Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
If you assume each stage of the sieve of Eratosthenes kills roughly $1/p_k$ of the remaining numbers then this would imply the Goldbach conjecture @BalarkaSen
I dunno exactly what bounds we would need on that to make it still work
There was a guy at my university who was convinced he had proven the Collatz Conjecture even tho several lecturers had told him otherwise, and he sent his paper (written on Microsoft Word) to some journal citing the names of various lecturers at the university
Here is one part of the Peter-Weyl theorem: Let $\rho$ be a unitary representation of a compact $G$ on a complex Hilbert space $H$. Then $H$ splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of $G$.
What exactly does it mean for $\rho$ to split into finite dimensional unitary representations? Does it mean that $\rho = \oplus_{i \in I} \rho_i$, where $\rho_i$ is a finite dimensional unitary representation?
I attended only first week of the two week long workshop, and Gouzel's talks were in the second week, so I couldn't attend them. I'll watch the uploaded talks sometime
But I know that these three people laid down the foundations of the theory and whatnot
Sometimes my hint to my students used to be: "Hint: You're making this way too hard." Sometimes you overthink. Other times it's a truly challenging result and it takes a while to discover the right approach.
@anakhro I got a response from the string diagram person
> I'm not aware of a way to use string diagrams as a full graphical programming language, but that's not what I want anyway -- I just use them for sketching module boundaries & abstractions
Once the $x$ is in there, you must put the $dx$ ... or else, nine chances out of ten, you'll mess up integrals by substitution. Indeed, if you read my blue book, you discover that it really only makes sense to integrate forms in the first place :P
$f : \mathbb{R} \to (\mathbb{R} \to \mathbb{R})$ and $x : \mathbb{R}$ then $f(x) : \mathbb{R} \to \mathbb{R}$... But I'm only kidding, no one sane would type $f$ and $x$ like that...
Using the recursive definition of the determinant (cofactors), and letting $\operatorname{det}(A) = \sum_{j=1}^n \operatorname{cof}_{1j} A$, how do I prove that the determinant is independent of the choice of the line?
Let $M$ and $N$ be $\mathbb{Z}$-module and $H$ be a subset of $N$. Is it possible that $M \otimes_\mathbb{Z} H$ to be a submodule of $M\otimes_\mathbb{Z} N$ even if $H$ is not a subgroup of $N$ but $M\otimes_\mathbb{Z} H$ is additive subgroup of $M\otimes_\mathbb{Z} N$ and $rt \in M\otimes_\mathbb{Z} H$ for all $r\in\mathbb{Z}$ and $t \in M\otimes_\mathbb{Z} H$?
I did, but I think I can't use it, I'm allowed to use only theorem that for linear $F: V \to W$, where $V,W$ are linear spaces, $\dim( \ker (F)) + \dim ( \mathrm{Im} (F)) = \dim (V)$
Oh, he's defining it by expansion by cofactors. Then it's very non-obvious, without proving the multilinearity properties and uniqueness, that you can switch rows.
@chandx: You're not answering my question. Tell me $F$, $V$, and $W$.
So, a question I've been thinking about: How difficult would it be to get a paper published on a topic which isn't exactly motivated by anything real? Like integration in the sedenions, or something.
you have 2 equations, 4 variables. use the augmented matrix associated with this system so you get the simplest solution and then the rank is your dimension
You haven't specified whether the paper had any correct content or whether anyone would be interested. Both of those are more important than "motivation" by "something real."
No augmentation, @Lucas. It's a linear map and he wants the kernel.
@TedShifrin what I've proved so far: row is sum of vectors $\alpha X + \beta Y$ is the sum of $\alpha$ time that matrix where the associated row is only X, same to $\beta$ and Y
Well, assuming that the paper is all correct (or at least to a reasonable point). I guess what I'm asking would really be "how much does 'motivated by real world application' affect whether people would be interested in the contents of the paper?"
@Rithaniel $2 + 2 = 4$ is a true statement. Would you publish that in a paper? Maybe... On the surface it seems dumb, but if you can convince me the proof is actually hard... then maybe I would reconsider.
Although not the only route, can you tell me something contrary to what I expect?
It's a formula. There's no question of well-definedness.
I'm making the claim that there's a unique function with the 4 multilinear properties. If you prove that your formula satisfies those (with any row), then it follows that they all give the same answer.
It's old-fashioned, but I've used Ahlfors. I tried Stein/Stakarchi and disliked it a lot. I was going to use Gamelin's book, but I ended up with cancer and didn't teach the grad complex course that time.
Lang's book actually has some good things. I like things in Narasimhan's book, but it's pretty sophisticated.
You define the residue to be $1/(2\pi i)$ times the integral around any suitably small smooth curve around the singularity. Of course, then you can calculate $\text{res}_0\big(\sum a_nz^n\,dz\big) = a_{-1}$ and check this is independent of coordinate system.
@A.Hendry: It looks pretty sophisticated, so I don't know the answer(s) off-hand. The things on $u$ at endpoints look like the dual boundary conditions. I vaguely remember this from teaching the material 30+ years ago.
@Eric: If you go eastward, we'll never cook! :(
I'm also making a spinach soufflé tomorrow — I don't think I've done that in 30+ years. Crazy ridiculous.
@TedShifrin Thanks for the help! Dual boundary conditions, eh? I'll look that up. I'm mostly concerned about $u(a)=0$ in the term $u'/u$ appearing in $h'-\frac{u'}{u}h$ (and also for $w=-\frac{u'}{u}P$)
@TedShifrin It seems to me like $u$ can't be zero, or else $w$ would be infinite.
@TedShifrin I know the Jacobi accessory equation is a type of Sturm-Liouville problem, from which Fox demonstrates in his book that $u$ and $u'$ cannot simultaneously be zero, but that doesn't stop $w$ from blowing up when $u(a)=0$ in the denominator
@Ted i think i’m gonna wait until i’m back in chi and able to talk to my letter writers before i think any harder about my decision, i’m getting nowhere on it
