We have that A, B, S are collinear. If S is in $\vec{AB}$ then does it hold that $\vec{A}-\vec{S}$ is a multiple of $\vec{B}-\vec{S}$ ?
And does it hold that $|SA|^2|SB|^2=((\vec{A}-\vec{S})(\vec{B}-\vec{S}))^2 \Rightarrow |SA||SB|=(\vec{A}-\vec{S})(\vec{B}-\vec{S})$, or can we not take the square root?
@SteamyRoot i proved it the way you suggested but i dont feel comfortable with $p \ ^ {-1} (p (U) ) = U$. i get that $U$ is contained in the left set. but if we take $z \in p \ ^ {-1}(P(U))$ which is not in $U$, then $z \in V$ where is the contradiction?
Okay so, if all loops are nullhomotopic, they are homotopic to each other. So traverse one path and then another to get a loop, and homotope that to the path which just goes forwards and backwards
@Daminark @BalarkaSen So if $f$ and $g$ are two paths in $X$ with initial point $p \in X$ and terminal point $q \in X$, then $f \cdot \bar{g} \sim c_p$, that part I get but what I don't get is how to homotopy that to the path which goes backwards and forwards
Usually one only uses them when they either abut quickly (like on page 3 or 4) or for the fact that in some nice cases they lead to a 5-term exact sequence
Lol I mean, I had a very brief excursion with Hatcher since we did some stuff in difftop along those lines and I dunno, it didn't resonate well with me in my memory
Maybe I should see more of both it and Concise to really tell which is more my speed, and also check out the other one Eric mentioned, but as of now I just prefer Concise
Which means that I will make everyone use it because no one is allowed to have different preferences
awake midnight, this night dream has this integral loat across in one scene: $$\int \sin (\theta)\sin\left(\sin\left(\frac{\pi\tan \theta}{2}\right)\right)d\theta$$
So the problem was showing that if $P\in \Bbb R[X]$ and $(\cos(P(n)))_{n\in \Bbb N}$ has a finite number of limit points, then $P-P(0) = \pi Q$ where $Q\in \Bbb Q[X]$
So the way she wanted me to try was to introduce $\Delta : P\mapsto P(X+1)-P(X)$
Like, if you have a sequence $(u_{n})$, if $\phi : \Bbb N \to \Bbb N$ is strictly increasing, $(u_{\phi(n)})$ is said to be an extracted sequence in french (suite extraite)
At first I thought of proving $(Q(n)-\lfloor Q(n)\rfloor)$ (where $Q$ is defined as above) had a finite number of limit points and to conclude that $Q$ was in $\Bbb Q[X]$
And also that it's a bijection from $X\Bbb Q[X]$ into $\Bbb Q[X]$
Typhon: Oh, is there some stipulation that $\cos(nθ)$ is dense in $[−1,1]$ if $θ$ is an irrational multiple of $\pi$ ... or something like that? ... WTF is unclear?
@HenningMakholm I recently have been having problems with the same user! Although the question did make sense in this case (with some clarification about some "bad english"). I am starting to think this guy is a troll (looking at the \sqrt 2 question or the ridiculous edits for the one you participated in) or growing into a crank and deeply affected by Dunning-Kruger.
In the text "Complex Analysis", by Elias M Stein and Rami M. Sharachki i'm attempting to take take a Taubrian Route to verifying the following properstion in $(1)$
Let, $F(z)$ be the following series:
$$F(z) = \sum_{n=1}^{\infty}d(n)z^{n} \, \, \text{for} \, |z| < 1$$
$$\text{Remark}$$
One can...
In the text "Complex Analysis", by Elias M Stein and Rami M. Sharachki i'm attempting to take take a Taubrian Route to verifying the following properstion in $(1)$
Let, $F(z)$ be the following series:
$$F(z) = \sum_{n=1}^{\infty}d(n)z^{n} \, \, \text{for} \, |z| < 1$$
$$\text{Remark}$$
One can...
