Given a subgroup $H $, find a normal subgroup of $G$ with index less than or equal to $n!$
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
Low wifi signal in 14.04 LTS Is there something like Linux console in Mac OS X? Like "terminal", but without running graphics environment. Unable to install VLC This is what happens:  sudo apt-get install vlc Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package vlc   Please help. I would also like to know when this error occurs, I've seen it before too. How can I test two static methods (one invokes the other) with PowerMockito? What is strain? Why does DNA coil upon itself? I have been reading Lehninger's Principles of Biochemistry to understand DNA supercoiling (pg-). Supercoiling is the twisting of a DNA upon itself and this happens when the two helically wound strands of DNA are either underwound or overwound. The book says that this act of underwinding or overwinding generates a structural strain  which is then accommodated by coiling of the DNA upon itself (supercoiling). Here the word strain has been used in the way stress of mechanical physics would be used. I wonder if it means stress. Besides I would like to know how underwinding or overwinding cause the DNA to coil upon itself. Excerpts from the book:  DNA supercoiling is generally a manifestation of structural strain. Most DNA are negatively supercoiled (which means that the supercoiling is left handed). In almost every instance, the strain is a result of underwinding of DNA double helix. Supercoiling is not a random process; the path of the supercoiling is largely prescribed by the torsional strain imparted to the DNA....  Torsional strain, again is it talking about ? P.S. The description of the question has been changed recently. Font corruption: lines through characters I have a problem where a one-pixel high white line is displayed through certain letters of text. Almost always only a single character type is affected at once, and only in one font size (though that size can change over time). Most of the time things are fine, but this happens several times a day. It happens frequently in Firefox, but system dialogs, menus, etc. can also be affected. It's been happening ever since I upgraded to 10.10. Only restarting seems to fix it.  Two examples (in the first, only the larger "r" is corrupted):     Any help would be appreciated, thanks.  Edit: I've since noticed that I can (sometimes) make the fonts appear correctly by zooming in/out all the way and returning to the default in Firefox (one or two steps, then going back, does nothing) seems to make the problem go away temporarily.  Edit 2: I figure it's worth noting that the solution below doesn't fully work for me any longer (I'm now using Ubuntu 11.10). It does significantly reduce the frequency of this problem occurring.  Edit 3: Still around in 12.04. The fix provided by Vanni works perfectly now for me, but apparently doesn't for some.  Edit 4: This seems to be an xorg problem, with Vanni's answer fixing it for many Intel video cards. I've had this occasionally happen with AMD graphics cards in 13.10 and 14.04, but very rarely (only two or three times). Unexpected results when mounting two drives on a single mount point I was running out of /tmp space on one of my servers today so I attached a new 100GB disk to the system and did the following without unmounting /usr/tmpDSK from /tmp:  mkfs.ext4 /dev/xvde mount /dev/xvde /tmp   and now df -h outputs as:  Filesystem      Size  Used Avail Use% Mounted on /usr/tmpDSK      98G  4.5G   89G   5% /tmp /dev/xvde        98G  4.5G   89G   5% /tmp    The size of /usr/tmpDSK shows :  [root@server2 usr]# ll -h /usr/tmpDSK -rw------- 1 root root 2.3G Jan 27 17:32 /usr/tmpDSK    Different forums suggested using mhddfs to mount multiple drives on a single mount point, but I just did it directly and it seems to have worked -- or am I missing something?  I am hoping to better understand what actually happened. Sometimes I wonder if he is actually/really  as lavish as he   pretends to be.   I know that this sentence can work without either actually or really, but if we were to use one of them to make an impact on the listener, which one should it be? Find all primes $p$ such that  $ p^3-4p+9 $ is a perfect square.   I tried a few different values for $p$, namely $2,3,5,7,$ and $11$. The prime $p =2,7,11$ all worked but $p =13$ didn't so it makes me wonder. How can I find all primes such that it is a perfect square?  EDIT: it turns out this is problem P25 in post number #63 at Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a&gt;1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition.   proof:  given $a_n=a^{\frac{1}{n}}$ for $a\in\mathbb{R},a&gt;1$.  