I have been working on this for a few days. I can show the conditions that $SO_2$ is a group, but I cannot show the normal subgroup condition.
Show that SO(n) is a normal subgroup of O(n).  A normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. SO(n) is the set of orthogonal matrices of determinant 1. O(n) is the set of real matrices whose inverses equal their transposes (orthogonal matrices). I'm simply bad at writing proofs. Please help?
How do I install python imaging library (PIL)? How do I install the python imaging library (PIL) on Ubuntu?  I tried doing  sudo apt-get install python-imaging   but now when i run     python selftest.py (a script I got from somewhere on the  website) I get (among other warning messages):  ... *** JPEG support not installed *** ZLIB (PNG/ZIP) support not installed ... *** 1 tests of 57 failed.   Have I somehow messed up the PIL -- how do I fix that?  Is maybe the PIL just fine, but that "selftest.py" is not really the right program for checking to see if PIL is installed properly -- how else can I tell if PIL is installed properly or not?  (I'm using Ubuntu 12.04 LTS "Precise Pangolin").  (What I'm ultimately trying to do is to add a 2D barcode generator to some python code, and all the 2D barcode generators I could find that were written in python all seemed to use the PIL). In first.bat, I use  var5=C:\Program Files (x86)\Microsoft Visual Studio 10.0\VC\bin\x86_amd64\ CALL scripts\vc64.bat %var5%   And in scripts\vc64.bat, I use  SET var6=%1vcvarsx86_amd64.bat CALL %var6%   But I get : 'C:\Programvcvarsx86_amd64.bat' is not internal or external command..... error.  If the path assigned to var5 has no space, then it is fine.  I tried several combination of quotations and %1vcvarsx86_amd64.bat, but no change.  How do I make it works with path with spaces?    To be more precise, suppose it is C:\a b c\  In first.bat:       SET var6=C:\a b c\  =====&gt;  '"C:\avcvarsx86_amd64.bat"' is not.....`      SET var6="C:\a b c\" ====&gt;   There should not be a b      SET var6=C:\a\ b\ c\ ====&gt;    Can't find the specified path      SET var6=C:\a b c\ + Using "%var6" ====&gt; There should not be a b     Update: Here is an example. Change a b to ab works.  call.bat in C:\  @ECHO OFF  SET var5=C:\a b\ CALL C:\1.bat "%var5%"  pause   1.bat in C:\  @ECHO OFF  SET var6=%~1Test.bat CALL %var6%  RMDIR /S C:\NoWorry   Test.bat in C:\a b Test.bat in C:\ab  @ECHO OFF  RMDIR /S C:\ThereIsNoSuchFolder Hibernate @Enumerated mapping How do I download torrents? What do spacelike, timelike and lightlike spacetime interval really mean? Suppose we have two events $(x_1,y_1,z_1,t_1)$ and $(x_2,y_2,z_2,t_2)$. Then we can define $$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2,$$ which is called the spacetime interval. The first event occurs at the point with coordinates $(x_1,y_1,z_1)$ and the second at the point with coordinates $(x_2,y_2,z_2)$ which implies that the quantity $$r^2 = \Delta x^2+\Delta y^2+\Delta z^2$$ is the square of the separation between the points where the events occur. In that case the spacetime interval becomes $\Delta s^2 = r^2 - c^2\Delta t^2$. The first event occurs at time $t_1$ and the second at time $t_2$ so that $c\Delta t$ is the distance light travels on that interval of time. In that case, $\Delta s^2$ seems to be comparing the distance light travels between the occurrence of the events with their spatial separation. We now have the following definitions:  If $\Delta s^2 &lt;0$, then $r^2 &lt; c^2\Delta t^2$ and the spatial separation is less than the distance light travels and the interval is called timelike.  If $\Delta s^2 = 0$, then $r^2 = c^2\Delta t^2$ and the spatial separation is equal to the distance light travels and the interval is called lightlike.  If $\Delta s^2 &gt;0$, then $r^2 &gt; c^2\Delta t^2$ and the spatial separation is greater than the distance light travels and the interval is called spacelike.   