To me when you say "rational thought", I get an entirely different perception than some enjoyable intellectual activity like music or math or whatever.
$$r^{d-1} f(r)|f(r)|^{d-1} - A\frac{d}{dr} \left( r^{d-1} f'(r) |f'(r)|^{d-1}\right) = 0$$
solve for function $f:\mathbb{R}\to\mathbb{R}$. $d \in \mathbb{N}$, $A \in \mathbb{R}$, $A,d$ are constant.
Has there been any work on this equation?
I tried Wolfram but its not able to, in the computati...
Oh by the way on prodding from Dylan we did do a couple computations here and there. The first class we had was the bonus, which was basically just review. The second one was a kinda random QA session, in which stuff happened
@MikeMiller Do you know it's a group off the book? $K(G, n)$ is at best an H-space with homotopy-associative product and has identity and inverse upto homotopy, right?
I didn't understand Neves's construction of the Hopf map too well but I dunno, this whole toying with spheres things sounds either toral or $\mathbb{RP}$ type
(This is why people use $\rho$ for spherical radial coordinate. But I hate that---rho is already density in physics, so it conflicts with one of the main applications of multivariable calculus. I like $r$ for spherical and $s$ for cylindrical, but that's a matter of personal taste. I know that doesn't sway math people, though...)
Seriously, one of the main reasons to do multivariable calculus is for physics/engineering computations. And one of the main reasons to do so is to compute things involving charge or mass distributions.
@BalarkaSen You can make K(G,n) a group on the nose reasonably easily. It's Omega K(G, n+1), and you can model the loop space with Moore paths [0,s] -> K(G,n+1) that gives it a product associative on the nose. I guess I still need to do inverses but I'm sure there's some similarly stupid trick.
@Mike for some reason I can't isolate, "group on the nose" reminds me of the justification we got for cofibers giving you the exact sequence of pointed sets
ahh , i must get better at this. the question that do not need to find this regions im ok with , but this questions that we need to find the boundaries for the coordinates im not sure sometimes :/
Say, population can be modeled by $p(t)=p(0)e^{kt}$, right? Are the units for $k$ given by 1/time? If not, then how does $e^{\text{sec}}$ comes out, as far as units.
@Ted Sorry, to clarify, none of us are asking for arguments. Daminark is learning this for the first time and I was asking if he actually had an argument or was going off memory.
Oh wait a second, checking back through notes we never actually proved the Hopf map stuff, he just said it and then proved that higher homotopy groups of $S^1$ were trivial
The point I was going ot make earlier is that, if you've got $z\in [r,\sqrt{1-r^2}]$, then that interval only makes sense when $r\leq \sqrt{1-r^2}$. @liad
Hmm. I'm embarrassed to admit: I'm looking at a PDE problem on the main site, and I can't for the life of me remember when the separation constant is allowed to be positive vs. negative vs. zero.
My recollection was "allow any of them and let boundary conditions sort them out."
$$r^{d-1} f(r)|f(r)|^{d-1} - A\frac{d}{dr} \left( r^{d-1} f'(r) |f'(r)|^{d-1}\right) = 0$$
solve for function $f:\mathbb{R}\to\mathbb{R}$. $d \in \mathbb{N}$, $A \in \mathbb{R}$, $A,d$ are constant.
Has there been any work on this equation?
I tried Wolfram but its not able to, in the computati...
