In an analysis class I took, mollifiers were used to prove smooth partition of unity, which was used to prove the change of variables theorem. Lots of applications! The harmonic function proof went a little over my head but the parts I did understand were beautiful!
The hardest part of this course except the really clever calculations. I understant the proofs line by line after your explanations but my main problem is i literally would not have a chance of thinking them by myself.Like the intuition for this course is ??? Everything is like magic. I wanna show differentiability ohh look ill introduce a new product and then a certain function and then prove my function is equal to this new product and this new function i came up with and this product is smooth so is my initial function. Like damn all the course doesnt feel like it has a "natural" way of thinking. Feels like i have to find ingenious ways to make things work with really weird machinery.Im really greatfull for your explanations tho!!!!
I never knew this actually, we only proved in my analysis class that functions in C0 are dense in Lp space, and the teacher just told us, btw also C infinity functions are dense but we won't prove it. So this video really filled a hole in my knowledge 🥰
a little correction about "seeing atoms": you only see the light scattered/emitted from those atoms. This is already a huge smoothing process, as visible wavelengths are much longer than the size of a typical atom. Sorry I had to play annoying physicist lol :) interesting video! Now I know what a mollifier is and you explained it really well.
The bit on Laplace's equation is really surprising. I did not expect you could convince me that was true. Thanks a lot! Learned something new about my favourite differential operator :-)
Peyam, why is that in math a derivation that would be called "formal" would probably be described in lay english as "informal" ? I've always wondered this.
From my non-academic understanding, formal math can be written completely in symbolic logic. Thus all the theorems/proofs can be verified via computer. Where as informal math sacrifices some rigour for a more intuitive approach.
That "aider" of yours, the m-ollifier, is that a lowercase η? Anyhow, the mollifier indeed _is_ an "aider" as it provides instantaneous analyticity as well as the expected convergence properties. Great video, as always. 👍
Turning a sharper function into something smooth, never really thought about that! Thanks!
0:12 ...so you're saying the function f is kinky?
LOL
I was so close to saying that LOL
In an analysis class I took, mollifiers were used to prove smooth partition of unity, which was used to prove the change of variables theorem. Lots of applications!
The harmonic function proof went a little over my head but the parts I did understand were beautiful!
I was reading Evans's PDE book and it started talking about mollifiers which I'd never heard of before... glad I found this video!
The hardest part of this course except the really clever calculations. I understant the proofs line by line after your explanations but my main problem is i literally would not have a chance of thinking them by myself.Like the intuition for this course is ??? Everything is like magic. I wanna show differentiability ohh look ill introduce a new product and then a certain function and then prove my function is equal to this new product and this new function i came up with and this product is smooth so is my initial function. Like damn all the course doesnt feel like it has a "natural" way of thinking. Feels like i have to find ingenious ways to make things work with really weird machinery.Im really greatfull for your explanations tho!!!!
You’ll get used to it, then things won’t look weird any more and everything becomes more natural
@@drpeyam You sir really helpin with that. !!!! Greatfull .
Title these days be going Real Smooth!
Mollifiers are Smooooooth operators....
while True:
print("great!")
Did it stop yet?😂
@@rjbeatz
"Ctrl+C"
Now, it stopped.
cha cha real smooth
Thanks a lot...you make hard topics look much easier
Hi, when we integrate over the boundarie at 12:28 it should be r instead of epsilon ?
Indeed, that should be r
I never knew this actually, we only proved in my analysis class that functions in C0 are dense in Lp space, and the teacher just told us, btw also C infinity functions are dense but we won't prove it.
So this video really filled a hole in my knowledge 🥰
Loved your channel! Subscribed!!
Thank you!!!
Interesting!
a little correction about "seeing atoms": you only see the light scattered/emitted from those atoms. This is already a huge smoothing process, as visible wavelengths are much longer than the size of a typical atom. Sorry I had to play annoying physicist lol :) interesting video! Now I know what a mollifier is and you explained it really well.
The bit on Laplace's equation is really surprising. I did not expect you could convince me that was true. Thanks a lot! Learned something new about my favourite differential operator :-)
Sir, you should get atleast 100k subs
I wish this video existed when I took PDES last year
Please
What is the name of the app you are using.
this is why blackpenredpen doesn't have a math phd
LOL but u have a PhD in integrals haha
Again...😶😶
Ok. Thanks.
Peyam, why is that in math a derivation that would be called "formal" would probably be described in lay english as "informal" ? I've always wondered this.
No idea at all, it’s so weird
Probably because when you're formally writing it you have to write it carefully, but if you're explaining it in a hand-wavey way, it's informal
From my non-academic understanding, formal math can be written completely in symbolic logic. Thus all the theorems/proofs can be verified via computer. Where as informal math sacrifices some rigour for a more intuitive approach.
This is insane I wrote a tune called Mollifaction its good to know its infinite
That "aider" of yours, the m-ollifier, is that a lowercase η? Anyhow, the mollifier indeed _is_ an "aider" as it provides instantaneous analyticity as well as the expected convergence properties. Great video, as always. 👍
Yes, eta!
Hahahahaha I get it now 😂
is |x|^2 not equivalent to x^2? why is the absolute value needed in the eta function?
In R^n x^2 is undefined
You say mollifier, we say "Glättungskern" 😄
I love this!!!!
in m^ε(x), what is N?
f is given as a function on R^N
low pass filter, cool
A kinky function. My kind of function.
ちょっとこれは難し過ぎた
2回生くらいでやる事なんだろね
@Dr Peyam Please Dr. Check your gmail inbox. I sent you a pdf file with the demo I did.
The only instance where abuse is nice.