/sci/ - Science & Math

bump

>>4340617
would fuck

dem titties

>>4340627
Amen to that brother.

Yes, it's transitive; don't know about the others though.

>>4340654
I could of guessed that, can you give a outline for the proof?

>>4340650
Well that's not beautiful but it's certainly very very interesting

>>4340675
>Confirmed for never doing any mathematics ever
You think
<span class="math"> 10^{k}\sum _{n=1}^k \frac{(10-n)}{10^{n}} = k+8*10^k\sum _{n=1}^k \frac{n}{10^n} [/spoiler] is interesting....

bump

>>4340713
Or in general if we're not in base 10:
<div class="math">b^k\sum _{n=1}^k \frac{b-n}{b^{n}} = k+(b-2)b^k\sum _{n=1}^k \frac{n}{b^n}</div>
Also getting <span class="math">b^k[/spoiler] in the sums and maybe switching <span class="math">k[/spoiler] for <span class="math">n-k[/spoiler] would make this look better.

bump

Doesn't it depend on X a bit. I can't see it being the same for compact and non-compact X for example, as compact reduces the problem to polynomials, whilst non-compact would make transitivity a bit harder to get.

>>4340786
Only compact <span class="math">X[/spoiler] is fine too.

>>4340592
I severely lol'd at your image

>>4340839
Me too. Couldn't stop for a good 20 seconds when I first saw it.

bump

>>4340654
>>4340659

How can it be transitive?

Consider <span class="math">f_1(z) = 1[/spoiler]. MTs are functions of the form <span class="math">\phi(z) = (az+b)/(cz+d)[/spoiler].
Thus, <span class="math">Gf= \{z\mapsto (a+b)/(c+d)\,:\,a,b,c,d\in\mathbb{C}\} = [/spoiler] constant functions.
Thus, <span class="math">Gf[/spoiler] cannot equal <span class="math">U(X)[/spoiler].

What am I missing?

>>4340713
>summing over n instead of over k
what the fuck is wrong with you

>>4340868
Sorry, <span class="math">U(X)[/spoiler] is supposed to contain no constant functions.

>>4340877
The replace <span class="math">f[/spoiler] with <span class="math">f(z)=z[/spoiler]. <span class="math">Gf[/spoiler] clearly doesn't contain exp().

?

>>4340883
It doesn't need to for compact X. In the compact case all meromorphic functions are rational functions.

>>4340883
<span class="math"> exp() [/spoiler] isn't holomorphic when codomain is the Riemann sphere.

>>4340888
?exp() isn't holomorphic when codomain is the Riemann sphere.

It isn't *bi*holomorphic, but unless I'm totally confused, it is holomorphic. [Here I can take the Riemann surface to just equal the complex plane.]

Did you mean to define <span class="math">U(X)[/spoiler] to be the set of all biholomorphic functions?

>>4340913
Sorry, my brain is full of fuck. (3 am where I live). It's obvious that it isn't transitive. Abandon thread, thanks for the help.