@onepotatotwopotato Yes, so if $\omega$ is a nontrivial meromorphic differential p, then $[\omega]$ is the canonical class and $[\omega^{\otimes k}]$ is the $k$-canonicsl class.
$\newcommand{\u}{\mathfrak{U}}$I have a question about the proof of Riemann-Roch theorem in Farkas&Kra Riemann surfaces.
Theorem. The Riemann-Roch theorem holds for every divisor on a compact surface $M$.
Proof. We write the divisor $\u$ as
$$\u = \u_1/\u_2,$$
with $\u_j(j=1,2)$ integral and th...
@PrateekMourya try writing out what the matrix $e_{i,j}$ would look like for a small $m$ and $n$. for example, write out what the matrices look like for $m = 3$ and $n = 2$
@PrateekMourya Don't worry about the general $e_{ij}$ just answer my question with $m = 3$ and $n = 2$ tell me what some of these matrices look like. Write out a couple
for the general case it does get messy. So don't get too frustrated if you can't present it perfectly. But write out the idea of what is happening and why
NOw I can't help you specifically with the question because I haven't taken measure theory, but you have to show what you have thought of if you want help @PolineSandra
@TedShifrin 8 hrs of math will have you forgetting proper English
ted: munchkin met a younger child at the pool today. the kid tried to talk to her and she wouldn't respond. after the kid left i asked "why didn't you talk to that kid?" "he was talking wrong." "he was two and a half." "he was saying, bah bah bah mah mah." "he said his name and the name of the toy he had." "he couldn't talk right."
that's not the easiest thing to explain by example. if A' is made by applying row operations to A, then the rows of A' are linear combinations of the rows of A. i would understand that by case analysis, one case for each of the various types of row operation (some books categorize these differently, but there are usually only two or three or four of them).
it also looks kinda like you've even maybe cropped out an attempt at an explanation of the thing you haven't cropped out, but whatever.
If $f:[0,1]\to\Bbb R$ is a smooth function then $\sum_{k=1}^n|f(x_k)-f(x_{k-1})|\leq\int_0^1|f(x)|dx$ where RHS is a Riemann integral? $0<x_0<\cdots<x_n =1$ is a partition of $[0,1]$.
yeah, i dunno. this looks like some differential equations thing. i never really studied that. i don't recognize the names of these function spaces, let alone understand the problem.
someone was asking about something similar earlier, it might be in scrollback. or there was question on MSE about it.
@leslietownes It's from math.stackexchange.com/q/2082709/668308 If $f$ is absolutely continuous on $[c,d]$ then $T_f = \int_{c}^d|f'(x)|dx$. $T_f$ is a total variation of $f$.
@TedShifrin got a question for your linear algebra brain: can you think of a good way visualize the action of this matrix? $M=\begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&0&1 \\ 0&0&1&0 \end{pmatrix}$
It’s easy enough to do this algebraically but I’m trying to find something more geometrically intuitive
@shintuku Well, if $A$ and $B$ are two dedekind cuts then their product $A.B=\{pq| p\in A\space \text{and }\space q\in B, p,q\geq 0\} \cup \Bbb{Q}^-$. This is the definition I use...
This is the definition of cut I use (I am posting this to avoid any confusion):
A set of $\alpha$ of rational numbers is said to be a cut if : (i) $\alpha\neq \emptyset, \alpha\neq \Bbb Q$ (ii) if $p\in\alpha$ and $q\in \Bbb Q$ with $q<p$ then $q\in\alpha$
(iii) if $p\in \alpha$ then $\exists q\in\Bbb Q$ with $q>p$ such that $q\in\alpha.$ The condition (ii) shows that if $\alpha$ contains a rational number then it contains all rational number preceeding it. The third condition shows that the set has no largest rational. If $\alpha=\{x\in \Bbb Q: x<r \space \text{and}\space r\in\Bbb Q\}$ then $\alpha$ is a cut and we denote $\alpha=r^*.$
Now, I want to prove, The product of two cuts is a cut as well, but I am unable to do it
@shintuku If q=0, then we know that, p>q, so, p=ab where a\in A and b\in B such that a,b>0. Thus, 0<a and 0<b due to which, 0\in A and 0\in B. So, 0\in A.B
@shintuku one thing, that bothers me is that, A and B are non-empty, then following our definition of a product, Q^- is always in A\cdot B, irrespective whether A , B are positive or negative
@shintuku I dont get it. Why? What I tried to mean is: since, Q^- is always in A\cdot B, irrespective whether A , B are positive or negative so, A\cdot B is always non-empty.
