prove a number is irrational
Can an irrational number raised to an irrational power be rational?
Two-finger swipe 'breaks' heads-up notifications on Lollipop? I have a Nexus 7 (2013) and a Moto X (2014) running Lollipop. On both of these, if I get a heads-up notification (i.e., those that show up on top of the screen when the device is unlocked) and swipe down on it with 2 fingers, it 'breaks' further heads-up notifications:   I can't tap on them to open the responsible app For, e.g., an incoming phone call with 'answer' and 'dismiss' buttons, tapping the buttons does nothing. I can't answer phone calls anymore!   I can still swipe to the side to dismiss the notifications, and they still disappear on their own after a few seconds. I haven't been able to find a way to get the correct behavior back aside from a restart.  Yeah, the obvious workaround is to not two-finger swipe on them, but it's second nature as that same gesture expands notifications in other contexts.  I'm guessing the problem is generic, but I've triggered it on Hangouts and Phone notifications.  Do others experience the same thing? Any known 'fixes' aside from restarting? Can every non-simple group $G$ be written as a semidirect product? When people want to classify a group with certain (small) order, they seem to find a normal subgroup $H$ and a subgroup $K$, and then consider how many distinct $H \rtimes_\varphi K$ are there.  My question is: For any group $G$, does it always exist normal subgroup $H$, subgroup $K$ and $\phi : K \rightarrow Aut(H)$ s.t. $G\cong H \rtimes_\varphi K $?  Sorry if this question is too basic...  Thanks the comment and the answers. Then if the $G$ is not simple, can we have can $G\cong H\rtimes_\varphi K$ for proper subgroup $H$, $K$, with $H$ normal?  EDIT:  If we have a group $G$, that is not simple (then $G$ has at least one normal subgroup), are there always a non-trivial normal subgroup $H$ and a proper subgroup $K$ s.t. $G \cong H \rtimes K$. (Note that if $G$ is simple this is clearly not possible).  But the quaternion group and $Z_4$ given in answers have shown that we cannot always have it. To my amazement I just discovered that the C99 stdint.h is missing from MS Visual Studio 2003 upwards.  I'm sure they have their reasons, but does anyone know where I can download a copy?  Without this header I have no definitions for useful types such as uint32_t, etc. I just bought an Xbox 360 game with wrong region. What should I do? I just bought a  game (Forza 4) for my Xbox 360 (bought in the US). When I tried to play the game, it wouldn't load because of region issue. Is there a legal way to solve this (no crack)? Did Voldemort refer to Snape as one of the missing Death Eaters at his rebirth? In Goblet of Fire, Voldemort says of the missing six Death Eaters:     “And here we have six missing Death Eaters… three dead in my service. One,   too cowardly to return… he will pay. One, who I believe has left me forever… he   will be killed, of course… and one, who remains my most faithful servant, and   who has already reentered my service."   The answer to  speculates about who the three dead Death Eaters were.  We know who the last one is. I’m guessing one of the remaining two is Karkaroff. Is the other Snape? If so, which one? Is there a way to include an entire text file as a string in a C program at compile-time?  something like:   file.txt:  This is a little text file  main.c:  #include &lt;stdio.h&gt; int main(void) {    #blackmagicinclude("file.txt", content)    /*    equiv: char[] content = "This is\na little\ntext file";    */    printf("%s", content); }    obtaining a little program that prints on stdout "This is a little text file"  At the moment I used an hackish python script, but it's butt-ugly and limited to only one variable name, can you tell me another way to do it? Compute the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$ What would be the sum of the following series?  $$\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$$  Thanks I'm hoping someone can help! I recently downloaded a torrent directly onto an external hard drive, (using a 2009 MacBook Pro and 1TB Seagate Expansion portable Drive) somehow during the download the drive was accidentally ejected, since this incident when I plug the drive back in it’s having problems reading.  The drive works fine for about 10 minutes, I can read/copy/paste and access files but then starts to wreak havoc; apps hangs, it hangs the Finder, etc.  I’ve tried repairing it in Disk Utility, which reads and repairs this disk but problems inevitably continue as described above.  I’ve deleted the Torrent file also. All my files on this external drive so re-formatting is not an option until I can retrieve them. Should I buy a Canon 55-250mm lens or 70-300mm for my Rebel T3i? I have the basic 18-55mm lens that came with my camera but I need a next step up that's not too expensive.   What are the most common types of photos shot with either lenses? Or what is each lens best for? $\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \mathrm dx$ Evaluate Integral Evaluate $$\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \ dx$$ How to fairly select a random number for a game without trusting a third party? Where in Europe can I purchase a car without residency? I am not a resident anywhere and plan to travel for the foreseeable future, mostly in Europe. My country has good visa-free access to most of Europe, so it is conceivable I could remain without any residency for quite some time.  