Let $$N=\{0,1,2,3,...\}$$ be the set of all non-negative integers and $A$ the set of all bijections from $N$ to itself.  Prove that   $i)$ $A$ is an infinite set.   $ii)$ There exist no bijection from $N$ to $A$
I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not.  I tried to prove it is uncountable somewhat mimicking the Cantor's diagonal argument but failed.
I need to install more applications - can I use my external HD (samsung T5) and add an additional applications folder? I have loads of space on it and it the speeds are great. What is the difference between I play and I do play?  For example:  If someone were to ask to me, do you play soccer? I have a USB drive that I can see/access/modify/etc in Linux/Ubuntu but it doesn't show up on my Mac OSX.  I did sudo fdisk /dev/sdb (where my USB is) and then g to create a new GPT partition table, n to make a new partition, used all default settings, then w to save and exit.  Then I did sudo mkfs.fat -F32 /dev/sdb1 to format the partition with a FAT32 filesystem.  I ejected the USB and put it in my Mac and it doesn't show up / get detected.  What did I miss? Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? Automatically redirect anonymous user from unanswered duplicate question to corresponding answered version Raised index of partial derivative I am having a really hard time wrapping my head around component notation for tensor fields. For example, I do not know exactly what the following expression means $$\partial_\mu\partial^\nu \phi, \tag{$\#$}$$ where $\phi$ is a scalar field. On the one hand $\partial^\nu=g^{\lambda\nu}\partial_\lambda$ where $g_{\mu\nu}$ is the Minkowski metric, and hence we could write explicitly $$\partial_\mu\partial^\nu \phi=\sum_{\mu,\nu,\lambda}g^{\lambda\nu}\partial_\mu \partial_{\lambda}\phi=\sum_{\mu,\nu}\partial_{\mu}\partial_{\nu}\phi=\partial_\mu\partial_\nu\phi. \tag{$*$} $$ On the other hand, we may think of $\partial_\mu\partial^\nu=g(\partial_\mu,\partial^\nu)=\delta_\mu^\nu;$ so that $\partial_\mu\partial^\nu\phi=\phi?$ Maybe? I am actually not sure of what this would mean. I am really confused. Any help is appreciated. Edit: To give context of where this expression comes from: I was computing the Lagrangian $$\mathcal L=\frac{1}{2}(\partial_\mu\phi)(\partial^\mu \phi) $$ considering an infinitesimal spacetime translation $x^ \mu\to x^\mu-\alpha a^\mu$. The scalar field thus transforms like $\phi(x)\to \phi(x)+\alpha(\partial_\mu\phi(x))a^\mu.$ Plugging thins into the Lagrangian yields the term I am referring to. Edit 2: The change in placement of indices are actually my doubts. I try to elaborate. I do not have any background in using indices to talk about tensors. I am used to interpret the expressions $\partial_\mu$ as the local vector field defined in some chart (local coordinates). I think about vector fields $X$ as abstract section of the Tangent bundle, which restricted to local coordinates can be expressed as $X=X^\mu\partial_\mu$. In the context of QFT, as far as I understand, the symbol $\partial_\mu$ denotes  $(\partial_t,\nabla)$ in the local coordinates $(t,x,y,z)$. So that $\partial_\mu\phi=(\partial_t \phi,\partial_x \phi,\partial_y\phi,\partial_z\phi)$. This was supposed to be my justification on why I wrote the summation on $\mu$ and $\nu$ in $(*)$, but now I note that this only applies when $\mu$ or $\nu$ appear twice, indicating the scalar product; which leads me to the last remark. I think of $g_{\mu \nu}$ as the component of the matrix $$g=\begin{pmatrix} 1&amp;0&amp;0&amp;0\\ 0&amp;-1&amp;0&amp;0\\ 0&amp;0&amp;-1&amp;0\\ 0&amp;0&amp;0&amp;-1\\ \end{pmatrix}$$ which represents the pseudo-Riemmanian metric, which by definition acts on tangent vectors, i.e. linear combinations of the $\partial_\mu$ applied to a point. This is where my doubt comes, in which was the right way to interpret the notation; in particular what is the expresion $(\#)$ in explicit coordinates? How to find infinite sum How to find infinite sum $$1+\dfrac13+\dfrac{1\cdot3}{3\cdot6}+\dfrac{1\cdot3\cdot5}{3\cdot6\cdot9}+\dfrac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $$   I can see that 3 cancels out after 1/3, but what next? I can't go further. Tools for tracking NPC/PC relationships and attitudes I have always had difficulty keeping track of NPCs, their attitudes and their relationships with each other and the PCs. I find relationship webs/diagrams are helpful, but find them difficult to keep up to date and accurate from session to session.  Are there any tools that would help with this task? I am aware that some general mapping software exists, but am more interested in tools specifically designed for RPGs. Also, I would prefer a graphical interface if software is suggested, and to not have to learn a text based language to define the diagram. However, if it turns out that the only tools available are more generalised, and/or complex, I would still like to hear about them as answers. Environment for two columns with lstlistings In order to reduce copy-pasting, I want to create an environment that will produce a two-column table with headers and two listings in different languages, like below:  \begin{tabular}{l l} Python &amp; Haskell \\ \hline \\ \lstset{language=Python} \begin{lstlisting} A \end{lstlisting} &amp; \lstset{language=Haskell} \begin{lstlisting} B \end{lstlisting} \end{tabular}   I want to use it like below, hiding everything but code:  \begin{comparison} A \vs B \end{comparison}   So far I tried making two new environments (instead of a \vs command inside) with \lstnewenvironment, where first environment had \tabularx command and second \endtabularx. I also tried creating normal \newenvironment for this tabular, &amp; for \vs command and manually inserting \begin{lstlistings} and \end{lstlistings} with code, but it stopped compiling when I put lstlistings inside.  