So we define $H_n(X,A):=C_n(X)/C_n(A)$,the quotient space, where $C_n(X):=$ free Abelian group generated by all singular $n-$maps $\Delta^n\to X$.
I don't understand the following:
Elements of $H_n(X,A)$ are represented by relative cycles: $n-$ chains $\alpha \in C_n(X)$
such that $∂α \in C_{n−1}...
@shintuku If this lemma: "if p>q and p=ab such that p,q,ab\in \Bbb Q then there exists c,d \in \Bbb Q such that c<a and d<b and cd=q" holds good, then it's trivial.
I should rephrase that as a conjecture though.
But the question is, does this statement really hold ?
@shintuku well, that's a nice way to start, but even if we do it, I admit, we do get some sort of an idea of what's going on, but then again we have to prove it for a more general proportion.
I might have a fun experiment with Chat GPT. Let's see...
@Astyx it is. What I have in mind is that, under an appropriate 4d rotation, it should instead be reflection along the $x_1’-x_2’=0$ hyperplane. But said that way it feels pretty weak sauce
That corresponds to the vector (1,-1,1,-1) in RR^4, so the M that I cited originally maps it to (1,-1,-1,1). But that’s $e_1\otimes e_1-e_1\otimes e_2-e_2\otimes e_1+ e_2\otimes e_2=(e_1-e_2)\otimes (e_1-e_2).$
By contrast, $M$ instead maps $e_2\otimes e_1$ to $e_2\times e_2$. So in one basis the action is seemingly on the second factor whereas in another it’s seemingly on the first.
Yeah. Controller-not gate: if you run it twice, the target bit gets flipped twice or not at all. But the point is that which bit is the target vs the control bit swaps under the rotation
Elliptic integrals don't have closed-form solutions in terms of the standard elementary functions. However, Gauss & others found expressions for them in terms of the AGM (Arithmetic-Geometric Mean), so they're easy to evaluate to high precision, assuming you can do sqrt. But they were a bit painful to evaluate back in the days of log tables.
With respect to the radius, I assume.
We had a question along those lines a week or so ago:
The derivative of the volume of a sphere, with respect to its radius, gives its surface area. I understand that this is because given an infinitesimal increase in radius, the change in volume will only occur at the surface. Similarly, the derivative of a circle's area with respect to its radius g...
@TedShifrin Oh, ok. Not the radius, the semi-major axis.
Or maybe some function involving the semi-major axis and the eccentricity.
That covers ellipses and simple ellipsoids of revolution. Fortunately, in astronomy we rarely need to deal with triaxial ellipsoids. But there are a couple of bodies in the solar system with sufficiently high rotation rates that they are triaxial.
Gauss & Lagrange did some lovely stuff with ellipsoids. They were trying to figure out how to do geodesy on an ellipsoidal model of the Earth, rather than on the traditional sphere. That paved the way to more accurate surveying & map-making, and we still use an ellipsoid, WGS84, in GPS. en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid
I'm looking to define a new function that is based on the perimeter not of an ellipse, but on the perimeter of $u^2+y^2=R$ where $u=\log x$ and $v=\log y.$ Mainly I just want to explore this via numerical techniques/programming.
with the help of Thorgott I calculated the volume of this
but the surface area is probably radically different
Letting $B_n$ denote $\{x_1^2+\ldots+x_n^2\leq 1\}$ our integral equals
$$\int_{B_n}\exp\sum x_i\,d\mu = \int_{B_n}\exp(\sqrt{n} x_1)\,d\mu=\int_{-1}^{1}\exp(\sqrt{n}x)\frac{\pi^{n/2}}{\Gamma(1+n/2)}(1-x^2)^{(n-1)/2}\,dx $$
which has a reasonably simple closed form for any odd $n$. Its maximum is...
Can anyone tell me what might be a canonical intuition for the number $\ln(x)$ for integers $x$?
Specifically regarding its numerical representation. I suppose I could ask also for the intuition behind irrational representations and what the pattern is with their numerical representations in general.
I reason that there must be a way to recover the other coefficient from $\ln(x)$ exclusively from its non-integer part since however many factors of the other number are present, this part remains unchanged and is therefore in some way identical with that factor in $x$.