Lately I've been thinking of asking on here every single question of Magnus et al.'s "Combinatorial Group Theory [. . .]", no matter how easy I find it - I could always use the proof verification tag - so that I'll get a better understanding of the area I'm doing my PhD in, instead of self-studying it and wasting time.
@Daminark, I am trying to understand this answer. What I have got is this: if $F$ is a finite dimensional field, then
@Daminark, I am trying to understand this answer. What I have got is this: if $F$ is a finite dimensional field, then there is a bijection which is linear, between row space and column space. Otherwise, there is a bijection between row space and column space, but there may not be any linear bijction.
Keep in mind the question, it's whether one can show bijectivity without the linearity bit. The point is that there is a bijection between two finite vector spaces over a given field iff their dimensions are the same
So the fact that they are isomorphic as vector spaces isn't saying anything more than the fact that they have the same size
In the case where you're field is R or C, all non-zero finite dimensional vector spaces have the same size, so the fact that there is a linear isomorphism between the row and column space says a lot more than the fact that they have the same size
These polynomials in principle may not be homogeneous so I'm guessing here the point is that we want to homogenize them in order to define a projective variety
So $\mathcal{X}$ is gonna be a variety over $\mathbb{Q}^{T+B+N}$ defined by having the $f_i$ (which homogeneous polynomials in the $z_i$) being $0$ as well as the $p_i$ (polynomials in the $y_i$?) being $0$, and I'm guessing this projection is just choosing out the latter points
Because those guys are gonna somehow "come from" $S$
(This is just me feeling it through, double check with someone who knows AG)
So, they define $\mathcal{X}$ as a variety over $\mathbb{Q}$ but there are no $p_i$ to work with here, so I'm not seeing how it's a variety over $\mathbb{Q}$
1) There is no non-constant map from $\mathbb{P}^n \rightarrow \mathbb{P}^m$ when $n>m$! To see this - think about what does it mean to give a map to $\mathbb{P}^m$ --- this is to give $m+1$ homogeneous polynomials $f_0,\ldots,f_m$ on $\mathbb{P}^n$. But you don't want anything to map to $[0:0:\ldots:0]$ (because this is not a point on $\mathbb{P}^m$) - but since $n>m$, $n \ge m+1$, and the intersection of $m+1$ hypersurfaces in $\mathbb{P}^n$ is non-empty!
(To parse this I guess you should probably think over $\mathbb{C}$, but the same claim is true generally (i.e. scheme-theoretically))
On this note - I found the claim "$X_{s_1}$ and $X_{s_2}$ are indistinguishable when $s_1,s_2 \not \in \overline{\mathbb{Q}}$ a bit confusing at first. At first glance I found the claim confusing because, over $\mathbb{C}$, elliptic curves are classified by the j-invariant, and certainly giving two different trascendental elements give you different j-invariants in general.
I realised that this doesn't contradict anything. One can think of this as saying that the two elliptic curves are actually "isomorphic", and that you can give a map from one to the other -- but this is not going to res…
Writing down the $\wp$ is probably one of my biggest struggles atm
Yesterday I gave a quick talk and introduced it, literally just said "I have no idea how to write it, the tex code is \wp, I'll just use $\varphi$ instead"
I was very confused when I heard people "use FOIL" for the first time
@Daminark sounds great!
we don't explicitly teach multiplying two binomials here in Germany and we don't have a mnemonic for that. We still teach people how to use distributivity, of course
There's a common technique in couting/tiling problems in Olympiad which is to use complex numbers. I'm giving some examples:
(IMO 1995 P6) Let $p > 2$ be a prime number and let $A = \{1, \cdots, 2p \}$. Find the number of subsets of $A$ each havinig $p$ elements and whose sum is divisible by $...
Hi @Adeek, I have to leave shortly but I'll be back in an hour or so
but let me just say that I'm also slightly confused by what he wrote later about projectivising the equation. I don't think it's meant to be interpreted as "Here you have a bunch of equations in variables $x_i, y_j, z_k$ (say you have $n,m,l$ of them respectively), now consider your variety sitting in affine space $\mathbb{Q}^{n+m+l}$ and take its projective completion", because if you do this I don't think you necessarily have a map to $S$ basically for the same reason I said earlier about 1)
I think, in the example for elliptic curves if you think of it sitting in $\mathbb{P}^2 \times \mathbb{P}^1$ it'd make sense, and I suspect in general when he says projectivsation perhaps he's also thinking something along these lines - but maybe I'll have to think/google a bit and see :p
@Adeek Right yeah you have $y^2=x(x-1)(x-a)$, if you projectivise this equation in $\mathbb{P}^2$ you'd get $Y^2Z = X(X-Z)(X-aZ)$, and you're suggesting that we're varying $a$, so we're looking at the family defined in $\mathbb{P}^2 \times \mathbb{A}^1_a$ (where I'm using $a$ to denote the coordinate in $\mathbb{A}^1$) by the same equation, and actually we want $\mathbb{P}^1$ instead because we want something projective. Is that what you were saying?
@Adeek Yeah I agree. I'm pretty sure this is what he meant when he talks about projectivisation later - for the general construction - also because that way you do always get a map to $S$, and if you interpret the statement as take my polynomials in $\mathbb{Q}^N$ for some big $N$ and then take its projective closure I don't think you always get a map to $S$, contrary to what he suggests...
@Semiclassical This is a nice read. And yeah, I don't know these diagrams exists until now. It does shows how orderly the chaotic region actually is, with the saddle node bifurcation being the hard to predict things
i sat in a hodge theory course last year but i didn't really put enough effort to it to understand much - basically i think nowadays if someone talks about hodge theory i can nod my head and kind of follow :p but i absolutely have to review it at some point
@G.Ünther Well according to this reading material Semi found, the chaos is actually a dense collection of period cycles appearing and they are all highly repelling
@Secret On that site you've linked there's a lot more interseting stuff I didn't know. Thanks for sharing. Now I'm kinda interested how coloring periodic points would look with King's Dream fractal
@LeakyNun Suppose $f = e^g$ where $g$ has an essential singularity at $a$. Then $f'/f = g'$. But I can cook up functions with essential singularities and whatever residue I want there.
@LeakyNun So it seems the answer is yes for functions with power series in $\frac 1 z$, but it is not clear to me for arbitrary functions with essential singularity
so we know the right-infinite case. for the left-infinite case, let $f(z) = z^m g(1/z)$, then $f'(z) = mz^{m-1} g(1/z) - z^{m-2} g'(1/z)$, and the required residue is $m$ again
Let $f : D \to \Bbb C$ be meromorphic where $D$ is an open connected subset of $\Bbb C$ that contains $0$.
Assume that $f$ has an essential singularity at $0$.
Is there any way to find $\operatorname{Res}\left(\frac {f'} f, 0\right)$ from the Laurent series of $f$ at $0$?
There's a common technique in couting/tiling problems in Olympiad which is to use complex numbers. I'm giving some examples:
(IMO 1995 P6) Let $p > 2$ be a prime number and let $A = \{1, \cdots, 2p \}$. Find the number of subsets of $A$ each havinig $p$ elements and whose sum is divisible by $...
Let $f : D \to \Bbb C$ be meromorphic where $D$ is an open connected subset of $\Bbb C$ that contains $0$.
Assume that $f$ has an essential singularity at $0$.
Is there any way to find $\operatorname{Res}\left(\frac {f'} f, 0\right)$ from the Laurent series of $f$ at $0$?
