ok. i can find an eigenvector, but then i need to pick a basis for V with the eigenvector as the starting vector in the basis. what if dimV is large? i wont be able to just "pick". am i misreading?
@Danu I have an algebra appendix mostly done except for two technical lemmas. The first I figure I'll ask an algebraic topologist who will probably know. The second, though, I had an incorrect proof and thought it would be easy to fix and I'm on week 2 of that easy fix.
@MikeMiller The first comment in the mail I got from Humphreys yesterday was that the first proposition in my paper, which I have a very detailed proof of, was probably just a special case of a fairly easy and well-known phenomenon
Hmm, I don't know what kind of results you are looking at, but I once read "Geometry Revisited" by Coxeter and he proves many high school geometry results there.
Classic geometry can be quite challenging. I also recommend Pedoe's book (easier than Coxeter, but still beautiful) "Geometry: A Comprehensive Course." He has amazing stuff in there.
My friend is studying to be a high school teacher so she'll have plenty of time in her life for elementary geometry books; the more recommendations the better ^^
Hi all, I've got a puzzle that I'm not sure about how to solve. There are three cities and I am to connect them by railroad. I want to use as little rail as possible, so I'm going to make three straight lines from the cities and have them intersect at some point in the middle. Is that middle point the center of mass of the triangle?
I didn't see any flag abuse during the time that I've spent here today, and I don't think it's a good idea for me to try to intervene after the fact given that I wasn't there.
@Perturbative: So you got it sorted out that for an abstract manifold it doesn't make sense to talk about smoothness of the charts? That's one of the main advantages of the simplification in Milnor and G&P.
@TedShifrin Ahh I see you saw my question on main, yep I didn't know that one can only talk about diffeomorphisms after we know that the object we're defining is a smooth manifold
I was actually meaning to ask you yesterday, if I'm losing out on much by working through G&P and having all my manifolds sitting in Euclidean space, am I?
Because I was thinking of working through Lee's book on the side
Is there a reason why the degree of a map $f : M \to N$ is only defined when $M$ and $N$ are oriented $n$-dimensional manifolds without boundary, and where $M$ is compact and $N$ is connected?
Well orientability has to be one of the conditions met because the sign of $df_x$ at a point $x \in M$ is defined to be $+1$ or $-1$ depending on whether it preserves or reverses orientation
Yeah I haven't seen that yet. But this is a bit elementary. Basically, let's say RP1 = f^{-1}(0). Since RP2-RP1 is connected and f can't be 0 on it, then 0 has to be either a global max or min
In order to prove $\sigma$-additivity implies finite additivity, essentially, I want to show $\sigma$-additivity for any $n$, with $A_{1},A_{2}, \cdots , A_{n}$ in my $\sigma$-algebra, where $A_{i}\cap A_{j} = \emptyset$ for $i\neq j$ and $A_{i}=\emptyset$ for $i\geq n+1$ and reduce it to finite case?
so, do I need to use induction for the finitely many $A_{i}$ that are nonempty?