I enjoy driving, and maintaining cars, and I would like to purchase a car somewhere in Europe.  Where, if anywhere, in Europe (or outside of Europe, but with the ability to drive the car in Europe) can I legally purchase and insure a car as a non-resident? I am open to creative solutions like setting up a company to own the car, if the capital and ongoing paperwork requirements for the company are not too onerous.    I am aware that I could save money, get places quicker, etc, by taking trains or flying. I have a passion for cars and driving. Suggesting that I not buy a car is out of scope of this question. Why use swap when there is more than enough free space in RAM? Using swap space instead of RAM can drastically slow down a PC.  So why, when I have more than enough RAM available, does my Linux system (Arch) use the swap?  Checkout my conky output below:    Also, could this be the cause of speed and system-responsiveness issues I'm having?  Output of free -m :  $ free -m              total       used       free     shared    buffers     cached Mem:          1257       1004        252          0         51        778 -/+ buffers/cache:        174       1082 Swap:          502        144        357 Is it possible to group several objects and then manipulate them (scale, rotate, translate) as one? I want to get a 360 and don't know wether to get the bigger (250GB), newer, more expensive Elite, or get one of the older, smaller (120GB), last gen ones. I know the arcade is going to be too small to save anything useful  How much hard disk space is recommend? What else is stored besides games? Merge two data frames while keeping the original row order Simple past or present perfect? Which is preferable, simple past or present perfect, in situations like this one: I had a conversation with my friend and we planned to do something; shortly afterwards I call him back to announce the accomplishments. My biggest doubt about simple past is that perfect aspect is used to indicate past action (accomplishment) with current relevance, at the same time I'm not sure if I'm allowed to use present perfect, because there's a gap between accomplishing and the current moment, which is a characteristic of simple past. How to exceed left margin with longtable I'm trying to make a longtable that is wider than the textwidth at a certain moment. I've tried different things always ending up with a different problem:  First possibility: If I use the adjustwidth environment the longtabu does not align correctly. The left margin stays fixed and the right margin gets out of the page. This in contrast, if I do the exact same thing to make a tabu (instead of longtabu), it works perfectly. I think using the adjustwidth is the best way to go, but I can't manage to get it working.  Second possibility: I've found a solution using the fullwidth package. That package does the trick on small longtables, but freezes when I'm trying to do it on my big longtable. The command \setlength\LTleft{-2cm} didn't work as well.  The final option I tried was to use geometry. But that changes the margin of the whole page thus making a page break which I'm not interested in. The code I used for this was  \newgeometry{left=1.2cm} \begin{longtable} long table stuff \end{longtable} \restoregeometry   Here's an example of what I want contra what I get  \documentclass[a4paper]{article} \usepackage{longtable} \usepackage{chngpage} \usepackage{lipsum} \usepackage{tabu}  \begin{document} \section{This is the original textwidth.} \lipsum[1]  \section{This is the width I need for my longtabu} \begin{adjustwidth}{-1in}{-1in} \begin{tabu} to \linewidth {*{13}{|X[l]} |} 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ \end{tabu} \end{adjustwidth}  \section{This is what happens} \begin{adjustwidth}{-1in}{-1in} \begin{longtabu} to \linewidth {*{13}{|X[l]} |} 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ \end{longtabu} \end{adjustwidth}  \end{document} The limit is   $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$  where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times:  $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$  For $n=1$ the limit is $\frac{1}{6}$, $n=2$, the limit is $\frac{1}{3}$ and so on.  Can we define a recurrent relation upon that given hypothesis? Also, how do I involve $n$ into calculation, because the final limit will depend on it?  Any suggestions on how to tackle this?  Thank you! "Never say die?" What's the deal with this idiom? I know what it means, but cannot wrap my head around its grammar structure. The sentence structure of     "Never say die"   looks like that of     "Never admit defeat"   but it does not make the same grammatical+semantical sense.  Is it an archaic use? Is it a contraction of a longer phrase? Was it originally misspoken by someone of note and survived memetically? What are its roots?  Edit: Well, the comments made me realize that I worded my question poorly. I'm not saying I think this phrase is ungrammatical. I'm asking how the heck "saying 'die'" came to mean what it means.