@ShivangiBhatnagar You should try asking it on the main website, math.stackexchange.com instead of in this chat ... looks like a cool question, I will upvote it when you ask it :)
@Chris'ssis Here's the exercise : $\Omega=\mathbb{N}^*,P(\omega=n)=\dfrac{1}{2^n}$, let $A_k$ be the event $k\mid\omega$. 1) Find $P(A_k)$ 2) Let B be the event "$\omega$ is prime", show that $\frac{13}{32}<P(B)<\frac{209}{504}$
@Chris'ssis I have found that $P(A_k)=\dfrac{1}{2^k-1}$
What starting point would you recommend me for the one below?
$$\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx $$
EDIT
Thanks to Felix Marin, we know the integral evaluates to
$$\displaystyle{\large{\ln^{2}\left(\, 2\,\right) \over 2\pi}}$$
@Hippalectryon, I think it is possible for infinite-dimensional vector spaces only, right? Because finite dimensional itself means that there are basis to define its dimension.
@Chris'ssis Here's the exercise : $\Omega=\mathbb{N}^*,P(\omega=n)=\dfrac{1}{2^n}$, let $A_k$ be the event $k\mid\omega$. 1) Find $P(A_k)$ 2) Let B be the event "$\omega$ is prime", show that $\frac{13}{32}<P(B)<\frac{209}{504}$
@MikeMiller $\mathbf Z_p$ acts on the solenoid (the inverse limit of inverse system of $S^1$s). Question is, whether the action is free and properly discontinuous. Fixed point free part is obvious, but it's not clear if the action is prop. disc. as the solenoid $X$ not only has the profinite topology but is also inherited with the topologies from the $S^1$s. In fact I guess it's less likely for the action to be prop. disc.
Although if it is, then you have the striking (to me?) identity $\mathbf{Z}_p \cong \pi_1(X/\mathbf Z_p)$
No you don't, @Balarka. You only have the theorems about $\pi_1$ in the case that that your spaces are path connected and locally path connected. Solenoids are neither.
@MikeMiller Isn't it true that complement of non path connect spaces inside a path connected space is path connected? Then how about thinking about $\prod S^1 - X$?
$\Omega=\mathbb{N}^*,P(\omega=n)=\dfrac{1}{2^n}$, let $A_k$ be the event $k\mid\omega$.
1) Find $P(A_k)$
2) Let B be the event "$\omega$ is prime", show that $\frac{13}{32}<P(B)<\frac{209}{504}$
One can see easily that $P(A_k)=\dfrac{1}{2^k-1}$
To find $P(B)$, I did the following :
$$\begi...
I don't have suggestions to patch it up, @Balarka. For screwed up spaces, like solenoids, the fundamental group is just not the right tool to study them.
What starting point would you recommend me for the one below?
$$\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx $$
EDIT
Thanks to Felix Marin, we know the integral evaluates to
$$\displaystyle{\large{\ln^{2}\left(\, 2\,\right) \over 2\pi}}$$
Say I have $n$ urns and $m$ balls. The distribution of the balls in the urns is equiprobable. What's the proba that one given urn has exactly $k$ balls ? @skullpatrol @MikeMiller
can anyone tell me: is a complex projective variety always a complex manifold? Presumably we need the variety to be smooth also? It seems like Huybrechts book says that we dont though which confuses me...
I need help. Say we got 2 fixed points $\pi_1,\pi_2$ and denote their distances from a point $\sigma$ $r_1,r_2$ respectively. If they satisfy $|r_1-|r_2|=\lambda k, k\in\Bbb Z(1)$ for some known $\lambda$ and create a coordinate system such that the perpendicular bisector is the y-axis and $\pi_1,\pi_2$ is on the x-axis, do the points on the locuses formed by (1)(hyperbolas) form a group?
First of all how can I denote the set with set builder notation of the family of hyperbolas?
Secondly, if they form a group, what are some isomorphic groups to it?
> The scheme has earned Vamplew some online recognition, but sadly his main aim remains unfulfilled. “They still haven’t taken me off their mailing list,” he said.
Here is one line proof using beta function
$$\int_0^{\frac{\pi}{2}}\tan (x)\log(\sin x)dx=\lim_{a\to1}_{b\to-1^{+}}\frac{1}{2}\frac{\partial B}{\partial a}\left(\frac{1}{2}(a+1)\frac{1}{2}(b+1)\right)=-\frac{\pi^2}{24}$$
Q.E.D.
What is the name of the law with polytopes that defines Planes, Dots and vertices relationship?
Or in graph theory.
I am trying to deduce Gibs' rule $F=N-P+2$ (thermodynamics where N is the number of components, $P$ is the number of phases and $F$ is the number of degrees of freedoms)
(it was some very basic rule, now cannot remember its name: Euler rule, too many of them...)
Found it: Euler's formule with Planars graphs, here. Yes it is a direct corollary from Graph theory :)
@Daniel isn't it true that if $AB$ is invertible than $A,B$ are invertible?
Because we can get from $AB$ being invertible that $B$ is invertible, and then the above argument, $A$ is invertible too - so the case of both being singular isn't possible.
@Studentmath That depends on the setting. If $A,B$ are square matrices, for example, then yes. If $A,B$ are non-square matrices, then neither of $A$ or $B$ can be invertible, but $A,B$ could be. Another example are the shifts on the sequence spaces $\ell^p(\mathbb{N})$, with $A$ the left shift, and $B$ the right shift. Then $AB = \operatorname{id}$, but neither $A$ nor $B$ is invertible.
I think you wanna see this:
Ramanujan’s Notebooks Part I
Click me and try Entry $14$ (ii) / pag 146 where you set $x=\ln2$
Chris.
$\Omega=\mathbb{N}^*,P(\omega=n)=\dfrac{1}{2^n}$, let $A_k$ be the event $k\mid\omega$.
1) Find $P(A_k)$
2) Let B be the event "$\omega$ is prime", show that $\frac{13}{32}<P(B)<\frac{209}{504}$
One can see easily that $P(A_k)=\dfrac{1}{2^k-1}$
To find $P(B)$, I did the following :
$$\begi...
