if $\vartheta(t_k) = k\pi$ and $\vartheta(t_k) = (k + 1)\pi$ then following $\cos(x-k\pi) = (-1)^k \cos(x)$, $Z(t)$ is expected to change sign in $[t_k, t_{k + 1}]$. does it make sense?
I feel that some users ask too many questions on SE. If one asks too many things, one cannot learn math effectively. One must learn to think deeply by oneself.
There was once I said a question was difficult, and someone said it was not difficult, just tedious. I bet he did not even attempt the question himself. So although some may think he is smart to say that, I actually think the opposite, lol.
@robjohn :P lol ... at one point of time that was the bedtime story for my bro .. he just wouldn't go to sleep without listening to it once :P .. heaven knows why :P :P (he even made my buy a colored booklet of that story :P )
Hi everyone I have a little problem with Algebra, I hope someone could give me a hint. It's not to difficult to open a question, I´m not sure if here is a correct place...
this says: Let $A,B,C$ subgroups of a finite group $G$. If $B\subset A$, then $A\cap C:B\cap C]\le [A:B]$ I honestly can´t see how to construct the correct injective map... i know is kinda stupid...
I have thinking in something like $x(B\cap C)\mapsto xB$ I didn't have any problem to show that is well defined but on the other hand the injectivity part...
and so y^{-1}x\in C and in B, i.e., $y^{-1}x\in B\cap C$ which shows that x(B \cap C) = y(B\cap C) and hence the map is an injecton as desired. @MikeMiller
You could probably a numerical method of calculating conformal maps but I am not competent enough at Pari to do it there
@Balarka You might look up "How explicit is the explicit formula?" It was a project of Mazur's and Stein's I was fortunate enough to see them speak about
Tiring ... Teaching new students to become mathematicians and having to recode web homework because new versions of the software are not backwards-compatible.
i found an interesting problem today. it asks for the probability of having 1 as the length of the straightline joining two random point on a unit circle.
let there be two points with distance 1. join them. join the two points with the center. the angle formed is 60 degree. 60 + 60 degres out of 360 degrees gives 1/3.
Naive argument: take the convex hull or the image of a loop. This is contractible. Of course, this doesn't work. More sophisticated idea: homotope the loop so that it's an immersion and intersects itself transversely. (Is this possible?) Then this probably has a coherent notion of interior.
Take the union of the curve and its interior... and it's probably contractible.
You do need it but imagine a lemniscate - not simple, but there's a "smallest" contractible set that contains it (just draw out the bigger heart shaped curve and take its interior)
The original question was that H_1 zero implies pi_1 = 0. But one can reduce it to this
The original original question was that an open connected subset of the plane with trivial homology is homeomorphic to the disc :)
Well, if we're using arsenals, why not use van Kampen to prove the contrapositive? Are we allowed Jordan curve? I dunno ... I can check old notes tomorrow. So many old course notes to scan ....
