According the  $3D$  $\in$ $SO(3)$ corresponding to a rotation by an angle $\theta$ about a fixed axis specified by the unit vector $\hat{\omega}=(\omega_x,\omega_y,\omega_z) \in R^3$ defined as follows  \begin{bmatrix} \cos \theta +\omega_x^2 \left(1-\cos \theta\right) &amp; \omega_x \omega_y \left(1-\cos \theta\right) - \omega_z \sin \theta &amp; \omega_x \omega_z \left(1-\cos \theta\right) + \omega_y \sin \theta \\ \omega_y \omega_x \left(1-\cos \theta\right) + \omega_z \sin \theta &amp; \cos \theta + \omega_y^2\left(1-\cos \theta\right) &amp; \omega_y \omega_z \left(1-\cos \theta\right) - \omega_x \sin \theta \\ \omega_z \omega_x \left(1-\cos \theta\right) - \omega_y \sin \theta &amp; \omega_z \omega_y \left(1-\cos \theta\right) + \omega_x \sin \theta &amp; \cos \theta + \omega_z^2\left(1-\cos \theta\right)  \end{bmatrix}  does there exist $n$-dimensional analogue of this formula?  How to find $n$ dimensional rotation matrix which corresponds rotation of an angle $\theta$ around the $(n-2)$-dimensional subspace?
There is a 2d rotation matrix around point $(0, 0)$ with angle $\theta$.  $$ \left[ \begin{array}{ccc} \cos(\theta) &amp; -\sin(\theta) \\ \sin(\theta) &amp; \cos(\theta) \end{array} \right] $$  Next, there is a 3d rotation matrix around point $(0, 0, 0)$ and unit axis $(u_x, u_y, u_z)$ with angle $\theta$ ().  \begin{bmatrix} \cos \theta +u_x^2 \left(1-\cos \theta\right) &amp; u_x u_y \left(1-\cos \theta\right) - u_z \sin \theta &amp; u_x u_z \left(1-\cos \theta\right) + u_y \sin \theta \\ u_y u_x \left(1-\cos \theta\right) + u_z \sin \theta &amp; \cos \theta + u_y^2\left(1-\cos \theta\right) &amp; u_y u_z \left(1-\cos \theta\right) - u_x \sin \theta \\ u_z u_x \left(1-\cos \theta\right) - u_y \sin \theta &amp; u_z u_y \left(1-\cos \theta\right) + u_x \sin \theta &amp; \cos \theta + u_z^2\left(1-\cos \theta\right)  \end{bmatrix}  How it is possible to generalize rotation matrix on $N$ dimension around zero point and $N-2$ dimensional unit axis with angle $\theta$?
Duplicates of duplicates should be remapped I noticed today another example of duplicates of duplicates: that  (A) was closed as a duplicate of  (B), but one day later (B) was closed as duplicate of  (C). Sometimes I need to click many times from one question to another until I get to the master question.  I think that it's better if the question (A) is closed as duplicate of (C), and even if (B) has already (helpful) answers, (B) and (C) could be merged.  Is it possible to remap all duplicates to the master question, just like the tag synonyms? Is the use of "one of the" correct in the following context? Solve the functional equation $f(xy) = f(x)f(y) - f(x + y) + 1$ and $f(1) = 2$ I'm working on this problem already, the domain and co-domain are the set of Rational Numbers. we see that setting $x = 1$ and $y = n$, we obtain $f(n + 1) = f(n) + 1$  But I'm trying to prove that $f(n) = n + 1$ for all $n$, the case when $n = 1, 2$ is obviously true, how do I approach this via induction? Is (rest) mass conserved in special relativity? I don't understand why it is said that the (rest) mass of a system is not conserved in relativity. I mean, the momentum of a system is conserved (i.e.: it remains constant in a frame of reference without any external influence). Also the energy of a system is conserved in relativity: it doesn't change without any external agency  and the (rest) mass of a system is just  $$m^2=E^2-p^2$$  $E$ is constant, $p$ is constant, how can $m$ change?  For example, entropy is not conserved in a system, that means that the entropy of the system will increase spontaneously with time, which is really the case. But, is this true for mass?   here is the problem from Griffiths and exact solution as given in the Griffiths,  Two lumps of clay, each of rest mass $m$, collide head-on at $3c/5$ and they stick together. Question: what is the rest mass $M$ of the composite lump?  Solution: In this case conservation of momentum is trivial it is zero before and zero after. The energy of each lump prior to the collision is $$mc^2/\sqrt{1-v^2/c^2} =5mc^2/4.$$ The energy of the composite lump after the collision is $Mc^2$ (since its at rest). so the conservation of energy says: $5mc^2/4 + 5mc^2/4 =Mc^2$ and hence $M=5m/2$.  Notice that this is greater than the sum of the initial masses! Mass was not conserved in this collision; kinetic energy was converted into rest energy, so the rest mass increased. Get OS-level system information I'm currently building a Java app that could end up being run on many different platforms, but primarily variants of Solaris, Linux and Windows.  Has anyone been able to successfully extract information such as the current disk space used, CPU utilisation and memory used in the underlying OS? What about just what the Java app itself is consuming?  Preferrably I'd like to get this information without using JNI. Efficient way to check collision with potentially hundreds of objects I am trying to make a top down shooter that could potentially have hundreds of objects at once at one point. I need to check collision with projectiles, walls, and enemies. I am currently using ArrayLists which probably won't suffice later on and was wondering if someone had a good method or knows a good method for checking all of those objects at once. This is going to be a game for smartphones and is built in java. Any ideas would be great or if you could point me in some direction that would be fantastic.  