@TedShifrin I did this, as $z_0$ is an isolated zero there exist an integer $k$ such that $F(z)^{k}\ne 0$, noted $p$. So I have $F(z)=(z-z_0)^p\sum_{k=p}^\infty F^{(k)}(z_0)/k! (z-z_0)^{k-p}$.
@TedShifrin si je suppose que f est surjective comme F est de dimension finie alors je peux poser $\{f_1,...,f_n\}$ une base de F comme f est surjective alors il existe $\{e_1,...,e_n\}$ tel que $f(e_i)=f_i \forall i\in\{1,...,n\}$
I've got a projector $p$ of rank $r$ on a finite dimension vector space $E$ and $\phi(f\in\mathcal{L}(E))=(p\circ f+f\circ p)/2$. The eigenvalues of $\phi$ are $0,1/2,1$, how do I get the eigenspaces ? I tried constructiong them (for instance, $E_0(\phi)$ would be imo the set of functions $f$ such that $f(\ker\phi)\subset\ker(\phi)$ and $f(Im\Phi)\subset\ker\Phi)$ but when I add their dimension I don't get $n=\dim E$.
That's awesome. I didn't get that before. Cool! So then, in your example with $\mathbf{F}(\mathbf{x},\mathbf{y}) = \mathbf{0}$ does the 0 vector here have dimension $n+1$?
@r9m btw, thank you for giving me the Knuth's problem, I would have missed such a beauty! Unfortunately I'm not aware when such problems are out (or I find them after a long period of time).
@Chris'ssis you can check Tauraso's page! he posts solutions to old problems that he has attended as well! :D (he keeps updating the list as he solves newer problems ... he puts the solution pdf once the last date of submission is past though .. for obvious reasons :P)
Yes, I knew what I said wasn't right, but I didn't have the words / understanding to state it that well. So I said something subpar and figured we'd end up correcting it. Yay, now I know. :)
@Chris'ssis ah! that also is giving me neck pain! I dunno how to deal with it ... first I thought of faa de Bruno's formula .. but couldn't make it useful though ..
I will do that today and send it to you so that we can be sure I did it correctly :P I am going to take real analysis in the fall. I had to wait because I resumed school spring quarter and haven't taken placement tests.
Hmm...well, I dunno. I certainly don't think it seems as intense as the Calc I took at Caltech. and my brother took the intro calc sequence and he found it easy and he's terrible at math.
I really like your teaching in general. I haven't used spivak yet. Do you recommend him? The problem is, one of the books I went looking for (diffeo geo maybe?), he had like 5 volumes and that was an immediate turn off because I don't think I can read 5 volumes
I was referring to his Calculus. The 5-volume book is good, but too drawn-out scrambling and unscrambling notation. But plenty of great stuff in there.
The first volume has all the basics on manifolds. Volume 2 gets a bit much with four different definitions of connections on Riemannian manifolds, but Volume 3 has some wonderful concrete stuff in it.
Not bad, @TedS. I managed to walk to get my groceries before it thunderstormed out. And now it's sunny again, so I may go for a walk soon. :) What's for dinner?
Haha, I see @TedS. I enjoy him, but I certainly see that point of view. One of my friends shares that sentiment rather intensely. He tends to enjoy more modern composers, ala Ives, Bartok, etc.
A regular polygon with $100$ sides is inscribed in a circle. What is the probability that three randomly chosen vertices of this polygon form a right angled triangle?
@Chris'ssis it's 4 am here .. I'll get some sleep :) there's a nasty expression I'm stuck with .. with a bunch of $p^{th}$ derivative and stuff .. I need a fresh start .. g'night!
A regular polygon with $100$ sides is inscribed in a circle. What is the probability that three randomly chosen vertices of this polygon form a right angled triangle?
