This is Exercise 3.1.A. in      Suppose that $\pi: X\rightarrow Y$ is a continuous map of differentiable manifolds. Show that $\pi$ is differentiable if differentialble functions pull back to differentiable functions, i.e., if pullback by $\pi$ gives mpa $\mathcal{O}_Y\rightarrow \pi_*\mathcal{O}_X$.   Let $f: V\subseteq Y\rightarrow \mathbb{R}$ be a differentiable function on an open subset of $Y$. Let $(U,\phi)$ be a chart of $X$, $(V,\psi)$ be a chart of $Y$. Then $f\circ\pi(\phi^{-1})$ is differentiable and $f\circ\psi^{-1}$ is differentiable. We need to show that $\psi\circ\pi\circ\phi^{-1}$ is differentiable. I don't know how to prove this.   Can we say that since $(U,\psi\circ\pi)$ constructs a chart of $X$, by the compatibility, $\psi\circ\pi\circ\phi^{-1}$ is differentiable? This seems true but this does not use the fact that $f$ is differentiable.  Sorry I have no background of Differentiable Manifolds. Any help would be appreciated.
This is Exercise 3.1.A. in      Suppose that $\pi: X\rightarrow Y$ is a continuous map of differentiable manifolds. Show that $\pi$ is differentiable if differentialble functions pull back to differentiable functions, i.e., if pullback by $\pi$ gives mpa $\mathcal{O}_Y\rightarrow \pi_*\mathcal{O}_X$.   Let $f: V\subseteq Y\rightarrow \mathbb{R}$ be a differentiable function on an open subset of $Y$. Let $(U,\phi)$ be a chart of $X$, $(V,\psi)$ be a chart of $Y$. Then $f\circ\pi(\phi^{-1})$ is differentiable and $f\circ\psi^{-1}$ is differentiable. We need to show that $\psi\circ\pi\circ\phi^{-1}$ is differentiable. I don't know how to prove this.   Can we say that since $(U,\psi\circ\pi)$ constructs a chart of $X$, by the compatibility, $\psi\circ\pi\circ\phi^{-1}$ is differentiable? This seems true but this does not use the fact that $f$ is differentiable.  Sorry I have no background of Differentiable Manifolds. Any help would be appreciated.  I also saw this post: , but I don't understand how $\gamma$ is smooth in the answer. $\gamma=\beta\circ\rho$, where $\rho$ is defined to be smooth, but $\beta$ is not.
Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$ \forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$  Then how prove $G$ is an abelian group?  Some context: Some of these commutation relations often imply that $G$ is abelian, for example , or . This looks like another example of this phenomenon, but the same techniques do not apply. Topologist's Sine Curve: A rigorous proof of why $\overline{S} = S \cup V$ Let $S$ and $V$ be the following subsets of $\mathbb{R}^2$.  $$ S = \left\{ \ x \times \sin \frac{1}{x} \ \colon  \ 0 &lt; x \leq 1 \ \right\}, \qquad V = \left\{ \ 0 \times y \ \colon \ -1 \leq y \leq 1 \ \right\} = \{ 0 \} \times [-1, 1]. $$ Then how to show rigorously that the closure $\overline{S}$ of $S$ is given by  $$ \overline{S} = S \cup V? $$  By rigorously, I mean using the smme kind of argument as is used in real analysis or topology, such as in (most of) the proofs in Rudin or Munkres.   Intuitively, it is clear that as $x \to 0+0$, then $\sin \frac{1}{x}$ will fluctuate its full range of $[-1, 1]$, and for any point $0 \times y$, where $y \in [-1, 1]$, there will be points of $S$ arbitrarily close to it.   But I'm unable to rigorously show this. How to circumvent using an out parameter in an anonymous method block? The following method does not compile. Visual Studio warns "An out parameter may not be used within an anonymous method". The WithReaderLock(Proc action) method takes a delegate void Proc().  public Boolean TryGetValue(TKey key, out TValue value) {     Boolean got = false;     WithReaderLock(delegate     {         got = dictionary.TryGetValue(key, out value);     });     return got; }   What's the best way to get this behavior? (Please refrain from providing advice on threadsafe dictionaries, this question is intended to solve the out parameter problem in general). How to carry audio over SSH? Is there a way to carry audio over ssh?   Say after connecting to remote machine through ssh -X I opened a browser and want to watch a youtube video. Currently I could only able to watch the video without any audio. But I want to hear the audio too. How to do that? show that if $a | c$ and $b | c$, then $ab | c$ when $a$ is coprime to $b$. Given two numbers $a$ and $b$, where $a$ is co-prime to $b$,     Show that for any number $c$, if $a|c$ and $b | c$ then $ab| c$. Is the reverse also true? In other words, if $ab |c$ and $a$ is co-prime to $b$, then do we have $a | c$ as well as $b|c$? How to count number of bases and subspaces of a given dimension in a vector space over a finite field? Let $V_{n}(F)$ be a vector space over the field $F=\mathbb Z_{p}$ with $\dim V_{n} = n$, i.e., the cardinality of $V_{n}(\mathbb Z_{p}) = p^{n}$. What is a general criterion to find the number of bases in such a vector space? For example, find the number of bases in $ V_{2}(\mathbb Z_{3})$. Further, how can we find the number of subspaces of dimension, say, $r$?   