Thanks in advance! I have seen a "Universal Laptop Adapter" sold in a computer shop before. I am thinking of getting one for a laptop. Is it safe to use? What differences are there between one of these and a typical laptop adapter? When can I make opportunity attacks? Suddenly my Ubuntu stopped playing video. When I open a video, the player suggests that I install  gstreamer1.0-libav. However, it fails to install.  I tried installing it manually, here is the error message:  $ sudo apt-get install gstreamer1.0-libav Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  gstreamer1.0-libav : Depends: libavcodec54 (&gt;= 6:9.1-1) but it is not going to be installed or                                libavcodec-extra-54 (&gt;= 6:9.13) but it is not going to be installed                       Depends: libavformat54 (&gt;= 6:9.1-1) but it is not going to be installed E: Unable to correct problems, you have held broken packages. underline omitting the descenders I'm very well aware that it is considered best practice not to use underlining other than on typewriters and in handwriting. But sometimes it is needed nonetheless.  In my opinion the typographically least offending way to underline in print is to leave out the descenders. This is also how I learned to do it in handwriting. Very few fonts offer a pre-made underlined variant, but some text-processors (e.g. OS X text-engine) do -- depending on the font used -- an acceptable job at emulating this effect:    How would I tweak any of the existing ways/packages for underlining to do the same?  As a bonus, I'd like to be able to tweak the behaviour (extent of the omitted part, apply to which letters) per font and text style, preferably working with XeLaTeX. Why is "letter" not plural in "two letter words"? Why is "letter" not plural in "two letter words"?  For me it's very strange as the equivalent in French would be plural but my English friend finds it totally normal. I have a Linux CentOS server, the OS+packages used around 5GB. Then, I transferred 97GB data from a Windows server to two folders on this Linux server, after calculated the disk usage, I see the total size of the two folders is larger than the disk used size.  Run du -sh on each folder, one use 50GB, the other one use 47GB  But run df -h, the used space is 96GB. (50GB + 47GB + 5GB) > 96GB  Is there any problem? Those two folders contain lots of files (1 million+). Thanks. I'm setting up an Nginx web server and and want to perform load balancing using the round robin method.  This is what I have found thus far:  upstream backend  {    server backend1.example.com;    server backend2.example.com;    server backend3.example.com; }   According to the page they says i need to add the above lines to my nginx.conf file and upload it to server to load balance.  I do not understand what it means by server backend3.example.com;.  How do I setup a backend instance? Caps-lock remapping stops working How to read the following Itinerary What do the abbreviations mean in the following itinerary:     2  EK 641 L 29NOV 2 KBLDXB HK1       I  1630 1905   *1A/E*                      3  EK 344 L 29NOV 2 DXBKUL HK1       3  2140 0835+1 *1A/E*                      4  MH 704 V 30NOV 3 KULMNL HK1       M  1245 1630   *1A/E*                      5  UA 184 Y 01DEC 4 MNLGUM HK1       1  2255 0440+1 *1A/E*                      6  UA 155 Y 02DEC 5 GUMTKK HK1          0820 1008   especially, the numbers (2,2,3,4,5) before KBLDXB, KULMNL.... The letters and numbers before time (I, 3, M, 1) *1A/E*? With powershell get exactly the same application list as in Add/Remove programms There are various methods to find installed applications on the system but to my knowledge none returns exactly the same list as control panel item Add Remove Programs. 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Captchas not displaying in SO for me When I get a captcha on Stack Overflow, I don't see the image or the input box with Firefox. I originally had the "HTTPS Anywhere" extension installed but still have the same issue when it's disabled (This caused issues previously when Google's SSL certs were expired for an old domain IIRC).  The requests to Google seem to be never-ending; browsing to the URLs that are still loading results in some pages of JavaScript, but no errors as you'd hope.  The screenshot was taken after the page had been open for more than 3 minutes, and I'm not sure why the GETs are still running. The request popup visible in Firebug is for "GET challenge?k=...."  When browsing to the same captcha page in IE, there are no issues.  I'm aware this could be an issue with my Firefox configuration, as opposed to the site, but would appreciate it if someone could verify if it's just me, and suggest a fix.  EDIT   While visiting the Recaptcha site, I can see a Captcha with no problems If I right-click the problem URLs in FF and Open in new tab, the pages load fine. From the fact that FB omits IP Addresses for the problem ones, I assume that it's not even opening a connection (nor perhaps performing a DNS lookup) Other sites that use Recaptcha seem to work fine (I picked 3 at random from Google) On the importance of the i.i.d. assumption in statistical learning Why is jumping into water from high altitude fatal?