Let $R$ be a Euclidean domain with function $h$ satisfying the condition of the following definition:
Definition: An integral domain $R$ is a Euclidean Domain if there exists a function $h: R\setminus \{0 \} \to \mathbb{Z}_{\geq 0}$ satisfying the following conditions:
$h(xy) \geq h(x...
@ALannister: Part of what you posted for the first part of the proof seems wrong to me. I don't get how you saw that $h(x)\ge h(1)$. It has nothing to do with the Euclidean algorithm.
Is it incorrect though to say that if $yx = 1$, then, $yx + 1 = 2$, and so $r=1$, and then from the second part, where $h(r) < h(x)$ we get $h(1) < h(x)$?
I'm trying to recopy what I did that was good, add stuff that was better, and figure that out all at the same time listening to my husband asking me if we should add the vision plan to our insurance.
OH, some amazing visual stuff has been done by blind mathematicians. A blind French mathematician figured out how to turn the sphere inside out (which a theorem of Smale's said was possible), but no sighted person had seen how to do it explicitly.
But if you think about it yourself, when you try to pass the sphere through itself and turn it inside-out, you have points where tangent vectors collapse to 0.
So we assumed $x$ was NOT a unit and proved $h(x)>h(1)$. By contrapositive, if $h(x)=h(1)$, then $x$ IS a unit. (Remember that we can't have $h(x)<h(1)$.
Oh, so what you typed is what I just reminded you of. What did we already prove holds in general always?
Anyone here know of a fast algorithm for finding a minimal edge cover for an acyclic, directed graph? A brute-force solution I hacked together takes several minutes for a 460-node graph
Brute-force here as in taking what I believe to be exponential time
@Ted: I really just meant Hirsch-Smale. You can only invert spheres in dimensions 0,2,6. The reason I don't like the video proof is I don't understand why it's special to 2 dimensions
Hey everyone, I've been wondering, what exactly is the deal with Fourier/harmonic analysis? I've only seen it in the context of being convenient in physics, but is there something... I dunno, deeper about the subject?
It doesn't do everything just on groups in the abstract, it talks about Fourier analysis on the circle, on $(\mathbb{R},+)$, and (in typical Sally style) p-adics
Let $R$ be a Euclidean domain with function $h$ satisfying the condition of the following definition:
Definition: An integral domain $R$ is a Euclidean Domain if there exists a function $h: R\setminus \{0 \} \to \mathbb{Z}_{\geq 0}$ satisfying the following conditions:
$h(xy) \geq h(x...
