Mathematical induction divisibility
Why is $a^n - b^n$ divisible by $a-b$?
How do I set a static IP address in Windows? A problem of missing energy when charging a second capacitor A capacitor is charged. It is then connected to an identical uncharged capacitor using superconducting wires. Each capacitor has 1/2 the charge as the original, so 1/4 the energy - so we only have 1/2 the energy we started with. What happened? my first thoughts were that the difference in energy is due to heat produced in the wire. It may be heat, or it may be that this is needed to keep equilibrium. Where is the password stored for ssh authentication? It's known that when you're connecting to a wrong server via ssh and send your password  to it, your password comes in a plain text and a server saves it by default. Probably. I wonder, where I can retrieve it? I don't see it in /var/log/auth.log, although I do see all the attempts to authenticate. Functions satisfying $f\left( f(x)^2+f(y) \right)=xf(x)+y$ Solve $f(x)f(2x^2) = f(2x^3+x)$ The full question is to solve $f(x)f(2x^2) = f(2x^3+x)$, prove there are at most one solution per degree of $f(x)$ and then find all the solutions. (Here $f(x)$ is assumed to be polynomial.)  So far I have proven that $(x^2+1)^n$ works for even degree polynomials, and I'm pretty near certain there are no odd solutions.  Is the following a valid argument for there only being at most 1 solution per degree? Let $K$ be the degree of $f(x)$ then $f(x)f(2x^2) = f(2x^3+x)$ produces $3k+1$ linearly independent equations with only $k+1$ independent variables. Therefore for any degree $K$ polynomial there can be at most 1 solution.  Finally, is there any elegant way to prove that no odd solutions exist? What does active mean? How to center text vertically within a textbox in Photoshop It's pretty annoying that you can't center text vertically in Ps automatically, like you can in Indd. It's even more annoying that the bounding box of text can't be 'collapsed' or 'fit to text', like in Indd.  I could convert the text to shape, but then I lose coloring. I could convert to smart object, but then its not vector anymore, for when I export to pdf.  Is there a work around anyone knows of that would make it easier to center text vertically? What is the most elegant way to solve this:   open a file for reading, but only if it is not already opened for writing open a file for writing, but only if it is not already opened for reading or writing   The built-in functions work like this  &gt;&gt;&gt; path = r"c:\scr.txt" &gt;&gt;&gt; file1 = open(path, "w") &gt;&gt;&gt; print file1 &lt;open file 'c:\scr.txt', mode 'w' at 0x019F88D8&gt; &gt;&gt;&gt; file2 = open(path, "w") &gt;&gt;&gt; print file2 &lt;open file 'c:\scr.txt', mode 'w' at 0x02332188&gt; &gt;&gt;&gt; file1.write("111") &gt;&gt;&gt; file2.write("222") &gt;&gt;&gt; file1.close()   scr.txt now contains '111'.  &gt;&gt;&gt; file2.close()   scr.txt was overwritten and now contains '222' (on Windows, Python 2.4).  The solution should work inside the same process (like in the example above) as well as when another process has opened the file. It is preferred, if a crashing program will not keep the lock open. Define path for packages in the LaTeX file (analog of \input@path or \graphicspath for \usepackage) Using TIPA with fontspec How do I restore the default repositories? I have an ec2 instance on AWS which already installed a few Wordpress-es in it.  I want to set things up to let more people handle their own projects, with me also being able to go into the server using terminal if I still want to.  I have been following tutorials such as  and a few others.  I figured all of them just teach users how to sftp into their own home folder, also the steps keep saying addUser but some users already existed and removing them might cause some issue.  How can I give existing users the permission to use SFTP for Wordpress projects?  Let's say all my Wordpress projects are already under /var/www/ which has already been set up with www-data:www-data.  Really new with setting things like this.  Thanks in advance for any help. I would like to create a minecraft server for me and and my boyfriend to play on.  How do I set it up? Do I need any additional software? Do you have some links to walkthroughs or tutorials?  I have tried to set this up a few times but have failed every time. Does the speed of light vary in non-inertial frames? The speed of light is the same in all inertial frames.  Does it change from a non-inertial frame to another? Can it be zero?  If it is not constant in non-inertial frames, is it still bounded from above? Is there anything to do during an 8 hour after-midnight layover in Dubai? We are planning a trip with an 8 hour layover in Dubai spanning from 1am-9am. I am thinking about going to see the city, but is it worth it to do it given the very late night hours? Will at least the outside attractions accessible?   Will Burj Khalifa be visible in the middle of the night?  Will at least some of the malls be open?  Will there be anything to do at night?  I have seen other questions about layovers in Dubai (), but none of them is asking about a night-only stay. Bolzano-Weierstrass Theorem Proof Abbott (Ax. of Choice) In Abbott's Proof of the Bolzano-Weierstrass Theorem, does Abbott use any form of the Axiom of Choice ?   I think he is since he chooses an $a_{n_k} \in I_k$ where there are multiple such $a_{n_k}$. Every once in a while, we get a question asking for a book or other educational reference on a particular topic at a particular level. This is a meta-question that collects all those links together. If you're looking for book recommendations, this is probably the place to start.  All the questions linked below, as well as others which deal with more specialized books, can be found under the tag  (formerly ).  If you have a question to add, please edit it in. However, make sure of a few things first:   The question should be tagged  It should be of the form "What are good books to learn/study [subject] at [level]?" It shouldn't duplicate a topic and level that's already on the list   Related Meta: Best Azure Service for Large no of SQLServer Databases Is there a word for someone who has others do all their work for them? In a way like a slave master. More though like say someone who had robots do all their work and they were never themselves productive and had no drive. If you've read R.U.R. you'll understand the question better. I'm looking for a word with more definition than just lazy.  Update 1: It think I may actually go with drone if anything. I think that definition fits what I'm looking for the best. I also liked the definition of the British term layabout that one person mentioned, but I will probably use drone. Thanks for all the help and sorry if I was confusing or unclear.  Update 2: I needed this word for use in an essay I was writing which I have now handed in. Thanks for the help guys. No more suggestions needed. Converting a string to a tuple in python