@LeakyNun Nevermind, I actually figured it out. It is just a simple application of integral triangle inequality. Thank you for help on the previous one.
Where can I find a result that says "if g is meromorphic in a compact then g is constant or assumes every value of the sphere a finite number of times"?
One particular thing which you may or may not have done in group theory (my class wasn't going to if I didn't ask my professor) but which you'll need for Galois theory is solvability of groups
One thing I never understood about solvable groups is the terminology. why does a terminating central series have anything to do with solving something. What are we actually solving?
Yeah, this was something where even though you probably learn about separable groups first, the idea was originally coined for the sake of Galois theory
So, it's apparently the case that if $X$ is some space and we have two maps $f,g:X\to S^n$ such that $\|f(x)-g(x)\| < 2$ (distance in Euclidean space), then they're homotopic
The idea here should be that we want to simultaneously choose geodesic paths from $f(x)$ to $g(x)$ for each $x$ and not worry that we have multiple paths coming out of a point in different directions, right?
Does anyone know which experts are verifying Atiyah's proof of the Riemann conjecture? Where can I find something about it? I am bit curious. Thank you!!
Hmm, so given two points on the sphere which aren't antipodal, I think we can draw a line through them that doesn't cut through the origin. At that point we can just kinda project the linear homotopy
So, if you're happy with obtaining this from the statement that $C(X,\mathbb{R})$ is separable where $X$ is compact, that's proven by taking a countable dense subset $\{x_n\}$ of $X$ and applying Stone-Weierstrass to the $\mathbb{R}$-algebra generated by $\{1,d(x,x_1),\ldots\}$. Then the $\mathbb{Q}$-algebra is the countable dense subset
(I'm on my phone so do forgive me if I fucked up the TeX :P)
Hi All! Can anyone provide me with a sanity check. I have a vector differential equation given by $$ d/dt (\mathbf{a} \times \mathbf{b}) + d C/dt \mathbf{a} \times \mathbf{b} = 0. $$ where $$C = C(t)$$ . I get the following solution $$\mathbf{a} \times \mathbf{b} = \mathbf{d} \exp{-4C}$$, where $$\mathbf{d}$$ is a constant vector of integration. I treated this as a simple first order ODE and just used the usual integrating factor. Is this legit or am I going crazy?
I hadn't thought of that line of attack (though I did know the argument that algebra is separable). I don't see how to conclude yet but I'll take a shower and think
Yeah, you can do the usual integrating factor thing by multiplying by $e^{\int_0^t C(s)ds}$ and you get $$e^{\int_0^t C(s)ds}(\mathbf a\times\mathbf b)(t) = \mathbf d,$$ a constant vector.
Now where did you get your thing?
Oh, my mistake.
That's wrong.
It should be $e^{C(t)}(\mathbf a\times\mathbf b)(t)=\mathbf d$.
So I agree with you except for the $4$. @Rumplestillskin
@LucasHenrique That's why I like math. When I'm bored I like to think of all the crazy stuff you can do with math. I want to get a Ph.D. in Pure math one day.
No, @Rumplestillskin, only if it's a matrix differential equation. But you're just thinking of the vector in terms of its components, and you use the method on each component, hence on the vector as a whole. Nothing to be gained, though, by writing $\mathbf a\times\mathbf b$. Just $\mathbf x(t)$ would do.
I have a calculus test coming up. I'm not going to take it when I'm exhausted like last time though. I can't look at my grades without cringing at the 80.
There's this guy Jair Messias Bolsonaro, 30 years on the Congress. He's not exactly what I'd call a sympathetic person. He's a fascist and publicly racist, homophobic and sexist, even if he always get terrible explanations to say he's not.
That's good advice @Ted. I do feel less anxious now. I always get nerves with exams (and it's usually not the content itself but missing the exam or being late)
One time I worked late the night before a history final and my phone alarm never sounded. Luckily the professor was gracious since I had a high test average up to that point and let me retake the test later in his court office (he was a judge)
Yes, I remember once completely sleeping through an important midterm, forgot to plug phone in and died. Professor was very kind and let me retake it in her office that afternoon.
@TrostAft Lessons are learned. In my case it was the Android clock app bugging out and not sounding properly, so I make sure to have 2 or 3 independent alarms set
@TedShifrin I don't know if you're still interested, but if you want to meet up, I'm now in SD for a while. I believe you have my email? Admittedly, my math is now sort of rusty... If not, no worries.
@TedShifrin Yeah, I was shook. After I ate the fast food, I got hungry again like 2 hours later because that stuff never fills me up. I decided to make some pork chops and while I was trying to put the sizzling food on a plate, a knife almost fell blade-first onto my foot. I dodged just in time, but I had a HOT pan in my hand. It was scary and it would have never happened if I ate real food in the first place.
Things like cross-products, @JoeShmo. There's a duality going on there that allows a 2-vector to be viewed as a 1-vector in $\Bbb R^3$. So it doesn't transform (under change of coordinates) the way a vector should; rather, it transforms as a 2-vector should.
