Prove that the following function: $$r:x \mapsto \begin{cases} e^{-{1\over (1-x^2)}},  &amp; \text{if $|x|&lt;1$} \\ 0, &amp; \text{if $|x| \ge 1$} \end{cases}$$  is $C^{\infty}$  I found this problem on internet and i was interested to find a proof but i did't find any otheρ similar with this exercise.It would be very nice if we can have a proof for this.
Let $f:\Bbb{R}\to\Bbb{R}$ be a function given by  $$f(x)=\begin{cases} \exp\left(\frac{1}{x^2-1}\right) &amp; \text{if }\vert x\vert\lt 1\\  0 &amp; \text{if }\vert x\vert\geqslant 1  \end{cases}$$  I would like to prove that $f\in C^\infty$, that is, $f\in C^k$ for all $k\in \mathbb{N}$. I think that it can be done by induction on $k$. If $\vert x\vert\gt1$, the problem is trivial. On other points, the base case is the simplest and the only that I'm be able to do. Can someone help me?  Thanks.
Show *realtime* mouse cursor coordinates? (Cursor mod / overlay) Also, copy to clipboard? I'm looking for a solution that would display the current mouse cursor coordinates in realtime (i.e. NOT xdotool and NOT xev).  I need to move the mouse to a certain position, then press Alt-Tab to flip to another window and record the coordinates there. (This would not move the mouse, so the coordinate display would stay the same).  There's a Windows program that works BEAUTIFULLY for this purpose -   ...but it doesn't even start up in Wine.  Alternately, instead of displaying the coordinates, if this solution could copy the coordinates (in XXX,YYY format) to the clipboard, upon pressing a hotkey, that would be even better.  Any suggestions would be much appreciated!  P.S. I'm running Ubuntu 12.04 LTS. How to install software or upgrade from an old unsupported release? I have often read that using RSA for encrypting bigger size plaintext is not good solution.  What I understand is that using RSA encrytion is best for small plaintext and most effective if it is smaller than 'bit size' of RSA. For instance, if I use RSA with a 2048 bit size then the plaintext that I can encrypt must be smaller than 256 bytes.  Is this true? What happens if the plaintext is bigger than 256 bytes? Why is there a problem with larger plaintext when performing RSA encryption? Overriding the java equals() method - not working? Keyboard shortcut for switching between different keyboard layouts in Ubuntu Mate Is there a pre-assigned keyboard shortcut for switching between different keyboard layouts in Ubuntu Mate? What are all the common undefined behaviours that a C++ programmer should know about?  Say, like:  a[i] = i++; can anybody tell me which one of these is the correct one? I have been searching all over the internet but I have still got no answer.  Types of books Types of book Why do ICs have NC (No Connection or No Function) pins? Drush installed and running; No Drupal site found even with uri specified After much trial and tribulation, I thought I had drush all ready to roar. Much excitement. But then, no joy...  Drush is working. The sites operate fine via MAMP. Regardless, a "drush cc all" command returns:   No Drupal site found, only 'drush' cache was cleared.                                          [warning] 'all' cache was cleared.                                                                       [success]   This question has stumped drupalcon attendees and very experienced developers. Any ideas?  I am on Mac OS 10.9.3 using MAMP (free) Version 3.0.5  Here is my drush status:   Drupal version         :  7.29-dev                                 Site URI               :  http://default                           Database driver        :  mysql                                    Database hostname      :  localhost                                Database username      :  db_user                             Database name          :  db_name                        PHP executable         :  /usr/bin/php                             PHP configuration      :                                           PHP OS                 :  Darwin                                   Drush version          :  7.0-dev                                  Drush temp directory   :  /tmp                                     Drush configuration    :                                           Drush alias files      :                                           Drupal root            :  /Applications/MAMP/htdocs/mysite.dev   Site path              :  sites/default       Have tried executing drush from /Applications/MAMP/htdocs/mysite.dev; from /Applications/MAMP/htdocs/mysite.dev/sites/default; and using drush cc all --uri=http://localhost/mysite.dev  Tried running drush from sites/default. Same result. Image Alpha Background Renders Black when using "Import Images as Planes" I must be missing something here.  I'm doing an extremely simple Import Image as Plane using the AddOn and the imported image has an alpha background but it renders (and even previews in the material preview) as black where the alpha background should be.  