Eigen decomposition of Householder operator
Eigen-Decomposition of Householder operator
lstlisting: how to set code color and frame color Plotting graphs using bash difference between "function foo() {}" and "foo() {}" I can define bash functions using or omitting the function keyword. Is there any difference?  #!/bin/bash  function foo() {   echo "foo" }  bar() {   echo "bar" }  foo  bar   Both calls to functions foo and bar succeed and I can't see any difference. So I am wondering if it is just to improve readability, or there is something that I am missing...  BTW in other shells like dash (/bin/sh is symlinked to dash in debian/ubuntu) it fails when using the function keyword. Differences between hash- and encryption algorithms? There is a lot of confusion between hash- and encryption algorithms; I would like to understand:   When do we use a hash algorithm and when do we use an encryption algorithm (a cipher)? How difficult it is to reverse hash- and encryption algorithms? From the theoretical/mathematical perspective, what differentiates a hash- and encryption algorithm? Why doesn't this perpetual motion machine using the buoyant force work? I realize this isn't possible, but I can't see why not, especially if you change the model a little bit so that the balls simply travel through a tube of water on the way up, rather than exactly this model.  Please be clear and detailed.  I've heard explanations like "the balls wouldn't move" but that doesn't do it for me - I really do not see why the balls on the right would not be pulled/pushed up, and the rest of the chain wouldn't continue on. Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? Thanks! My (very simple) question to a friend was how do I prove the following using basic trig principles:  $\tan75^\circ = 2 + \sqrt{3}$  He gave this proof (via a text message!)  $1. \tan75^\circ$   $2. = \tan(60^\circ + (30/2)^\circ)$  $3. = (\tan60^\circ + \tan(30/2)^\circ) / (1 - \tan60^\circ \tan(30/2)^\circ) $  $4. \tan (30/2)^\circ = \dfrac{(1 - \cos30^\circ)}{ \sin30^\circ}$    Can this be explained more succinctly as I'm new to trigonometry and a little lost after (2.) ?    EDIT  Using the answers given I'm almost there:   $\tan75^\circ$    $\tan(45^\circ + 30^\circ)$   $\sin(45^\circ + 30^\circ) / \cos(45^\circ + 30^\circ)$   $(\sin30^\circ.\cos45^\circ + \sin45^\circ.\cos30^\circ) / (\cos30^\circ.\cos45^\circ - \sin45^\circ.\sin30^\circ)$    $\dfrac{(1/2\sqrt{2}) + (3/2\sqrt{2})}{(3/2\sqrt{2}) - (1/2\sqrt{2})}$   $\dfrac{(1 + \sqrt{3})}{(\sqrt{3}) - 1}$   multiply throughout by $(\sqrt{3}) + 1)$   Another alternative approach:   $\tan75^\circ$    $\tan(45^\circ + 30^\circ)$   $\dfrac{\tan45^\circ + \tan30^\circ}{1-\tan45^\circ.\tan30^\circ}$   $\dfrac{1 + 1/\sqrt{3}}{1-1/\sqrt{3}}$   at point 6 in above alternative This song by The Rolling Stones, "(I Can't Get No) Satisfaction". As a non-native speaker, I always wondered what's the exact meaning of this phrase?   Is it "I'm not getting any satisfaction" (this seems to be the most widely met translation; double negation used where single negation should have been used). Or is it "I want my satisfaction, I cannot leave without getting some" (makes much more sense to me).   Or is it something else? I've always had troubles understanding double negation as applied to English. First I was taught it's a mistake to use double negation at all, and then I see it everywhere. I am using Google Chrome. I re-installed it today as it was crashing when I start it. I guess may be that this was due to any of my Extensions or Google Update. I disabled my extensions but it was still crashing. Now I installed an old version of chrome from . However, Chrome automaticallyuppdates itself (although Windows asks me before installing and I don't grant it permissions, its still annoying to get Google update asking me permission after a couple of minutes.   I saw the question at  however, it doesn't seem to hlp me. i disabled the service and I stopped it as well, but the problem still persists. I also tried to look into the registry according to the following link:  but the "Policies" section of my registry doesn't contain any folder called "Google". I use Windows 7 - 64 bit Home Premium Service pack 1.  Could anybody help? Thanks a ton! I tried writing a shell script which can do automatic login into a ssh server using password which is mentioned in the script. I have written the following code:  set timeout 30 /usr/bin/ssh -p 8484 root@172.31.72.103 expect  {    "root@172.31.72.103's password"     {       send "password\r"     } }   This code is not running properly, still it is asking for the password. Can somebody please help me in solving this How do I setup an encrypted swap file? When I installed Ubuntu, I setup an encrypted home partition. Now I have come to an issue where I am running out of memory because I haven't setup swap. How can I setup an encrypted swap file?  