I know in a graph with n vertices, there are m = (n(n-1)/2) edges, but in a graph with m edges, how many vertices are there?
I am given this problem: Suppose you have a complete graph with $n$ edges. How many vertices does your graph have?  I have this math figured out so far: We know that a complete graph has $m$ vertices, with $m-1$ edges connected to each. This makes the sum of the total number of degrees $m(m-1)$. Then, since this sum is twice the number of edges, the number of edges is $\frac{m(m-1)}{2}$.  But I don't think that is the answer. I think I am missing a piece of the math, but I can't figure out what else I need to do. Can anyone help me with what I'm missing or doing wrong?
How to find what device a file is on (and use that in a script)? I want to find out what device my file is on so that I can use it in a script. I can get this far:  $ df  . Filesystem   512-blocks      Used Available Capacity  Mounted on /dev/disk0s2  498438976 294369520 203557456    60%    /   but this output feels too clumsy; is there a better way than parsing this to get the first 'word' of the second line?  What I really need is something like this so I can pipe it to the next command:  $ somecommand . /dev/disk0s2   How can I achieve this, preferably without resorting to string hacking the 'df' output? Apparently, my linux ubuntu setup of QGIS doesn't contain grass gis, although when I initially installed it last week I added grass gis to the command line. Anyway, I typed in once again and installed grass gis 7.0, however while GRASS starts, the plugin QGIS for it doesn't show up. Any ideas? Prove that $x-1$ divides $x^n-1$ In algebra &amp; polynomials, how do we prove that $$x-1 \mid x^n -1?$$ For one of the clients, I have spin up a modern communication site collection and on the landing page of the same, I am seeing 'Get the mobile app' showing up, I was able to hide the 'feedback' button using css but no luck with Get the mobile app button.  Can someone please share css to hide 'get mobile app' button, appreciate the help. Protect .NET code from reverse engineering? Obfuscation is one way, but it can't protect from breaking the piracy protection security of the application. How do I make sure that the application is not tampered with, and how do I make sure that the registration mechanism can't be reverse engineered?  Also it is possible to convert a C# application to native code, and  is too costly.  C# provides lot of features, and is the ideal language for my code, so writing the whole codebase again in C++ is out of the question.  Secure certificates can be easily removed from the signed assemblies in .NET. Tutorial on how to back up my full Ubuntu desktop and files? I would like to be able to back up all my settings and files to a dedicated external hard drive in a way that makes (1) all the backup files accessible and (2) would allow me to restore my system is there is a crash.   Is there an app for this or any pointers?  Thank so much. We know that if you (using Polyjuice), take the form of someone who had been injured, you assume that injured form (e.g. Barty/Moody). If you are injured, presumably you would be uninjured while under Polyjuice. My question is, is this in real-time? That is, if you take a Polyjuice, and while you're still under its effects, the person being transformed into gets injured (or even dies), will that injury be transferred onto you? And conversely, if you get injured while in someone else's body, will that transfer over to your real body after the Polyjuice wears off, will it disappear, or will it go to the person you turned into? What Android emulators are available? I will like to know some of the android emulators that work really well with Ubuntu 11.10 and above. I want to try 'whatsapp' on Ubuntu through an android emulator. Thanks in advance losing ligatures when switching font series or color in the middle of a word Prove that the Interior of the Boundary is Empty Suppose X is a Metric Space  Let S $\subset X$  Prove that if S is Closed then, the Interior of the Boundary of S is Empty  Totally stuck on how to solve this. How long should I keep my tax documents, and why? "Not a COM32R image" error when trying to install from a USB key Not a COM32R image boot   is a message that the console infinitely keeps spitting out when I try to install Ubuntu 10.04 from a USB. I created the USB using the StartupDiskCreator in 11.04.  What is the reason for this behavior? What do I do to set it right?   I have followed the steps perfectly, as enlisted in the . Ahlfors "Prove the formula of Gauss" He says:     Prove the formula of Gauss:   $$ (2\pi)^\frac{n-1}{2} \Gamma(z) = n^{z - \frac{1}{2}}\Gamma(z/n)\Gamma(\frac{z+1}{n})\cdots\Gamma(\frac{z+n-1}{n}) $$   This is an exercise out of Ahlfors.  By taking the logarithmic derivative, it's easy to show the left &amp; right hand sides are the the same up to a multiplicative constant.  After that I'm lost. It's easy using another identity when $n$ is even to use induction. But when $n$ is odd I am lost.  It's obvious when $n$ is a power of 2. How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$ Nice problem:  Let $a_{0}=1$ and $$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$  Show that  $$\lim_{n\to\infty}\dfrac{a_{n}}{n}=\dfrac{12}{\log{432}},$$  where $\lfloor x \rfloor$ is the largest integer not greater than $x$.  It is said this problem was created by Paul Erdős, and I can't find this problem's solution, does anyone have any nice methods? Thank you. Fix SD card that cannot be formatted Matrices A+B=AB implies A commutes with B For each positive even integer $n$, set   $$P_n = \displaystyle {{n} \choose {\frac{n}{2}}}\frac{1}{2^n}.$$  Show that $\displaystyle \lim_{n \to \infty} P_n$ exists and determine its value.   Here's what I have so far: each $P_n$ can be thought of as the probability of tossing a fair coin $n$ times, and obtaining exactly $\frac{n}{2}$ heads. My expectation is that $P_n \to 1$, since in any large trial, one would expect to record just about as many heads as tails. But I'm not sure how to mathematically justify this hunch.  Hints or solutions are greatly appreciated. Proving $\, a^b = (a\bmod n)^b\, \pmod{\!n}\ $ [Congruence Power Rule] I am working on a problem I am pretty close to solving but I can't figure out the last part. I used some algebraic manipluation to break the problem down. The problem is:     Show that the following congruence is true      a^b = ((a mod n)^b) (mod n)      for any positive integers a, b, and n.      Hint: Number a can be written as a = q*n + r, where q = floor brackets a/n, and r = a mod n.   My attempt it:     (q*n +r)^b = ((a mod n)^b) * (mod n)      (q*n)^b + (a mod n)^b = ((a mod n) ^b) * (mod n)      (a)^b -(r)^b = ((a mod n) ^b) * (mod n)      (a)^b = modn  ----> divided by ((a mod n)^b)   Am I doing this correctly and if so how can I prove this is true? Is there a trick I am missing? I think that by simplifying it further I would just keep going in a loop.  Thanks! Possible Duplicate:             According to , Win 7 only supports 4 GB of RAM. I, however have been reading some forums and some say they are using 8 GB RAM with good performance. I am wondering if I can use 8 GB before I purchase. If A is a finite set and B is an infinite set, then B\A is an infinite set Prove: If $A$ is a finite set and $B$ is an infinite set, then $B \setminus A$ is an infinite set.   I've tried proving the contrapositive, that if $B \setminus A$ is a finite set then $A$ is infinite or $B$ is finite, but I don't think this is the way to go.