When I import, I select Emission &amp; Transparent so the material setup is automatic.  I've read similar issues but people are typically missing the transparent node and connecting the Alpha channel from the image into the Mix Node factor.  But this is all here.  I must be missing something really stupid but I've wasted so much time trying to fix this thing.  Any help would be greatly appreciated.  Not sure if it matters but I'm using 2.74 RC2.  Many thanks!  I created this dummy blend file to test outside my project but problem still exists.  Here is the simple .blend file: why don't we define vector multiplication component-wise? I was just wondering why we don't ever define multiplication of vectors as individual component multiplication.  That is, why doesn't anybody ever define $\langle a_1,b_1 \rangle \cdot \langle a_2,b_2 \rangle$ to be $\langle a_1a_2, b_1b_2 \rangle$?  Is the resulting vector just not geometrically interesting? 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? Many spells have "At higher levels" that provides for more targets or more damage.  For example Magic Missile can be cast as a second level spell and shoot 4 missiles instead of it's 3 as a first level spell.  Can a spell without that verbiage, such as Firestorm be cast in higher level slots for the same effect as the lower level spell? biblatex - issue number of journal in date I am building a bibliography using biblatex and biber (\usepackage[backend=biber,minnames=2,backref,firstinits,sortcites]{biblatex}) with a lot of cited papers. Therefore I often have the case of a journal name, volume, issue, month and year, all necessary.   Now the strange thing is, if I have a bibitem such as:  @article{item,   title = {Some Nice Scientific Title},   author = {Doe, Jane},   journal = {Important Journal},   volume = {50},   issue = {20},   pages = {1877--1879},   year = {1984},   month = {Jun},   publisher = {Important Society},   doi = {00.0000} }   the bibliography looks like this:    Can you spot it? The issue number 20 is in the brackets with the date. Now it looks like it is the day in the date.  So, how do I get around this?  Here's the full MWE:  \documentclass{scrartcl}  \usepackage[backend=biber,minnames=2,backref,firstinits,sortcites]{biblatex}  \usepackage{filecontents} \begin{filecontents}{item.bib}   @article{item,     title = {Some Nice Scientific Title},     author = {Doe, Jane},     journal = {Important Journal},     volume = {50},     issue = {20},     pages = {1877--1879},     year = {1984},     month = {Jun},     publisher = {Important Society},     doi = {00.0000}   } \end{filecontents}  \addbibresource{item.bib}  \begin{document}  \printbibliography[title = {References},heading=bibintoc] \nocite{*}  \end{document} An .mov video that is upside down in Windows appears right side up in OSX Considering that sound can't travel through vacuum. What is the meaning of the "expansion of space"? When we say that "the space between galaxies is expanding," what do we really mean? For instance, if I think of space as being a Cartesian grid, then when space expands should I think of it as adding more grid-points or as making the distance between the grid-points larger? Or is this a flawed picture of space-time even when far from significant mass densities?  It seems to me that it must be the case that we are "adding more grid-points" because otherwise we would not observe the expansion since anything that occupied that part of the grid (e.g. light, my hand, etc.) would also expand by a corresponding amount (otherwise, after a sufficient amount of expansion, we would be able to observe sub-atomic scale processes as being macroscopic in scale). But if it is the case that we are adding more grid-points, then the "expansion" of space seems like a misnomer: shouldn't it be called the "creation" of space?  That being said, either case (adding grid-points or stretching distances) could conceivably be the same process depending on the nature of space-time (i.e. if it were discrete but dense - like the Rationals - rather than either strictly discrete (Integers) or strictly continuous (Reals)... I hope that analogy makes sense).   So what is really going on in intergalactic space? What is the meaning of its expansion? Does it correspond to creating more space, stretching the space that's already there, or something more subtle? In QGIS I would like to clip a raster precipitation layer using an admin boundary layer that is vector data. However the Geoprocessing tools seem to be usable only for vector data. How can I clip this precipitation layer? New tab in chromium is empty I installed Ubuntu 18.04, and I want to customize the fonts. I have already change all the setting by tweak tool. But the system default font not be changed, is anything I need to do?