What's everyone up to right now?

[Alex K]

P.s. hint: it works best with your original definition of C

[Kernel Sohcahtoa]

...

Amusing how the general solution seems to break when setting n=2, while your special solution shows that it's a log.

I got a teaser question for you (to which I know the answer bc. I encountered such problems in my work): can you nevertheless extract the seemingly special n=2 solution from your general solution using a (dirty and somewhat magical) trick?

I'm afraid not: I can't see a fancy, beautiful, or magical extraction.

All I can see is the following: If the general solution u(x1,...,xn)= -Cr^2-n +K  (1), then if we let n=4 and plug that into (1), we will obtain -Cr^-2 + K (2). Now integrating (2) yields (-Cr^-1)/-1 + K1 or Cr^-1 + K  (2), where K=K1.  Integrating (2) gives Cln( r )+K2 or Cln( r ) + K, where K=K2, which is the solution for n=2.

Hence, besides that, I'm afraid I cannot see the magical solution that you have mentioned in your post, sir.  With that said, I'm very interested in reading your thoughts/clarifications on this, especially since you are very well-versed in this stuff.

[Alex K]

Amusing how the general solution seems to break when setting n=2, while your special solution shows that it's a log.

I got a teaser question for you (to which I know the answer bc. I encountered such problems in my work): can you nevertheless extract the seemingly special n=2 solution from your general solution using a (dirty and somewhat magical) trick?

I'm afraid not: I can't see a fancy, beautiful, or magical extraction.

All I can see is the following: If the general solution u(x1,...,xn)= -Cr^2-n +K  (1), then if we let n=4 and plug that into (1), we will obtain -Cr^-2 + K (2). Now integrating (2) yields (-Cr^-1)/-1 + K1 or Cr^-1 + K  (2), where K=K1.  Integrating (2) gives Cln( r )+K2 or Cln( r ) + K, where K=K2, which is the solution for n=2.

Hence, besides that, I'm afraid I cannot see the magical solution that you have mentioned in your post, sir.  With that said, I'm very interested in reading your thoughts/clarifications on this, especially since you are very well-versed in this stuff.

Ok, I'm not 100% sure how this can be made into a rigorous argument, but instead of setting n=2 which is impossible, you can go to real n and choose a value near 2 and then take the limit n->2. A taylor expansion for small n-2 yields

r^(2-n)=e^(2-n)ln( r ) ~ 1+(2-n) ln( r )

Plugging in the prefactors the complete solution becomes

c/(2-n)+c ln( r ) + k

Now the constant part diverges as n->2, so it seems like we need to redefine c/(2-n)+k as the new k. After that you can take n->2...

It's black magic

[Alex K]

@Kernel

The appearance of that log in two dimensions has profound consequences for many things. I once had the privilege to lunch with renowned mathematical physicist Carl Bender and he told me that the fact that thunder claps have such a rolling spread out sound also comes from the fact that the lightning bolt has codimension 2, leading to a log in the power distribution

In type II String Theory, the branes of dimension 8 play a central role, and they have logarithmic solutions because they have codimension 2 in the 10 dimensions. That logarithm leads to a whole list of weird consequences and a whole subfield of string theory called F-theory which is more or less based on this effect.

/ramble

[Kernel Sohcahtoa]

@Kernel

The appearance of that log in two dimensions has profound consequences for many things. I once had the privilege to lunch with renowned mathematical physicist Carl Bender and he told me that the fact that thunder claps have such a rolling spread out sound also comes from the fact that the lightning bolt has codimension 2, leading to a log in the power distribution

In type II String Theory, the branes of dimension 8 play a central role, and they have logarithmic solutions because they have codimension 2 in the 10 dimensions. That logarithm leads to a whole list of weird consequences and a whole subfield of string theory called F-theory which is more or less based on this effect.

/ramble

You were not rambling, sir.  IMO, I've enjoyed the insights/thoughts that you have shared with me today.  However, I must admit that I've got a lot to learn and am somewhat rusty in my calculus knowledge.  With that said, the ideas that you've shared here today have definitely made me look forward to gaining a more fundamental understanding of PDEs and the role that they play in our reality.  Thanks for your time and attention, Alex K.

[Edwardo Piet]

I believe every home should have a Kernel.

[pocaracas]

I believe every home should have a Kernel.

Going nerd with this one...

https://www.kernel.org/

The kernel handles Demons!

---

---

[Alex K]

@Kernel

The appearance of that log in two dimensions has profound consequences for many things. I once had the privilege to lunch with renowned mathematical physicist Carl Bender and he told me that the fact that thunder claps have such a rolling spread out sound also comes from the fact that the lightning bolt has codimension 2, leading to a log in the power distribution

In type II String Theory, the branes of dimension 8 play a central role, and they have logarithmic solutions because they have codimension 2 in the 10 dimensions. That logarithm leads to a whole list of weird consequences and a whole subfield of string theory called F-theory which is more or less based on this effect.

/ramble

You were not rambling, sir.  IMO, I've enjoyed the insights/thoughts that you have shared with me today.  However, I must admit that I've got a lot to learn and am somewhat rusty in my calculus knowledge.  With that said, the ideas that you've shared here today have definitely made me look forward to gaining a more fundamental understanding of PDEs and the role that they play in our reality.  Thanks for your time and attention, Alex K.

Thanks, glad you enjoyed it. I'm always fascinated by how these concepts relate to endless fascinating other things, and I have trouble shutting up about it
The hardest part about teaching is always what I can't say.

[Joods]

It is absolutely exhausting thinking for another adult. All the time. Without a break.

[Silver]

Watching Gotham. I just started with season one the other night and I am only on episode three.