for $n\in\mathbb{N}^*,n+1&gt;n\Rightarrow \frac{1}{n+1}&lt;\frac{1}{n}$ and because $a&gt;1$ we gets $a^{\frac{1}{n+1}}&lt;a^{\frac{1}{n}}$, since $\frac{1}{n+1}&gt;0$, then $1=a^{0}&lt;a^{\frac{1}{n+1}}&lt;a^{\frac{1}{n}}\Rightarrow 1&lt;a_{n+1}&lt;a_{n}$.  then we proof that $\lim\limits_{n\to+\infty}a_n=1$ we need to show that $\forall\epsilon&gt;0,\exists N,\forall n&gt;N,|a_n-1|&lt;\epsilon$  then for $\epsilon&gt;0$, choose $N=\frac{1}{\log_a(\epsilon+1)}$, since $a&gt;1$ we gets that $$\forall n&gt;N,1&lt;a^{\frac{1}{n}}&lt;a^\frac{1}{N}\Rightarrow0&lt;a^{\frac{1}{n}}-1&lt;a^{\frac{1}{N}}-1\Rightarrow |a^{\frac{1}{n}}-1|&lt;|a^{\frac{1}{N}}-1|=|a^{\log_a(\epsilon+1)}-1|=|\epsilon+1-1|=\epsilon\Rightarrow |a^{\frac{1}{n}}-1|&lt;\epsilon$$  wich implies that $\lim\limits_{n\to+\infty}a_n=1\square$  my proof to the limit is correct? Should inner tube have excess length? I just re-watched the scene of Voldemort's death in the Deathly Hallows Part 2 movie. Voldemort and Harry's wands were locked with each other and then Neville killed Nagini. Suddenly, the green and red lights from Harry and Voldemort's spells disappeared. Both Voldemort and Harry noticed that the last Horcrux was no more.  Seeing that, why didn't Voldemort simply escape? He could have later returned with a new set of Horcruxes. Google map only loading half of itself Converting string into datetime I've got a huge list of date-times like this as strings:  Jun 1 2005  1:33PM Aug 28 1999 12:00AM   I'm going to be shoving these back into proper datetime fields in a database so I need to magic them into real datetime objects.   This is going through Django's ORM so I can't use SQL to do the conversion on insert. Solving "mv: Argument list too long"? I have a folder with more than a million files that needs sorting, but I can't really do anything because mv outputs this message all the time  -bash: /bin/mv: Argument list too long   I'm using this command to move extension-less files:  mv -- !(*.jpg|*.png|*.bmp) targetdir/ How do I repair a corrupted Firefox places.sqlite database? I had some problems with my RAM (bluescreen several times, Windows XP) and now are my Firefox databases damaged. Firefox is working, but my history is gone and it's reporting several inconsistencies and errors when executing pragma integrity_check on places.sqlite:     database disk image is malformed   Now the question, how do I repair SQLite-Databases? How to integrate $\int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx$ where $a,b > 0$. Movie featuring  space crew, murders, a forbidden door and ending with a revelation I searched wide and far and couldn't come up with a title for a movie I watched somewhere around the mid-90ies up to early 00's as a rerun in late-night television (in Germany, specifically on Vox; but I'm sure it was an American movie to begin with).  Basically, what I remember from the story revolves around a crew in a space station or maybe space ship of some sort. Some crew members murder other crew members and there is some kind of door they are not supposed/unable to go through. Late in the movie they do go through the door and I'm pretty sure one of the first views is a skeleton of a dead security guard. It turns out the whole thing was some kind of simulation gone wrong and they find themselves to be in a (nuclear?) wasteland. And that's pretty much the revelation/wrap-up of the movie, as I recall it.  I would be delighted if somebody had any idea what movie this could be :) Thanks!  EDIT: The whole movie was in color and I think it was feature-length, not just 45 mins. I have HTML code like this :  &lt;div&gt;        &lt;a&gt;Link A1&lt;/a&gt;        &lt;a&gt;Link A2&lt;/a&gt;        &lt;a&gt;Link A3&lt;/a&gt; &lt;/div&gt;  &lt;div&gt;        &lt;a&gt;Link B1&lt;/a&gt;        &lt;a&gt;Link B2&lt;/a&gt;        &lt;a&gt;Link B3&lt;/a&gt; &lt;/div&gt;   When user clicks a link from above HTML, I want to get the jQuery object of the corresponding &lt;a&gt; element, and then manipulate its sibling. I can't think of any way other than creating an ID for each &lt;a&gt; element, and passing that ID to an onclick event handler. I really don't want to use IDs.   Any suggestions? $f(z)=\frac1{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}d\zeta$ if $\lim_{z\to\infty}f(z)=0$ and $f$ is analytic in $|z|\ge1$ Let $f$ be analytic in $|z|\ge1$ and  $\lim_{z\to\infty}f(z)=0$. Then is $f(z)=\frac1{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}d\zeta$,$\forall |z|\ge2$, where $\Gamma$ is the circle of radius $2$ with centre at the origin.   I think yes, by the Cauchy integral formula. But, how do we ensure that the non analyticity of $f$ in $|z|&lt;1$ dosent affect the formula? Thanks beforehand.