How can I create such an environment?  EDIT: The most compact version I've came up with until now is  \newenvironment{comparison} {\begin{parcolumns}[nofirstindent,distance=0pt,colwidths={1=0.35\textwidth}]{2}} {\end{parcolumns}}  \lstnewenvironment{python} {Python \lstset{language=Python,frame=t}} {}  \lstnewenvironment{haskell} {Haskell \lstset{language=Haskell,frame=t}} {}   which is used like that:  \begin{comparison} \colchunk{ \begin{python} A \end{python} } \colchunk{ \begin{haskell} B \end{haskell} } \end{comparison}   I'm still unable to hide \colchunk{} within python or haskell environments. Does Windows 7 allow auto-login with a stored password, like TweakUI did? The TweakUI powertoy for Windows XP had an option to auto-login. It stored your password and did the login for you. This was useful, since you kept password protection, but had login programs loaded when you first booted.  When you switched users or hibernated, it would still ask for a password, which is the desired behavior.  Is there a way of doing this in Windows 7? Pure significance of line integrals of vector fields I can understand how the line integral of a scalar function has pure-mathematical significance, representing an area beneath a curve. But is there a pure significance to the line integral of a vector field? The only understanding I have of this is the physical example of work done on a moving particle. Thanks for any insight. LibreOffice toolbar icons went blank after application hanged How to only substitute lines that match several patterns in sed? $ echo -e 'CH12\nCH23au' | sed '/^CH/s=^=&lt;b&gt;=' &lt;b&gt;CH12 &lt;b&gt;CH23au   I know I can match lines starting with CH by ^CH but how can I match multiple patterns?   Example:  Input:  CH12 CH23au   Output:  CH12 &lt;b&gt;CH23au   How to only put &lt;b&gt; where there is ^CH and au in the line? Finding the Dimension of a Matrix Polynomial: $W$ = { $p(B)$ : $p$ is a polynomial with real coefficients} Let $W = \{ p(B) : p \text{ is a polynomial with real coefficients}\}$, where $$B= \begin{pmatrix} 0 &amp; 1 &amp; 0\\  0 &amp; 0 &amp; 1\\  1 &amp; 0 &amp; 0 \end{pmatrix}$$  Which of the following possibilities presents the tightest bounds on the dimension $d$ of the vector space $W$?     $4 ≤ d ≤ 6$    $6 ≤ d ≤ 9$    $3 ≤ d ≤ 8$    $3 ≤ d ≤ 4$ How to install brackets.io from the command line? For some reason the gnome installer that handles .deb packages on my system is not working. I'm not concerned.  How would one install the Brackets.io program from the terminal? $N$-body simulation in General Relativity How would one perform an $N$-body simulation in General Relativity (GR) for something like galaxy formation or galactic dynamics?  Suppose one wants to simulate the rotation curve $v(r)$ for galaxies with an $N$-body simulation, and wants to do it in the framework of GR / the Einstein equations. How to exclude jars generated by maven war plugin? Is there a nice example of integral domain $R$ and $s,t \in R$, with $Rs \cap Rt$ not principal but lcm($s,t$) exists? I am currently searching for an integral domain $R$ which allows lcm, but where the intersection of two principal ideals is not necessarily a principal ideal (or at least an example with two certain elements).   As converse example (where the sum of two ideals is not principal but $R$ allows gcd) is $\mathbb Z[X]$. It allows gcd, but obviously $\left&lt;X+1,2\right&gt;$ is not principal.  An example of an integral domain where the intersection of two principal ideals is not principal would be $\mathbb Z[\sqrt{-5}]$ with the ideal $\left&lt;2\right&gt; \cap \left&lt;1+\sqrt{-5}\right&gt;$, but this doesn't allow lcm.  Since I can't find a nice example, I put the question here. Maybe someone has already seen this? (I am not completely sure, one even exists, but pretty certain there is an example. In the case my hunch is wrong, I would be happy, if someone could proof me wrong.) Suppose we have an object in space. say gravity is negligible, and the thing has a rocket on both sides. the thing fires one rocket for, say, 5 seconds. now it's moving. The object has kinetic energy. Now the other rocket fires for 5 seconds. Then it's stationary. where does the energy go? I know that some of the energy from the rockets would just go into space as heat and kinetic energy of the hydrogen particles flying out, but how does there seem to be more energy in the situation if the first rocket just goes for 10 seconds than if one goes for 5 and the other goes for 5? Intellisense in IntelliJ no longer working