show it for a degenerate case, use that to develop an equation in the general case, then cycle between the variable singled out and add them together :)
In this question, rings are unital and associative by definition. I have read that there are $10$ commutative rings of order $8$, but I haven't found a concrete list of them. But I think that they are the following:
$\mathbb{F}_2 \times \mathbb{Z}/4$
$\mathbb{F}_2 \times \mathbb{F}_2 \times \ma...
@mahmoud I just realized something. Since $a\neq b\neq c$, you can assert that $a>b>c$ without loss of generality (i.e. if the order is different, then just relabel the variables)
In that case, the proof we had above in fact yields the stronger statement that $a^2+b^2+c^2>ab+bc+ca$
which asserts that if you've got two sets of $n$ real numbers $\{x_k\},\{y_k\}$, each arranged in increasing order, then any way of taking products of the two lists and adding them together is bounded below by $\sum_{k=1}^n x_k y_{n-k}$ and above by $\sum_{k=1}^n x_k y_k$
So there's a huge room for generalization of it. (plus, the rearrangment inequality is a handy tool in actual problems)
@TedShifrin a small question regarding SES's of holo. vector bundles:
Huybrechts told me that I can't always split them, but now he has the following:
First, he talks a bit about about Hermitian vector bundles that decompose $E=E_1\oplus E_2$ in orthogonal fashion (w.r.t. the Hermitian structure).
He also discusses the splitting of holo. vector bundles, saying that the complex-differentiable split is holomorphic iff $b_2$ is of type $(1,0)$. That's all OK.
Now he says: "Usually, one combines both situations. A SES of holo. vec. bun.'s can be split by choosing the orthogonal complement $E^\perp\cong E_2$ of $E_1$ with respect to given Hermitian structure on $E$."
This is purely about complex-differentiable splitting, right?
Because if it's about "holomorphic splitting" I'm confused by it
I use the shorthand notation (for exists and forall) on the board sometimes. Never in homeworks or notes or anything anyone else will read. I think I did as an undergrad in the first grad course I took and the grader penalized me significantly for it.
I was trained by Jim Munkres (as an undergraduate) never to use symbols. I use abbreviations, of course, and symbols like $\in$ and $\implies$. I told my students they could write shorthand in their notes, if they wanted.
But I cannot deal with sentences that are just strings of symbols.
But I hate reading proofs in books that make you look up every single Proposition and Lemma while you're reading. I tried not to overdo that in my books, but a certain amount is necessary so as not to be ponderous.
"Since for every $U\in\mathcal O_X$, $f(U)\in \mathcal O_Y$, and by lemma 3.4.21 $f(X\setminus U)=f(X)\setminus f(U)$ for $f$ bijective, we have for every $U\in \mathcal O_X$, $f(X\setminus U)=f(X)\setminus f(U)$ where $f(U)\in\mathcal O_Y$."
Actual sentence from the notes on what I told her about today :D
@TedShifrin It was the classic compact-to-Hausdorff proposition. You need to establish that $f$ is open, but first show that that is equivalent to showing $f$ si closed.
No, I mean if a result pertaining directly to the subject matter, i.e. not something assumed as a prerequisite, is not proven, should it be referenced?
@TedShifrin For instance, Hawking & Ellis use a nontrivial result from Hodge theory, and say "by a theorem of Hodge" without any reference or actually stating the result.
@TedShifrin We're talking about a book on J-holomorphic curves. I think it's safe to say we're not complaining about sequential compactness properties.
I think most of the fundamentals from measure theory are distinct enough from the fundamentals of the kind of classical analysis that there isn't really that much needed to cover.
@TedShifrin oh please. I could request the library scan the whole thing and send me a PDF. That's 100% legal but getting it from a Russian server isn't?
In this question, rings are unital and associative by definition. I have read that there are $10$ commutative rings of order $8$, but I haven't found a concrete list of them. But I think that they are the following:
$\mathbb{F}_2 \times \mathbb{Z}/4$
$\mathbb{F}_2 \times \mathbb{F}_2 \times \ma...
