How to prove $(\mathbb{R}\backslash \mathbb{Q})\cap (x,y)\neq \emptyset$ for $x,y\in \mathbb{R}$ and $x<y$?
Proof that the set of irrational numbers is dense in reals
Picking random points in the volume of sphere with uniform probability Story about a trip to a Valley of Sound I read a short story, can’t remember if it was part of a larger narrative or just on its own.  It was about a crew that was taking a bunch of space tourists to a planet that had plants with seeds in glass-like structures that make certain sounds. The main character in this story feels connected to this planet because there’s a Valley of Sound, where there’s a large number of these glass plants that make all sounds imaginable.  People can almost pick out a loved one’s voice for instance. I absolutely love this concept but can’t for the life of me remember what the story/book was called! I have a few satellites and stations orbiting my planets, but I'd like to add onto them. I've seen numerous videos on using docking clamps to attach my new ship to the existing station, but can I do that on a part-by-part basis? For example, I take an extra fuel tank up there, can I attach just the tank to my station, then send my vessel back to Kerbin? I'm not talking about refueling, but rather transferring individual parts from my ship to the station in orbit.  I'd prefer to not have to use an unmanned probe and jettison everything but the desired part in space after docking if possible, but if that's the only way I guess I don't have a choice. Why am I getting zero divergence of function $\vec{f} = \frac{1}{r^2} \hat{r}$, where $r$ is the distance from the origin and $\hat{r}$ is the unit vector in the radial direction. The divergence of this function over a sphere of radius $R$, which includes the origin.  $$\nabla \cdot f = \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 f_r) = \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{1}{r^2}) = 0$$ babel shorthand "| doesn't work in macros The package babel with the ngerman option (and most other languages as well, but not english) provides the shorthand "|, which prevents a ligature at the given position (see p. 81 of the , e.g. Auf"|lage will result in the word Auflage without an fl-ligature (as is desired because, in German, you don't want ligatures across morpheme borders).  However, when placed in a macro, "| doesn't work anymore. Why is this and how can I circumvent it?  \documentclass{article}  %\usepackage[T1]{fontenc}% doesn't help \usepackage[ngerman]{babel}  \newcommand{\foo}{"|}  \begin{document}  ff  f"|f  f\foo f  \end{document} Adverb position in "Listen carefully to what I say" How do you integrate $\int \frac{1}{a + \cos x} dx$?  Is it solvable by elementary methods?  I was trying to do it while incorrectly solving a homework problem but I couldn't find the answer.  Thanks! Books to learn physics, being a math major I am currently writing an essay on artificial moral agents, and I need to explain the difference between ethics and morals. However, I am finding it really hard to find a good book or research paper which contains a citable definition.   Could someone point me in the right direction? Using Custom Labels in a Class I have the below If statements using some hard coded strings and I want to make them use as custom labels. I have created custom labels with the same String names as given below. So how can I make these statements to use the Custom Labels I created ?  if(monitoringFrequency == 'daily'){             return dateValue.addDays(1);         }else if(monitoringFrequency == 'weekly'){             return dateValue.addDays(7);         }else if(monitoringFrequency == 'monthly'){             return dateValue.addMonths(1); I have a lot of 3D objects of various sizes, shapes in blender 2.76b and I would like to set the origin to the bottom center of the objects bounds.  I cannot select the bottom edge and snap to cursor etc as each shapes edges are in different X/Y positions.  Is there a script or tool that will allow me to set the origin to the lowest edge z position? Eigenvalues and power of a matrix The Node Editor won't work for the object selected Debian : Verify disk encryption password for already mounted system How can I safely move a module to another folder? I created a module through Drupal Console, unfortunately it wasn't placed in a folder of its own, rather straight into /modules/custom. Before I noticed where it had gone, I enabled the module.   In Drupal 6/7, this was easy to fix - I could literally just put the files into the correct path, clear the cache (maybe rebuild the registry, it wasn't an exact science), and all was well.  The same isn't true for Drupal 8; I moved the files into a subfolder, rebuilt cache, and all requests tell me that Drupal is failing to include a required file. The particular file is a block class, and the path Drupal is trying to find it is where it used to be, not where I've moved it to.  The bit of code objecting is in Symfony's ApcClassLoader, so I guess I need to clear something that's slightly above Drupal's head to get this path reference updated.  How can I do that?  Uninstalling and re-installing the module does work, but once there's important data stored that's not going to be an option.  I was wrong, uninstalling and re-installing is not a viable option. After uninstalling and reinstalling, Drupal can no longer pick up the custom blocks the module defines. I'm guessing that reference is cached somewhere that isn't cleared when the module is uninstalled. Just need to find out where that is and how to clear it... Brackets in if condition: why am I getting syntax errors without whitespace? Switching a DIV background image with jQuery How are connections differentiated from behind NAT routers? I get this message while trying to install Ubuntu 12.04 as a guest in VirutalBox 4.1.18, on an Ubuntu 10.04 host.         This kernel requires the following feature not present on the CPU: pae   Some host specs:     The host's kernel is: Linux 2.6.32-41-generic-pae GNU/Linux   lscpu (host): Architecture: i686, CPU op-mode(s): 32-bit, 64-bit   grep --color=always -i PAE /proc/cpuinfo   does show pae in its output.   The 12.04 iso used is: ubuntu-12.04.0-desktop-i386.iso      As a comparison/check, I downloaded and installed Linux Mint 13 Cinnamon to the same host on the exact same VM (I just changed the .iso image). It worked fine. Its iso is: linuxmint-13-cinnamon-dvd-32bit.iso   It seems (to me) that I have pae.. what is going on here?   Update: I had assumed that Linux Mint also required pae (being Ubuntu based), but I've just run;   grep --color=always -i PAE /proc/cpuinfo   in the Mint VM.   It showed no output.   So it seems the issue may lie with VirtualBox.   If that is the case, how can I get Virtualbox into pae mode? Block any query in BIND