Define a equivalence relation For any (x,y)∈N...
Define a relation and find its equivalence classes.
I read Griffiths EM today and it says something very interesting but a little bothering to me. It states for an atom, the position of center of mass of an electron cloud lies in the center of the proton.  But my question is why? What will happen if this is not true? How do I call the parent function from a derived class using C++? For example, I have a class called parent, and a class called child which is derived from parent. Within  each class there is a print function.  In the definition of the child's print function I would like to make a call to the parents print function.  How would I go about doing this? What is the difference between range and xrange functions in Python 2.X? Can't execute terminal commands: "the command 'sudo' is available in '/usr/bin/sudo'" I have been using Ubuntu 12.04 for quite a while and have never had a problem come at me out of the blue like this. I'm trying to install python 3.2 on my Ubuntu system and every time I run the code to compile the python source, I get this error:  'Command 'sudo' is available in '/usr/bin/sudo' The command could not be located because '/usr/bin' is not included in the PATH environment variable. sudo: command not found'   I've also tried running gksudo command to view what my environment shows but I pretty much get the same error. How much Valyrian steel is there that is unused for link making and why aren't people making weapons out of it? I was reading  that stated Maesters can have Valyrian steel links:     Valyrian Steel links (for the study of Magic + the Occult) are exceedingly rare and held by less than one Maester in a hundred. Ironically, they aren't seen as especially prestigious but rather viewed with suspicion.   This made me wonder how much Valyrian steel there is lying around for Maesters to be adding them to their chains. It seems like such  resource like this would be used for crafting weapons not being kept as ingots ready for Maesters to make links.  Indeed the Wiki I linked above claims:     the amount of Valyrian steel in the world is finite and extremely rare   How much of his extremely rare resource is being kept in reserve for link making and why hasn't it been used for more useful stuff like making weapons? How can I type special symbols on my Android device like ☺ or ♥ as a regular character? On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?) How to configure extra buttons in Logitech Mouse I use a Asus laptop with dual-boot Ubuntu and Windows 8.1  Wifi is on but I am not able to detect Wifi networks.  When I run :  lspci -knn | grep Net -A3; rfkill list   This is what happens:    I get  acer-wireless: Wireless LAN     Soft blocked: yes I used the upgrade manager to upgrade from 17.04 to 17.10. When I start the machine, I only get a black flickering screen now. I was using the nvidia binary drivers before the upgrade.     NOTE: This question (and answer) relate to a specific issue when upgrading from 17.04 to 17.10 while using Nvidia or AMD binary drivers. You might experience a completely different issue. For that please see the more general question . How to completely remove kubuntu-desktop from Ubuntu I installed kubuntu-desktop just to experiment with its interface.  Then I decided to remove and did the traditional apt-get purge and autoremove but it seems elements of kubuntu are still there. For example interface elements, scrollbars, bootscreen etc are still there.  Is there any method I can completely remove without installing a fresh copy of ubuntu? Diagonal lines in table cell I need this table:    (each row and column should have the same height and length, last cell should be divided by diagonal line). I tried the following code:  \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{array} \usepackage{makecell} \newcolumntype{x}[1]{&gt;{\centering\let\newline\\\arraybackslash\hspace{0pt}}p{#1}} \begin{document} \setlength{\extrarowheight}{0.1cm} \begin{tabular}{|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|}\hline &amp;&amp;&amp;&amp;20\\ \hline &amp;&amp;&amp;&amp;30\\ \hline &amp;&amp;&amp;&amp;45\\ \hline 15&amp;12&amp;18&amp;50&amp;\diaghead(-3,2){\hskip \hsize}{$a_i$}{$b_j$}\\ \hline \end{tabular} \end{document}   but text in the last cell is displayed incorrectly and cells are not the same.     