then you take a family of disjoint sets $F$ and consider the function $g:\bigcup F\to F$ that sends $z\in\bigcup F$ in the only $A\in F$ such that $z\in A$ (this requires no choice), this is a surjective function and then you need to show that its right-inverses are choice functions for $F$
Okay wait let's not look at it as a product, $A$ is a collection of disjoint sets, so map their union to it such that a point is mapped into the set containing it, the right inverse is a choice
Now, assume AC over disjoint sets is true, maybe you can take a general collection and try to force it to be disjoint by taking products with ordinals?
4 of us did manifolds, and had a more general treatment of forms from there (rushed, but we did have a pset dedicated to it), and from just talking around to friends a lot of people were just playing it by ear wrt even Schlag's plane treatment
Well, I don't think you guys should sacrifice actual differential geometry for days of learning about forms, much as I love forms. I think it's better to learn more curves and surfaces and develop geometric intuition. But Eric thinks you're all baby Erics :P
This is the problem when people don't teach what they should in the important formative courses ... and go off the deep end too much into fancy analysis.
Like one day he just went through a bunch of definitions of stuff, one day where he did arclength and parametrization, and one day he proved that conformal mappings satisfied C-R equations
only me and a few other people who had seen the stuff in high school could even remotely follow what was going on @Daminark. I think it put a lot of people off
So, the sequence's description says this: Maximum m such that there are no two adjacent elements belonging to the same n-th power residue class modulo some prime p in the sequence 1,2,...,m
The way you termed it, it has 3 numbers: the power, the remainder, and the base. But n-th power residue class modulo some prime p only has 2 numbers. What does that mean?
DogAteMy: Yeah, you got the tetrahedral symmetries. Now the matrix representations are cool (you could even say 3-dimensional representations) :P
Demonark: Read through the first two sections of my notes. I can recommend particular exercises if necessary. You should do a few computations (with chain rule, even) to make sure you understand basics.
It's a less abstract treatment than you'll find in chapter 4 of G&P ... also, I hate their numerical convention that has the unit cube in $\Bbb R^k$ having volume $1/k!$.
@HyperNeutrino The offset means the number input to the sequence that generates the first result. For example, in A00478, it doesn't make sense to put fewer then 6 balls in 3 containers with at least 2 balls per container, so the sequence isn't defined for inputs less than 5, and thus the offset is 6.
No, Demonark, I mean to correct so that you get the right thing for the volume of the cube. G&P try to save some factorials in formulas and that's the price they pay. I'm not willing to pay it.
Well yeah, I mean he introduced a $k!$ type of term into the formulas early so that down the line when he got to $\mathbb{R}^n$ he wouldn't have to scale the volume of the cube as such
You see it occasionally in physics as well, e.g. whether $T_{[ij]}=\frac{1}{2}(T_{ij}-T_{ji})$ or not (though that's the simplest possible case of course)
