Apparently some people think its really important that the Lorentz Metric for 4-vectors can be though of as arising from a symplectic form on $SL(2, \mathbb{C})$. I fail to see why this is exciting
@TedShifrin I can't split the product, I can't commute the matrices, and I don't find any arrangement that trivialize the problem, since one has $ABA^TC$, $BA^TCA$, $A^TCAB$, and $CABA^T$, so I can't annihiliate $A$ and $A^T$... am I missing something obvious?
Not that I can see. Because $A^T B A$ is similar to some matrix $D$ with the same eigenvalues. But now you've split up the eigenvalues and so when you multiple by $C$ and trace, you get $\Sigma \lambda_{c,i} D_{ii}$ where $\lambda$ are the eigenvalues of $C$ and $D_{ii}$ are the diagonal entries of D. BUt that doesnt obviously factor into anything nice
That is a good exercise you for you, though. Show that the product of two positive definite matrices is positive definite when they commute, but not in general.
That sounds good becasue its open-ended. I was wondering the other day how you can teach math grad students by always doing problems that say "Prove X" and then they get to research and theres no one telling them what to prove
If we want to use geometric series do we write $x=0,010101010101\ldots=\sum_{k=1}^{\infty}a_k\cdot \frac{1}{2^k}$, where $a_k=0$ if k is odd and $a_k=1$ if k is even ?
So, $x=\sum_{i=1}^{\infty}a_{2i}\cdot \frac{1}{2^{2i}}=\sum_{i=1}^{\infty} \frac{1}{4^i}=\sum_{i=0}^{\infty} \frac{1}{4^i}-1=\frac{1}{1-\frac{1}{4}}-1=\frac{4}{3}-1=\frac{1}{3}$
At least in other disciplines you can give a problem and not say what the answer is. Then whent he student gets an answer they have to think on whether its right or not. For me at least, proofs have to be iron-clad. I think our grader might hate me because I run down some trivial shit in my homework solutions so I can be omre confident in my answer.
Oh, I have a standard problem I assign in grad manifolds, Demonark. Griffiths once gave a totally false proof (without realizing it was false) and it bothered me for ages. So I started assigning it as a "What's wrong with this?" problem.
It's about invariant differential forms when you have a manifold $G/H$.
@Kevin: The issue that bothers me — and bothered me a lot when I graded students — is belaboring trivial details and totally handwaving the hard ones.
@Eric that, the problem about whether typical continuous functions map first cat to measure zero, and the one from the final about showing Besicovitch in R^2 fails with constant 4 are my favorite problems from that class
Multiplication by 10^2 would mean that we shift the comma two places to the right. So, if this the same in base 2 do we get $4\cdot x=1,0101010101\ldots$ ? Then we subtract from x and we get $4 x-x=1,0101010101\ldots-x$, so $3x=1 \Rightarrow x=\frac{1}{3}$.
I'm worried about time with this upcoming difftop final. Etnyre said it was going to be like a comprehensive exam. And doing the homework takes me ages. I get full marks, but only because I spend a lot fo time and go to office hours.
@Ted I struggle with how much to mark off when the students in this Group Theory in Quantum Mechanics class hand-wave hard stuff. If it were a pure physics class I wouldnt care so much, but its sort of an applied math class.
Let $\varphi(n)$ denote the sentence "$n$ does not encode a proof of $0=1$". Then, $\forall n \varphi(n)$ is basically the consistency statement of PA. By second Godel's incompleteness theorem, PA does not prove $\forall n \varphi(n)$, so $PA \cup \{\exists n \neg \varphi(n)\}$ is consistent assuming that $PA$ is consistent, so by completeness theorem, there is a model of PA such that $\exists n \neg \varphi(n)$ is true, but clearly $\varphi(n)$ is true for every natural, so what's wrong?
I was talking to the prof at UGA who's teaching undergrad real analysis, and he was given a grad student grader for a class with 35 students for a total of 3 hours of grading. Useless.
@TedShifrin Ya I like this Professor, but being the TA for this has been odd at times. I've never gotten a handle on exactly what his expectations are for the course and the students. Per usual, in hindsight I shouldve tried to communicate about it more.
Yeah, like I've found at the beginning of the quarter, since Laci would never tell us how much to knock off without asking us how much we'd knock off, for a number of problems I was quite a bit off
My name is Kevin Driscoll, and for the course I'm TAing right now, I am a bad TA. I don't really know the material. And the solutions provided to me are not very detailed. And I don't really have the time to learn the material and write my own solutions, so I kind of do my best. But overall I'd still give myself low marks on an instructor survey.
@LeakyNun In a model for ${\sf PA}\cup\{\exists n:\lnot\phi(n)\}$, $~n$ will be a nonstandard element (larger than all the standard elements, aka infinitely large)
