In the European system you can get paid to do some (hopefully interesting) research for a few years, and then get a real career later if you're not immediately getting overwhelmed by tenure-track offers.
@ZhenLin Sure, but since you get paid a salary while studying it's not as if you'll emerge with the PhD deeper indebted than when you got your master's.
Of course the first step of getting a real career after a PhD can be a bit iffy. You'll need to find an employer who you can convince you're not bringing your own portable ivory tower into the workplace.
Anyway, sleep, and then 9am Elliptic Curves (followed by 10am Local Fields, 11am Algebraic Topology, 12pm Category Theory, 2pm Category Theory seminar...)
Anyone know what natural logic is? Google finds mostly references to specialized logics for natural language processing, which is probably not relevant here.
I have a quick question about Riemann surfaces. Why are chart transitions symmetric? That is, if phi_1 compose phi_2^(-1) is holomorphic, why is phi_2 compose phi_1^(-1) holomorphic?
Is it just because they are homeomorphism, so the derivative is nonzero, and the inverse of a holomorphic functions with nonzero derivative is homomorphic?
Could you refresh my memory on why holomorphic chart transitions necessarily have nonzero derivatives in their domain? I keep thinking of the inverse function theorem, but that's the converse...
If the derivative vanishes at point then your function is not injective there: You can write f(z) - f(z_0) = (z-z_0)^n g(z) with g(z_0) \neq 0. Now g admits an n-th root, so f(z) - f(z_0) = h(z)^n for some function. Now remember that z -> z^n is not injective in any neighborhood of 0.
@t.b. Consider this other approach, just for kicks. Let our transition function be T. Let S be its inverse. S(T(z))=z, so S'(T(z))T'(z)=1, and T'(z)/neq 0.
However, this can be made into a proof by showing that 1/T'(z) is in fact the derivative of S at the point T(z). Note that this is exactly how the inverse function theorem is proved.
I don't know but I think so. In any case I could cast a vote. It takes three votes by non-mods for deletion of answers. It happens very rarely, but it happens.
@tb good foggy morning. I woke up too late and cannot see Pedja's answer now, I guess he deleted it because of downvotes. For the first time I see such situation with an accepted answer
@Gortaur: It was deleted for him by some high-rep users. He couldn't have deleted it himself because you can't delete accepted answers. It is funny to see a deleted answer with a green check mark.
I'm not sure what you mean by eliminating. For each x in S^(n-1) you can choose a hyperplane that halves the volume of the nth set. Do this in such a way that the hyperplane depends continuously on x. Now associate the sum of volumes of the parts of the n-1 other sets that lie on the positive side of that hyperplane. This gives you a continuous function on S^(n-1).
Sorry, not the sum: the (n-1)-vector consisting of the volumes of the parts on the positive side.
where A_i^+ and A_i^- are the parts of the set lying on the positive and negative sides (respectively) of the hyperplane we picked for the given direction
Oh, I see. Slightly different than what I had in mind, but yes, this works. Alternatively just take f_i(x) = \mu(A_i^+) then for -x you get the volume of the other part of A_i. Now antipodal points for which f(x) = f(-x) are exactly the points where both halves coincide in size.
Holy crap monkeys! Now the administration sends me an e-mail that I cannot do my final presentation because not all my grades are known! Damn professors.
"so i have to utilize borsuk-ulam... hm... i can parametrize hyperplanes by points in RP^{n-1}, but S^{n-1} is only a double covering of this thing... but i can mark the planes and exploit this later... orientation!"
my rough thought process :D
of course, I knew n = 2 version very well (it's a very nice puzzle in itself) so it wasn't a stroke of a genius or something :D
$f(x)$ and $g(x)$ are positive nondecreasing functions. $\sum 1/f(n)$ diverges, so does $\sum 1/g(n)$.
(Why) must the sum $\sum 1/(g(n)+f(n))$ diverge ?
Reminds me of the professor who asks: Is the following problem correct? ... If it is correct, suggest proofs. If it is not, change it so that it becomes true.