I need a justification with proof. I have a formula, but I am unable to understand the basic idea behind that formula. jquery mobile external panel not taking on styling The answer is, according to the book: $(m-1)!\cdot \binom{m+r-1}{r}$ I get why this is true. You arrange $m$ people in their seats in $(m-1)!$ ways and then you put $r$ empty spots inbetween them.I was thinking - choose $m$ seats out of $m+r$ to put the people on => $\binom{m+r}{m}$ Arrange them once you've chosen the seats $\to (m-1)!$ So in total: $\binom{m+r}{m} \cdot (m-1)!$ This is obviously not true. What's wrong with the way I'm thinking? Can't change mouse influence in proportional editing mode I press O and I see it change to Proportional Editing mode, but when I press G and mouse wheel down/up, or arrow keys down/up, it doesn't change the influence area.    And I am, I press G and then I tried to scroll to change influence, but it's not working.    Done this aswell and it's not working Why Faddeev-Popov ghost cannot exist in external line? Reference for $\text{Var}(X)\le (b-a)^2/4$ I am not a statistician, but am working a proof for the upper bound of an expression which contains the variance of a variable which obtains its values from a closed interval, [0,1]. I have seen in several places (mostly forums and web pages) the result that $\text{Var}(X)\le \frac{(b-a)^2}{4}$ for $a\le X\le b$. I would like to use, and cite, this result but am having an unusually difficult time finding a solid reference. Could someone here recommend one for me? Chapterwise page range in the TOC Can you give me some advice for how to obtain the following format of the TOC?        Chapter’s heading 1-7   Chapter’s heading 9-10   Chapter’s heading 11-15 I will be attending a summer school which will last 10 days.  I want to know: for applying for a Schengen visa, is the purpose of my trip considered as touristic or study? I understand monsters only spawn in a 9 x 9 grid of chunks centered on the player, and I assume they stay around while they remain within that distance of you. But do they despawn as soon as you move away from them? Or do they remain? If so what (if anything) causes them to despawn? Are the rules any different for passive mobs?  Will running far away from an area and then coming right back clear it of monsters? Support for N-Gons (polygons consisting of more than 4 vertices) has been in Blender for a while now, but many of us still may not know when to use them, and how to do so effectively.  In what situations do N-Gons really come in handy? When should they be just plain left out? Do they have any place in finished models? Does there exist a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$ such that $\mathbf{A}^{2}=-\mathbf{I}$? How do I join two separate objects so I can edit them as one mesh? How was the torque-force relationship discovered Identify $P(2014)$ if $P(i)=1/i$ for every positive integer $1\le i\le2013$ Clean URLs and php extension on nginx I've looked at dozens of other questions and references on the web - and by all my calculations, my setup should work, but it doesn't.  I have nginx installation with php-fpm.  If I try to access a .php file, it runs correctly and I get the correct results.  I got this in my config file:  location ~ \.php$ {     try_files $uri =404;     fastcgi_pass unix:/var/run/php5-fpm.sock;     include fastcgi_params;     fastcgi_param SCRIPT_FILENAME  $document_root$fastcgi_script_name; }   Now, I want to setup my web app so that /somedir/file automatically executes /somdir/file.php while still displaying /somdir/file in the browser's address bar.  So I modified my config to contain the following:  location / {     try_files $uri $uri/ $uri.php?query_string }   This kind of works, that is, the server does access the .php file.  Yet, instead of executing it using the existing location ~ \.php$ block above, it simply spits the php source code as the download into the browser.  If I append the .php manually to the requested URL, then the php is executed.  It feels as if once the server matches try_files to $uri.php, it then does not do another pass at locations to see that what it needs to do with the php files.  I tried putting the php block above and below the location /, but it makes no difference.  How can I get the php to be executed?