made a big social gaffe. My mind quickly raced...,oh maybe she must be a celebrity of some kind. Perry Perry...which Perry do I know...So previous night I was watching some show on TBS and suddenly a Perry name flashed into my brain, so I instantly said - Oh she must be the daughter of Tyler Perry! The room just doubled up in laughter. Finally I did a wiki lookup and found out who she was. But for days, people would look at me, laugh and say he’s the Katy Perry Tyler Perry guy ha ha ha. Finally I was so embarrassed and pissed off I asked them - Do you know the difference between between Jacobian and a Hessian ? Nobody knew. Well, I said triumphantly, I might not know your katy perry from tyler perry, but atleast I know what I need to know to fo my work. So yeah, some of us do know that a Gaussian is its own Fourier transform, derivative of an exponential is the exponential itself, etc.
- jordigh 9y ago
  You were one of that day's lucky 10,000: https://www.xkcd.com/1053/ https://www.xkcd.com/1053/
  - BoiledCabbage 9y ago
    Actually, you were one of today's lucky 10,000 - and I won't make fun of you for it.
- whatshisface 9y ago
  You've given a good example of what well-known really means: "well-known among the expected audience." Aside, I honestly wonder if I would be able to recognize Katy Perry in person...
- 9y ago
jacobolus 9y ago
It is “well known” as in “this was worked out 200 years ago and is a foundational idea for several branches of mathematics and every undergraduate studying science, mathematics, or engineering learns this several times in multiple classes before the end of college, and readers of a mathematician’s blog post about his new research results can be expected to be well familiar with the idea, and if they aren’t the research will not be meaningful to them anyway”.
tzahola 9y ago
What? This is college-level knowledge...
vinchuco 9y ago
This is not a case of the turbo encabulator [1] which you may seemingly enjoy. What he likely meant is that it is well known from the literature. Or there's an easy explanation that supersedes the verbose notation. [1] https://www.youtube.com/watch?v=rLDgQg6bq7o https://www.youtube.com/watch?v=rLDgQg6bq7o
dang 9y ago
Please don't post this sort of empty dismissal here. I know it feels funny at the time, but it's boring in the end. There's no insight here, and as jordigh and others pointed out, it's a cheap shot. Since people sometimes wonder: humor is welcome on HN, lame humor not so much. It chokes out the rarer things, and there are plenty of places to get it elsewhere.

by kleiba 9y ago

photon-torpedo 9y ago

Better not tell Mrs Cringleton about the hyperbolic secant...

waqf 9y ago
Most likely Mrs. Cringleton knew about the hyperbolic secant and just assumed it was unique up to linear factors.

jordigh 9y ago

All jokes aside, Terry is writing to an audience where this is utterly elementary. The phrase "it is well known" is seldom misused. Yes, anyone who knows anything about the Fourier transform will know the Gaussian is a fixed point, or can easily verify it for themselves. It would actually look very amateurish if Terry provided a citation for that. In fact, it would look borderline cranky. Most of the cranky "proofs" of the RH begin by dedicating a whole section or two to elementary results, such as analytic continuation, followed by three or four sections of misunderstandings or utter gibberish.

meuk 9y ago
Agreed. "It is well-known" should be read as "It is well-known among specialists". On the other hand, I have some professors who abused the word "trivial" to mean "it can be proven by me". One professor spent the better half of an hour thinking about a question, only to declare that the answer was trivial. His explanation followed and filled the rest of the hour.
- WhitneyLand 9y ago
  That’s funny. I wonder if the origin of the concept of something being trivial only in hindsight can be dated, or has it always been a truism?
- vostok 9y ago
  I think trivial means that the proof requires no serious amount of creativity.
  - zeth__ 9y ago
    >Takes less than a week to complete.
- alphydan 9y ago
  Remember that problems in high school typically can take 1/2h to solve, in college a couple of hours ... as a PhD typically a couple of years and as a professional researcher ... many years. So for him, a problem where he was confused 20 minutes and solved it in 10 minutes is probably "trivial".
- contravariant 9y ago
  There's a reason there's a somewhat famous joke about a professor who, when asked to explain an step he claimed to be trivial, furrowed his brow deeply, spent the best part of an hour looking at notes and scribbling on the blackboard only to declare afterwards "Yes! It is trivial!'.
  - SAI_Peregrinus 9y ago
    Or the numerous results which are obvious in hindsight.
- ajkjk 9y ago
  (Not even specialists -- 2nd-3rd year undergraduates in engineering/math/physics!) In that case the word 'trivial' means it doesn't require complex machinery to solve. A simple consequence of arithmetic manipulations, say, may be extremely difficult to think of yourself, but once you know it it's trivial to see the validity of. I believe that counts as 'trivial' in mathematical terminology.
munificent 9y ago
> All jokes aside, Terry is writing to an audience where this is utterly elementary. One of the things I find fascinating about the Internet is the effect it has on writing when you have so little control over who your audience is. Every now and then, a blogger or Twitterer used to writing for some niche audience catches the larger eye of the Internet zeitgeist and all of the sudden momentarily finds themselves with a much larger, very different audience, who perceive their work entirely differently because they lack the context the author presumed. I wonder how much of the modern trends of political correctness, trigger warnings, etc. come from this experience where you can no longer safely make any assumptions about your audience. At the same time, it's not possible to write effectively when you assume the audience could be anyone, even a potentially malicious aggressor.
- msla 9y ago
  It's the result of opening all the echo chambers to the world, including parts of the world who, for whatever reason or none at all, simply don't like you, who you are, what you are, or what you do. It also opens them to people who are just not going to get it, but who still feel compelled to comment and get a reaction, whatever kind of reaction they can elicit, and will not stop until they get a reaction or are forced to leave/stop participating. So, can we recreate the good echo chambers, such as academia, but do it right this time, such that irrelevant characteristics no longer matter but the people inside of them no longer have to deal with nutballs and assholes coming in and disrupting anything?