@tb: could you apply your gift to the problem when one consider series of min{1/f(n),1/g(n)}? this is sufficient for the problem of sum but not necessary. I cannot find a counterexample
@robjohn Yes, and I think that's it. Lion has the stix fonts installed, but there seem to be some problems. I'm not sure if they've been resolved by now.
Given some integer $k$, define the sequence $a_n={n\choose k}$. Claim: $a_n$ is periodic modulo a prime $p$ with the period being the least power $p^e$ of $p$ such that $k<p^e$.
In other words, $a_{n+p^e}\equiv a_{n} (\text{mod } p)$. But the period $p^e$ is smaller than I'd have expected (it...
@tb done, thanks. do you have any ideas? question seems to be so simple, but the only idea I have is to relate the construction of the canonical event space to the space of trajectories of \xi
Hmm. Stein in harmonic analysis does H^p on R^d. In the paper H^p spaces of several variables they do this on R^d x (0, infty). In another one they do this on the upper half plane in the complex plane. Are these virtually all the same? In the R^d case they convolve with some function depending on t.
I have asked about my working hours. He said I should figure out for myself what works best. But if I decide to have a nice bike trip in the summer during office hours and I get hit by a car some people may ask questions. :D.
first days I tried to ask my supervisor if I can come later someday etc. and he told me that whenever we have no meeting he don't care much
though I don't like working from home (which is not my home and I cannot even put screws in the wall, arghhh) so I'm coming to uni from Mon to Fri, which also increases enjoying Friday nights )
could anybody tell me the following: I've found a possible typo in one journal paper from 2009. shall I report it to them? the typo is: talking about CDFs (cumulative distribution functions) the author writes PDF (usually probability density function)
there is nothing in it......just saying 'f' is a function and asking to prove it is periodic....no information at all........i wonder how one could write something like that
This was my past year question.
Question: How to prove that
f(x,y) = f(x+M,y) = f(x,y+N) = f(x+M,y+N)?
where f is a 2-D Signal.
I am not sure how to prove this.. Need some help...
Let $Z$ be a Markov process on $\mathbb R$ given in the form $Z_{n+1} = f(Z_n,\xi_n)$ where $\xi_n$ is a sequence of iid real-valued random variables. The canonical space of $Z$ is the space of trajectories given by
$$
\Omega = \mathbb R^{\mathbb N_0} = \{\omega:\omega = (Z_0,Z_1,...,Z_n,...)\}...
@robjohn I'm trying to work our British accent by watching HP without subtitles. The last week I watched HP7: Deathly Hallows p.1; certainly in your movie there is much more action
How can we show that the polynomial $f(x)=1+x+x^3+x^4$ is not irreducible over any field?
$f(x)$ and $g(x)$ are positive nondecreasing functions. $\sum 1/f(n)$ diverges, so does $\sum 1/g(n)$.
(Why) must the sum $\sum 1/(g(n)+f(n))$ diverge ?
Given some integer $k$, define the sequence $a_n={n\choose k}$. Claim: $a_n$ is periodic modulo a prime $p$ with the period being the least power $p^e$ of $p$ such that $k<p^e$.
In other words, $a_{n+p^e}\equiv a_{n} (\text{mod } p)$. But the period $p^e$ is smaller than I'd have expected (it...
This was my past year question.
Question: How to prove that
f(x,y) = f(x+M,y) = f(x,y+N) = f(x+M,y+N)?
where f is a 2-D Signal.
I am not sure how to prove this.. Need some help...
Let $Z$ be a Markov process on $\mathbb R$ given in the form $Z_{n+1} = f(Z_n,\xi_n)$ where $\xi_n$ is a sequence of iid real-valued random variables. The canonical space of $Z$ is the space of trajectories given by
$$
\Omega = \mathbb R^{\mathbb N_0} = \{\omega:\omega = (Z_0,Z_1,...,Z_n,...)\}...