How can I change that? Why must the eigenvalue of the number operator be an integer? My apologies if this question has already been answered. Using the notation of Sakurai, we have an energy eigenket N and an eigenvalue n.  $$N | n \rangle = n | n \rangle$$  For the number operator $N = a^\dagger a $. My question is precisely why does n have to be an integer. I do understand that it must be positive definite due to the norm condition on $ a | n \rangle $:  $$\langle n | a^\dagger a | n \rangle \geq 0$$  I can see that we can show:  $$ a | n \rangle = \sqrt{n-1} | n-1 \rangle $$ and similarly for $a^\dagger | n \rangle$  Sakurai argues that the sequential operation of $a$ operators leads to the form which he does not write, but I will write here for clarity:  $$ a^k | n \rangle = \left(\sqrt{n}\sqrt{n-1}...\sqrt{n-k+1}\right) | n - k + 1 \rangle $$   Since  $$ a|0 \rangle = 0 $$  the sequence must terminate, but can only terminate if n is an integer. How can we make that jump? Is there a formal mathematical proof that the sequence $\left(\sqrt{n}\sqrt{n-1}...\sqrt{n-k+1}\right)$ will terminate if and only if n is an integer? Why are heatmaps on qgis 2.8.1 produced in the wrong CRS? I noticed that the heatmap calculation on the new upgrade version of qgis is given the raster map without any coordinate system.   Have you got the same problem? Are all electrons identical? Why should two sub-atomic (or elementary particle) - say electrons need to have identical static properties - identical mass, identical charge? Why can't they differ between each other by a very slight degree? Is there a theory which proves that?  Imagine an alien of size of order of Milky-way galaxy examining our solar system with a probe of size of 10's of solar system dimension and concludes that all planets are identical. What is the trajectory of a photon moving through a vacuum? Since electromagnetic energy is carried by photons and moves in forms of waves, does it mean that a single photon when propagating through space doesn't follow the straight path but instead always moves up and down, up and down like a wave. If so another question arises the speed of propagation of light in vacuum is fixed meaning that it will always take the same amount of time for it to travel from point A to point B, but if a photon always moves up and down it will also mean that it travels longer distance than the distance between A and B and so it ill travel faster than light propagates, is it even possible, could you please clarify these concepts for me? Making a (simple) directory tree I'm trying to make a simple directory tree representation in LaTeX. Something like:  spam  |  +-- ham  +-- eggs      |      +-- more spam      +-- dead parrots   The tree won't be that long (say, 10--15 entries or so), with three or so levels, and is a one-off thing. So, quick-and-dirty solutions are welcome. When recommending tikz, please note that I have almost no experience with it, so I'd appreciate a clean simple solution, even if it doesn't look quite perfect.   On Tex.SE tikz is almost like the jquery recurring joke on StackOverflow..    Edit: As per Thorsten's suggestion,  is a nice and easy-to-use solution. The example from above could be written as:  \usepackage{dirtree} ... \dirtree{% .1 spam. .2 ham. .2 eggs. .3 more spam. .3 dead parrots. } Windows update doesn't progress beyond 0 KB 0% For a presentation with Beamer, I would like to have a slide where a long list of formulas (maybe mixed with text) scrolls automatically within a certain part of the slide. This should look like e.g. closing credits in movies.   I have tried the idea from  based on animateinline from the , but whenever I use environments like \begin{equation} ... \end{equation} or commands like \textcolor{red}{...} I get an error:  TeX capacity exceeded, sorry [main memory size=3000000]. \end{frame}   Ideally, the formulas/the text should be given within the document itself, but if a solution with an external PDF works fine, I'll take it!  Any ideas how that might work? How is Google Scholar compared to other academic search engines and catalogues such as ScienceDirect, Scopus, and universities' own online catalogues--in terms of the following?   comprehensiveness of search (i.e. can fetch all the ever published relevant titles, whether the user have access to full-texts or not) accuracy and precision of results advanced search features usability  Are there good reasons for (or against) using Google Scholar as the only literature search tool for PhD research? Does the answer of (2) depend on the research discipline?