eesmith 9y ago

Did Mrs. Cringleton at least know about !! in the shell? https://news.ycombinator.com/item?id=2983466 https://news.ycombinator.com/item?id=2983466 ;)

meuk 9y ago
That is actually very good to know!

waqf 9y ago

> mom > tea and bicuits Where are you claiming to be from?

adw 9y ago
The same place as Dick van Dyke, one suspects.
- munificent 9y ago
  That tiny little island in the middle of the Atlantic which, for several decades, was astonishingly highly populated. https://en.wikipedia.org/wiki/Mid-Atlantic_accent https://en.wikipedia.org/wiki/Mid-Atlantic_accent

dxbydt 9y ago

Back when I worked at Twitter, celebs used to show up randomly at the HQ at lunchtime. I was just your average Indian immigrant data scientist, not clued into American culture. The team I was on was trying to hire a lady for a front end eng role. I didn’t know what the lady’s name was, but I was told she was a very promising candidate, and we were all looking forward to the day when she joined because we had a ton of frontend work. Ok, so at lunch, my colleague said out aloud - Hey Katy Perry is in the lunchroom! Everyone got excited. So I said, Oh is that the new front end engineer ? The whole room exploded in laughter. I thought I must have made a big social gaffe. My mind quickly raced...,oh maybe she must be a celebrity of some kind. Perry Perry...which Perry do I know...So previous night I was watching some show on TBS and suddenly a Perry name flashed into my brain, so I instantly said - Oh she must be the daughter of Tyler Perry! The room just doubled up in laughter. Finally I did a wiki lookup and found out who she was. But for days, people would look at me, laugh and say he’s the Katy Perry Tyler Perry guy ha ha ha. Finally I was so embarrassed and pissed off I asked them - Do you know the difference between between Jacobian and a Hessian ? Nobody knew. Well, I said triumphantly, I might not know your katy perry from tyler perry, but atleast I know what I need to know to fo my work. So yeah, some of us do know that a Gaussian is its own Fourier transform, derivative of an exponential is the exponential itself, etc.

jordigh 9y ago
You were one of that day's lucky 10,000: https://www.xkcd.com/1053/ https://www.xkcd.com/1053/
- BoiledCabbage 9y ago
  Actually, you were one of today's lucky 10,000 - and I won't make fun of you for it.
whatshisface 9y ago
You've given a good example of what well-known really means: "well-known among the expected audience." Aside, I honestly wonder if I would be able to recognize Katy Perry in person...
9y ago

jacobolus 9y ago

It is “well known” as in “this was worked out 200 years ago and is a foundational idea for several branches of mathematics and every undergraduate studying science, mathematics, or engineering learns this several times in multiple classes before the end of college, and readers of a mathematician’s blog post about his new research results can be expected to be well familiar with the idea, and if they aren’t the research will not be meaningful to them anyway”.

tzahola 9y ago

What? This is college-level knowledge...

vinchuco 9y ago

This is not a case of the turbo encabulator [1] which you may seemingly enjoy. What he likely meant is that it is well known from the literature. Or there's an easy explanation that supersedes the verbose notation. [1] https://www.youtube.com/watch?v=rLDgQg6bq7o https://www.youtube.com/watch?v=rLDgQg6bq7o

dang 9y ago

Please don't post this sort of empty dismissal here. I know it feels funny at the time, but it's boring in the end. There's no insight here, and as jordigh and others pointed out, it's a cheap shot. Since people sometimes wonder: humor is welcome on HN, lame humor not so much. It chokes out the rarer things, and there are plenty of places to get it elsewhere.