@FreeMind I am sorry if I go away for a while. It can be that I am thinking about a problem, it could be that I have been called away from my computer. It could be that moderator duties have called me away. I apologize for that.
@robjohn No problem, but thank you for explaining what's going on, because it gives bad feeling when you expect someone to say something but they don't!
@Alizter I have already asked the problem check the link above!
@Ted Shoot, I forgot to send you an email about the problem I'm having trouble with!!! Do you know which problem I'm referring to when I say #2 on the 2410 Final? Can I share my answers here?
I had a risk management major crying (literally) that she can't graduate without a C in my class. I tried to tell her that a 33% test average and 40% homework average wasn't going to get her a C, and that she should have been worried about that from day 1.
@TedShifrin At the sites 36-37 from the link: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf there are the identities and a theorem that I tried to apply...
OK, that's a reasonable definition, @evinda. So you need to look at the quotient of the local ring at $(0,0)$ by the ideal generated by the two polynomials. Have you done that?
@Kaj: Only Julie and Dr. Graham and I have access to override prerequisites. It's because you never took 3200. Remind me tomorrow and I'll give you an override.
Oh rats, @teadawg. I see that this was not the ultimate version of the exam. There are problems missing. Ugh. I'll have to find the actual file at school tomorrow.
That's what is on p. 37 of Fulton, @evinda. You need to know some commutative algebra to read his book. But think about taking $k[x,y]/\langle f,g\rangle$ and try to what its dimension (as a vector space over $k$) is.
The direct limit is the term for what you want. And that's not isomorphic to the symmetric group of any set - the symmetric group of an infinite set is uncountable, while yours is countable. m
Also, @Ted, if it's no trouble, after they're all done with it and handed and graded and whatever is needed and if it's okay, I would like to practice on your final :)
@FreeMind doesn't equation $(39)$ answer the whole thing? That is what I was trying to get you to when suggesting the Fourier Transform. I thought there must be some simpler looking solution, but that is just using Fourier Inversion.
It's not, unless your notion of "really big" is not actually given by a definition you or I know, in which case you're not asking a math question that I can help with...
There was a horribly bungled answer on main last night, @Kaj. If $A\subset]\Bbb R$ has the property that $A\cap A' = \emptyset$ ($A'$ the set of limit points), prove that $A$ is countable.
@robjohn I'm looking for a real approach, the person who proposed this question told me that, there are bunch of ways on the internet, I want something else, but simpler from you! :((((
@Ted I'm just looking for verification on the answer I've gotten. "Let R denote the plane region bounded above by $y=e^{-x}$, below by $y=0$, and lying between $x=0$ and $x=1$.
@MikeMiller So my logic is that a mapping can be though of a a set of ordered pairs therefore the powerset of pairs of N is uncountable thus maps are uncountable
Is there any reasonable way to see for which z in (-2, 2) the sum $\sum_{j=0}^\infty |((z^2 + \sqrt{z^2 - 4} - 2)/2)^j + ((z^2 - \sqrt{z^2 - 4} - 2)/2)^j - 2|$ converges?
I've been trying to find the spectrum of an operator forever, and I think I can get something meaningful if I can figure out whether or not that converges.
@Alizter No, this is not a correct argument... There is an injection $\Bbb N^{\Bbb N}\to \mathcal P(\Bbb N \times \Bbb N)$. Not all subsets of the latter define a mapping. The better reason is that we know the latter is in bijection with $\Bbb R$, and as the cardinality of $\Bbb R$ is the cardinality of $2^{\Bbb N}$, we have an injection $\Bbb R \to \Bbb N^{\Bbb N}$, there is a bijection between the two by Cantor-Schoeder-Berenstein.
@TedShifrin I haven't yet. The spectrum I'm trying to find is that of $S_l + S_r$ on $l_2[0, \infty)$, which I know I can turn into the spectrum of what I actually want.
Hmm, I just realized something. When I'm confident with an answer, it tends to be slightly incorrect. However, if I question an answer I've gotten, it tends to be correct
@Ted That was the only question I had, since I'm a bit rusty with surfaces of revolution. I find double integration and triple integration much more useful
@TedShifrin Do you renember the exercise about proving that $cl(A)_{\Vert \cdot \Vert_1 }=C([0,1],\Bbb{R})$ where $A=\{f\in E: f(0)=0\}$ you said draw a picture but I don't know how can I do this (to find a suitable sequence (g_n)). Can you help? Thanks
So, @MarcGato, take your continuous function and choose $x_0$ very close to $0$. Join $(0,0)$ to $(x_0,f(x_0))$ with a line segment, and use $f$ by itself to the right of $x_0$. Show it works.
I took $f\in E$. I define a function $(f_n)\n ge 2$ as $f_n(t)=f(t)$ for $t\in [1/n,1]$ and I am not sure for the part $[0,1/n]$ something like $f_n(t)=f(1/n)t$ we have $f_n /in A$ I think...?
You don't need a sequence, @MarcGato, although one is OK. You only need to show that you can get somebody in $A$ as close as you want to a given function $f$.
@TedShifrin Coolio! I've had enough of single-variable calculus, I've pretty much "mastered" it. I'll start watching your lecture vids tomorrow (I was too sick over the weekend to do much of anything), I promise
@Alizter, Consider $A \subset \mathbb{R}$ with the property that $A \cap A' = \emptyset$, where $A'$ is the set of limit points of $A$. Prove that $A$ is countable.
This one is rather nice. Let $P$ be a sylow-p subgroup of $G$. We look at some $A$ subgroup of $N(P)$, and we know that $o(A)=o(N(P)/P)$. I want to show that we can conclude that $PK=N(P)$
@TedShifrin I am reading it now. Some friends said it is old fashioned. As far as I read, it seems just have a little different "face" on the proof. But ideas are identical to the lectures I attend.
@Ted It's categoey theory. That's what Balmer calls the theory of tensor triangulated categories (the most common example of which are derived categories).
