Let $x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{\alpha}{x_{n}^4}\right)$ with arbitrary $x_0$ and $\alpha$. Show that if the sequence converges, then it is eventually constant. Any idea?
Si on a $b\in [\pi,3\pi/2]$, on cherche $c\in [\pi/2,\pi]$ tel que $\cos c = \cos b$. !!
@Emolga: Interesting. So the limit would be $\root5\of\alpha$, if it exists. But the unique critical point of $f(x)=\frac12(x+\alpha/x^4)-x$ is at $\root5\of{-4\alpha}$.
Astyx, I'm trying to get her to use the unit circle and reflect from the third quadrant to the second.
We seem not to understand how reflection is calculated with the angles. :(
@TedShifrin Yes, I agree. Do we know anything "general" about what happens when there are no real critical points of $f(x)-x$? I assume my aim currently is to show preperiodicity
Can you good people frish up my memory. Can you multiply a row or Coulumn of matrix wih scalar null. I am reading couple of proofs and it is said that the scalar not equal to null. However when i went back to the definition it did not spesifically said it can not be null
Is there an example of groups $A$ and $B$ such that there exists non-trivial homomorphisms from $\prod\limits_{i=1}^\infty A$ to $B$, but only trivial homomorphisms from $A$ to $B$?
note that the element $(1,1,1,\dots,) \in \prod_{n=1}^\infty \Bbb Z/n\Bbb Z$ has infinite order, thus the subgroup it generates is isomorphic to $\Bbb Z$, so we can map it to any element in $\Bbb Q$
but $\Bbb Q$ is an injective $\Bbb Z$-module, so that homomorphism can be extended to the whole group
Could we just use a homomorphism that sends $\{1\}_{i>0}$ to $\mathbb{Z}$? (So the codomain would be $\mathbb{Z}$ instead of $\mathbb{Q}$) Or is that not necessarily a homomorphism?
Oh, wait, so we're sending $\mathbb{Z}$ to $\langle \{1\}_{0<i}\rangle$ and then using injectivity. Gotcha (I had the arrows reversed on the injective diagram. Had to look at my notes)
seems to consist of one finite and various infinite trees, every graph that's just one cycle, the triangle grid, square grid, and hexagonal grid, and just one more that I can find where each vertex neighbours four triangles and a rhombus
So, my understanding of Injectivity is that, if $f:A\to B$ and $g:A\to X$ are homomorphisms where $X$ is injective, then there exists a $\psi:B\to X$ such that $g=\psi f$. So, in this example we want to use $f:\mathbb{Z}\to\prod\limits_{i=1}^\infty\mathbb{Z}/i\mathbb{Z}$ and $g:\mathbb{Z}\to\mathbb{Q}$ to get a homomorphism $\psi:\prod\limits_{i=1}^\infty\mathbb{Z}/i\mathbb{Z}\to\mathbb{Q}$.
Hello all. I'm reading John Lee's Axiomatic Geometry. It includes a detailed treatment of Euclidean plane geometry with rigorous proofs from axioms. But it doesn't include Euclidean solid geometry and Cartesian coordinate system (Analytic geometry). Anyone know of books that cover these topics and are rigorous like John Lee's book?
If $G = \Bbb R / \Bbb Z$, I'm asked to prove that the action of $G$ on $C(G, \Bbb C)$ by translation is not continuous for the operator norm on $C(G, \Bbb Z)$
et $X$ and $Y$ have the standard bivariate normal density, with parameter $\rho,\sigma$ and $\tau$. Let $U=X\cos(\theta)+Y\sin(\theta)$ and $V=Y\cos(\theta)-X\sin(\theta)$. Find the values of $\theta$ such that $U$ and $V$ are independent.
@Lukas the only listed reference for the analytic number theory Vorlesung is Zagier's "Zetafunktionen und quadratische Zahlkörper" so looks like it's gonna be about Dirichlet/Dedekind zeta functions :D
So this year things are weird, the guy who's teaching both semesters of AG now is teaching "topics in AG" next fall and wants to make it an extension of this class
Lee's book was written for future high school teachers. No one teaches 3D geometry in high school, so you won't find a Lee-like source. Here's a fabulous book on all of geometry, by one of the great French differential geometers. Geometry (volumes 1,2) by Marcel Berger. Super rigorous.
And yeah I'm thinking that like, the action of R on C(R) definitely isn't continuous because you can take a bump and push it off to infinity, then translation pushes it back to 0
Yeah see I was originally thinking that they were asking to prove that T_t (translation by t) isn't a BLO but maybe they're referring more to the map G\times C(G,C)) -> C(G,C)
And the discontinuity of that is more plausible to me because if you take an extremely tiny bump and rotate it by some g that's real close to the identity, now you'll get two functions that are kinda far in operator norm
"Soit G=R/Z. On munit V=C(G) de la norme ||f||= supx|f(x)| et on considère l’action par translation de G sur V. Montrer qu’elle n’est pas continue pour la topologie de la norme d’opérateur."
I don't think putting the operator norm on $\mathrm{Hom}(V,V)$ gives you the right notion for a continuous action on a normed space $V$. In general a continuous action corresponds to a continuous map $G \times V \to V$ which is equivalent to a continuous map $G \to C(V,V)$ where we put the compact-open topology on the latter. Now I don't think the compact-open topology restricts to the operator norm
I have a theorem that states that a group morphism $G \to Aut(C(G))$ is a continuous representation iff both partial applications are continuous everywhere
But there aren't any $f$ close to 0 such that $\vert g\cdot f - f\vert$ is big are there ? And that would be the definition of continuity of $f\mapsto g\cdot f$
@anakhro I'm not following. You want to show that every chain map $f:A^{\bullet} \to B^{\bullet}$ is homotopic to itself. $h_n=\delta_n$ is not a chain homotopy from $f$ to $f$, because $h_n=\delta_n$ is not a map $A^{n} \to B^{n-1}$
But yeah so the way I was thinking about it earlier wasn't quite the way your prof wanted it to be stated since it's in terms of operator norm but the vague idea was like