A couple of times while booting I've seen some reference to cryptswap. I've also got these lines contained in /etc/fstab  # swap was on /dev/sda6 during installation #UUID=1ba8a53b-1842-4de7-8e38-2d6c79f3a1ab none            swap    sw              0       0 /dev/mapper/cryptswap1 none swap sw 0 0 I've lost my portal home, and I am invincible. How can I leave the nether? I've gone into the Nether and seemed to have lost my portal out of it, I also can't die in lava or by mobs attacking me, I'm appear to be invincible and can't die.  This is Single Player by the way.  Why is this happening? How do I call legacy PHP files? I am unable to call PHP files inside the Drupal directory. My task is maintaining staff's old pages on the new platform. With pages written on clear HTML, there is no problem: I place files in a folder I created (domain.org/sites/default/files/share/staff_folder).  In sites/default/files/.htaccess I added DirectoryIndex share and now the link to the domain.org/sites/default/files/share/staff_folder/index.html can be used. When I import pages written in PHP, e.g. calling domain.org/sites/default/files/share/staff2_folder/index.php, I get a 403 error:     Forbidden      You don't have permission to access /sites/default/files/share/staff2_folder/index.php on this server.   While /sites/default/files/share/staff2_folder/index.html still works. How to create automatic dual battery changeover/switching circuit for uninterruptible power? I've search for a solution to my problem both on this site and on Google but I did not find a complete and adequate response.  I need a circuit that switches two 12v sources (one that comes from a lead-acid battery powered by a solar panel and the other source comes from the output of a 220-12v switching power supply) without the final power output being interrupted during switching (this is very important!).  Notes:  - The final power output will be used to power 12v devices (30 Ampere maximum).  - The sources voltage range is 11-14.7v  How can I switch between two 12V sources? Consider the following example:  \documentclass{article} \newtheorem{Theorem}{Theorem} \begin{document} \begin{Theorem} On the other hand, this one has only one claim. \end{Theorem} \begin{Theorem} \begin{enumerate} \item First claim \item Second claim \end{enumerate} \end{Theorem} \end{document}   Typesetting it results in:    Notice there is extra horizontal space before the first item in the list of the second theorem.     Is there a sensible way to get rid of that?   Ideally, the solution would work with amsthm, thmtools, enumitem.  N.B.: this is related to  but not quite the same. I just reinstalled my Ubuntu system with 11.10, but ran into a problem when I tried to install Gimp 2.7.  I always get unmet dependencies (it's german, so I can't tell you the exact english translation):     Needs gimp-data (&lt;= 2.6.11-z) but 2.7.3-2011082002~oo should be installed   I already read the follow post, but the answers didn't really help me.  When I type in     sudo apt-get install gimp=2.6   it just says that there is no version 2.6 found. I get the same with 2.7.  I really need Gimp. But installing Gimp 2.6 isn't possible as well.  When I remove the repos for 2.7, I still get the same error. Order of an element in a finite cyclic group Let $G$ be a cyclic group of order $m$ generated by an element $a$.  I want to show that the order of $a^k$ is $m/d$, where $d:=\gcd(k,m)$.  I have a simple proof, but want to make sure I haven't overlooked anything since the other proofs I've seen (such as in Dummit and Foote) look more complicated.    The order of $a^k$ is the cardinality of the set $\{a^{ks}: s \in \mathbb{Z}\}$, which equals the cardinality of the set $\{a^{ks} a^{mt}: s,t \in \mathbb{Z} \}$ since $a^m$ is the identity.  But the set $\{ks+mt: s,t \in \mathbb{Z} \}$ is the set $d \mathbb{Z}$ by the basic properties of integers. So we have that the cardinality of the set is $| \langle a^k \rangle| = |\{1, a^d, a^{2d}, \ldots, a^{m-d} \}| = m/d$. Why there is no operator for time in QM? Hidden features of Python How would I connect to a proxy on Ubuntu Studio? So I am running Ubuntu Studio on my laptop and what do I do to connect to a proxy? I search for proxy and in network settings their is no PROXY tab. What are key words in the tabs to look for?