I just read that the algebraic closure of the field of laurent series over an algebraically closed field $F$ of characteristic 0 is the field of puiseaux series over $F$, which in my mind is pretty cool
I heard a comment recently along the lines that "Mathematicians would love to claim Newton as one of their own but he was really a physicist at heart" that made me smile
Can anyone help me with this problem? Roots of unity. "Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find \[\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}"
trying to make a firm distinction between newton as a mathematician v. physicist is sort of silly, since at that point the distinction between them wasn't there in the way it is now
heck, we remember gauss as a mathematician, but his professorship at gottingen was in astronomy, and he was the director of their observatory. plus there was his collaboration with weber on electromagnetism
Astronomy? That's interesting, @Semiclassical. I recall that Hershel (sp?) stumbled upon the normal (aka Gaussian) distribution for astronomical measuring errors; I wonder where Gauss found it...
Nevertheless, having multiple misspellings in every sentence makes a bad impression. For example, people are more likely to answer questions that are well-edited, etc. Your writing is the only window we have into you as a person, after all.
I would just take me much longer. To right something. And in a counstation, I can't do it. Because once I got done what I wanted to right. The window to say it would be over.
These kinds of chatrooms are a good place to practice "on the run"...but, as you said, you have to learn to walk before you can do it "on the run," my friend :-) @MathCubes
@DanielFischer how do we explain the rationale behind choosing $\mu_{b}\circ f \circ \mu_{-a}$ and applying Schwarz lemma to it to infer $|f'(a)| \le \frac{1-|b|^2}{1-|a|^2}$ for an analytic $f$ that maps the unit disc to itself and satisfies $f(a) = b$? .. where $\mu_a (z) = \frac{z-a}{1-\bar{a}z}$ is the mobius transform?
@TedShifrin this is great. I am going to finally learn what compactness is. I think I will try to understand how to prove $[0,1]$ is compact. And prove myself $(0,1)$ is not compact.
@TedShifrin your intro didnt seem to emphasize covers and subcovers. Was this intentional or am I just blind and missed it?
How would I figure out what resolution at (x PPI) would about equal around (x inch(es))? Here is a example what I mean; persay you have a FHD monitor at 40 inches, you want to have 39 inches of the screen to be a color. At what resolution would you need the rectangle to be?
Story goes here (...
I have a speech speacialist at school that I see ever week about. Well that is use too because it is summer now. I tell her, and well she get what I am saying, but again don't.
I didn't truly know if I have or not dyslexia. I think I do since my sister told me about it. How my wrighting stley fix it. and I look into it, and it fix me.
@BalarkaSen Yeah, though obviously actually figuring out a project and getting the stipend will be hard (especially the last part, as it is quite competitive)
@r9m The idea is to reduce it to a known case. We know something about the case $f(0) = 0$, and the composition with $\mu_x$ produces exactly that case.
@skillpatrol They support "Danish research" which is interpreted in the broadest possible sense, meaning that it can either be research performed in Denmark by anyone or research performed somewhere else by Danish citizens
@user1658028 the text is getting scattered between other posts ... please write the question in one go and post it ... so that we don't have to keep scrolling and looking for the pieces of text .. thanks!
I got two circles of same radius. Some time they are completely overlapped and some times far but for most r which is the radius. I need to know the relation of distance of centers of these two circles and the area of difference of them
@DanielFischer I'd say that $\sum_{n = 3}^\infty \frac{1}{n\log n \log \log n...\log\circ...\circ \log n}$ always diverges, but I don't see any direct proof (other than integrating which seems hard)
@LeGrandDODOM You can't start at $3$ if you have more than two logarithms composed. But if you compare to the integral, the substitution $t = e^u$ always gets you to the case with one factor (the last) less.
You can also use the Cauchy condensation test, but the integrals are IMO nicer.
@Chris'ssistheartist when I see "without L'Hôpital", I think "only use pre-calculus". Using power series when trying to avoid L'Hôpital seems very peculiar since L'Hôpital is usually introduced first.
@robjohn That's kind of my point. Here, Taylor expansions aren't more advanced than l'Hospital. Taylor expansions are bread-and-butter, l'Hospital is almost esoteric. The culture is different in different cultures.
@skillpatrol Don't ask me. I don't even know what calculus is. Ted once tried to explain it, but the only thing I remember is that it's kinda sorta vaguely like analysis, but without proofs.
@robjohn I don't know whether the blistering Bologna process changed it, but in the good old days, the first courses one took here were Analysis and Linear Algebra.
@Hippalectryon it might be considered outpowered only if you happen to know the power series at the given point. Otherwise, L'Hôpital is more efficient.
@Chris'ssistheartist shush... we are talking about high powered math like L'Hôpital and Taylor series... ;-)
yeah, i'd love to but let's do that in a couple of hours. need to do some complex analysis exam prep first
@DanielFischer hi daniel. can you tell me if my proof is correct? i asked you about this before but i was confused by the casorati weierstrass argument
Claim: Let $g$ be an entire non-constant function. Then for any $a\in \Bbb{C}$ there is a sequence $(z_k)_k$ such that $g(z_k)\to a$ as $k\to \infty$.
Attempt: Suppose there in an $a\in \Bbb{C}$ such that $$|g(z)-a|>\epsilon$$ for some $\epsilon>0$ and all $z\in \Bbb{C}$. Then $f(z)=1/(g(z)-a)$ is entire and bounded, hence constant: $f\equiv c\neq 0$. But then $g\equiv \frac{1}{c} + a$ is constant too, a contradiction.
anyway, I don't think I will study algebraic/geometric topology. that said, probably I wouldn't do much homotopy theory either, just basic things about fiber bundles.
@iwriteonbananas impressed prof by my recent discoveries about galois theory vs. covering space analogy. i think he'll get me into arith. geo. quicker than i thought.
of course, i'll have to study loads of comm. alg. before that, and alg. geo afterwards.
@iwriteonbananas the punchline of arith geo, as far as i understand, is "take ideas from alg. top, apply it in alg. geo., and solve number theoretical problems"
but i am not the right person to ask this question.
the most interesting link is galois theory-covering spaces, because there's no easy way to explain it. other topology-algebra links do exist, but the ones i know are easy.
and even though it's a link with algebraic topology, the study of the similarity no longer stays in the realm of algebraic topology
How would I figure out what resolution at (x PPI) would about equal around (x inch(es))? Here is a example what I mean; persay you have a FHD monitor at 40 inches, you want to have 39 inches of the screen to be a color. At what resolution would you need the rectangle to be?
Story goes here (...
