@EnjoysMath: $|z|^n$ attains a maximum $M$ in your compact domain, then $|\sum_{n = N+1}^{\infty} \frac{z^n}{n!}|\leq M \sum_{n = N+1}^{\infty} \frac{1}{n!}$. Then use the fact that the series for e converges.
@Daniel: You'll know the answer to a question that was raised the other day. Someone asked whether a continuous, monotone function (which you and I know is differentiable a.e.) must have left-sided and right-sided derivatives everywhere. My guess is that the Cantor function fails that. Do you know for sure? I've been too lazy to write it down. I even taught Pedro the Cantor function so that he'd work on it, but he didn't like it (plus he has a linear algebra final).
No, personally, I don't, @Daniel. :) So for the points of the Cantor set that are themselves endpoints of omitted thirds, the derivative on the interval side is clearly $0$, and I was guessing non-existent derivative on the other side. And what about the "limit" Cantor set points?
Actually, I'm not sure about the accent. I'm an expert on them in French, not Italian :P
Suppose I have a $\mathbb C$-vector space $V$. Then its underlying abelian group becomes a $C$-vector space $\overline V$ with the scalar multiplication given by $\alpha\cdot v$=$\overline\alpha v$. I have to prove that neither $V$ or $\overline V$ is a subspace of the other.
I'm not really sure if I actually understood the question
By dimension count (over $\Bbb R$) if $\bar V$ were a subspace of $V$, they'd have to be isomorphic. Right? Ugh: Both $\bar V$ and $\overline V$ look UGLY.
Starting in the center of a sphere of radius 1, draw a path with the shortest possible length that intersects every plane that is tangent to the sphere.
This question appeared as a generalization of the recently considered problem of the lost ant
Starting in the center of a circle of radius ...
Oh, well, it would have to hold on any finite-dimensional subspace, which is enough to mess it up. Hmm. The problem is that the isomorphism isn't going to be $\Bbb C$-linear, right?
@Ted: To be honest, I'm not really sure. The book defined "being a submodule" as being a subgroup stable under scalar multiplication. Since $V$ and $\overline V$ are the same as sets (and each one of them is stable under their own scalar multiplication) they are both stable under both scalar multiplications. Obviously there's something I'm misreading here.
My comp is really slow, can some 1 give me an estimate on $$\sum_{m=1}^\infty\frac{e^{2\pi i/3}\psi(e^{2\pi i/3}m)+e^{-2\pi i/3}\psi(e^{-2\pi i/3}m)}{m^5}$$ where $\psi$ is the digamma function
user97303
@N3buchadnezzar my initial reaction was, take the ant to be at the centre of the Earth and the plane is somewhere tangent to surface of the Earth, have the ant walk outward at the north pole and follow an "orange peel" path
user97303
but I have no idea how to calculate for optimal distance
@Fernando: Well, it's subtle. Why is $\bar{\Bbb C}$ not isomorphic to $\Bbb C$ as a $\Bbb C$-module? Isomorphism needs to be by a $\Bbb C$-linear map, right?
@DanielR, you should be out partying the last few hours of your youth away :P
Then again, there's the guy I just answered about baby Borsuk-Ulam who says his professor didn't teach him about homotopy lifting for maps $S^1\to S^1$ but yet applies what's needed in the proof. Sigh.
@TedShifrin I'm a bit more relaxed about that, but I'm not particularly fond of it either. Although it happens to me too that I give away more than I should.
@Fernando: Subspace inherently means that the module structures must be compatible.
Yes, @N3. I've often told bright students who aren't working up to their ability to take a semester off so that they'll come back rejuvenated and eager.
Nice try, @Pedro. If you're doing things like analytic combinatorics, you're still studying things like distributions of sequences/primes, and you do not get a free pass.
Because he basically knows almost all of the foundations (doesn't just think he knows it) and keeps being bored and reading ahead to more and more stuff.
My smart advisees who are prepared to skip our undergrad real analysis and go right to the graduate one are grateful. They work their butts off but they are very excited about it.
It's just that the book defines "being a submodule" set-theoretically (this set is closed under this operation), and doesn't talk about compatibility of structures if the set happens to be a module on itself. But I'm pretty sure I should think this through by myself.
I think they're assuming the module structures are automatically compatible, @Fernando.
Well, @N3, in this country, to be competitive applying to top Ph.D. programs with the kids coming out of the top 10 schools, who've taken Ph.D.-level courses their junior and senior years.
And so as to be excited and stimulated, not bored.
For example, Princeton doesn't even bother teaching beginning-level graduate courses. They assume all their students know it all already and they hold them responsible for them.
Thank you for a great answer. I see your point, I am also thinking that having a rock solid fundament is better than being a year ahead with a few gaps.
Also what are the hardest schools in the US ? I do not think that those with the toughest intake, are amongst the hardest to attend.
Well, you are of course responsible for filling in gaps, as you will be as a graduate student and professional. But taking courses that do not challenge you and in which you are bored doesn't necessarily fill in much in the way of gaps. This depends, of course, on the students and on the courses.
@N3, I would agree that most students I deal with are overconfident and do not know even the stuff they've taken nearly as well as they should. I'm talking students with exceptional ability and maturity here.
I admit I'm occasionally stupid. But I did it earlier today because someone asked a probabilistically phrased question about matrices of maximum rank, and some dope answered that you could have chosen the zero matrix.
@N3, I would rather teach a Spivak-type course to super-talented students, rather than have them be bored in a standard course. On the other hand, I also incorporate plenty of the standard applications problems and far more challenging integration problems than they'll get in the standard courses. Same goes for my multivariable/linear algebra course that I keep teaching.
@skull: Never let it be said that you don't bitch or stay silent.
The right level of the bar for students who intend to go get Ph.D.'s is, at least in this country, not appropriate for 95% of the students studying mathematics, including the huge majority of students majoring in mathematics.
We can't fail so many students in this country, @N3. The culture is totally different from yours.