These are just mathematical definitions. What, however, is the physical intuition behind them? What does an interval being timelike, lightlike or spacelike mean? What is the simplest SQL Query to find the second largest value? For years I've been trying to find this movie I saw on TV once.  I only saw the end.  It featured a teenage boy (about 18 years old) and his family hiding out in a cabin.  The kid is making molotov cocktails and preparing for something.  Sure enough two bad guys show up and I think one of them had a golden pistol.  They start fighting and one of the guys turns into a werewolf type creature.  The kid wounds it and tracks it through the woods to finish it off with his lever action rifle.  Anyone know the name of this movie?  I saw this probably mid to late 90s.  I'm guessing it was an 80's American film. It was live action. Wrong Java resource bundle loaded $f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$. Assume that $f_n → f$ uniformly on $S$ and each $f_n$ is continuous on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.  I'm stuck in thinking about it and I don't manage to prove it neither to find a counterexample. Thanks for your help. App Store asks for old Apple ID when installing updates I got a MacBook that a colleague of mine had used previously. It runs OS X 10.10.2. I created a new administrator account for myself and turned off the "admin" flag on the old account.  I also entered my iCloud / Apple ID account, so that it syncs my calendars and contacts correctly, and even in the App Store, I am signed in with my correct account.  But when I try to install pending software updates (that is, OS X system updates!), the window that pops up has the the e-mail address of my colleague filled in and greyed out—and I can't change it.     Note that I'm signed in with a different (correct) account in the App Store. When I go to Featured and click Account on the right, it has my correct e-mail address filled in:    How do I make the App Store "forget" the old Apple ID?  I  where it just says "it's not possible", and that apps are glued to the original account the Mac was set up with, but that doesn't make a lot of sense, especially in a company where laptops may be handed over from one person to another. What is the right ffmpeg output format for a v4l2-loopback device? Following the guide in  I did:  $ sudo modprobe v4l2loopback $ ffmpeg -f x11grab -r 15 -s 1280x720 -i :0.0+0,0 -vcodec rawvideo -pix_fmt yuv420p -threads 0 -f v4l2 /dev/video0   The device /dev/video0 shows up as expected but the ffmpeg command fails with following output:  $ ffmpeg -f x11grab -r 15 -s 1280x720 -i :0.0+0,0 -vcodec rawvideo -pix_fmt yuv420p -threads 0 -f v4l2 /dev/video0 ffmpeg version 4.0 Copyright (c) 2000-2018 the FFmpeg developers ... [x11grab @ 0x559767794cc0] Stream #0: not enough frames to estimate rate; consider increasing probesize .... [v4l2 @ 0x55976779e6c0] ioctl(VIDIOC_G_FMT): Invalid argument Could not write header for output file #0 (incorrect codec parameters ?): Invalid argument Error initializing output stream 0:0 --  Conversion failed!   I already tried to follow the suggestions below the answer to no avail. I noticed that v4l2 is listed as demux only in ffmpeg -formats. Does that mean I can't use it for output to /dev/video0? If so, what should I use instead? I tried h264 but that crashes after ~1000 frames with following error message:  $ ffmpeg -f x11grab -r 15 -s 1280x720 -i :0.0+0,0 -vcodec rawvideo -pix_fmt yuv420p -threads 0 -f h264 /dev/video0 ffmpeg version 4.0 Copyright (c) 2000-2018 the FFmpeg developers ... [x11grab @ 0x55da61e29cc0] Stream #0: not enough frames to estimate rate; consider increasing probesize ... File '/dev/video0' already exists. Overwrite ? [y/N] y .... Error writing trailer of /dev/video0: Invalid argument .... Conversion failed! I'm working with jQuery and looking to see if there is an easy way to determine if the element has a specific CSS class associated with it.  I have the id of the element, and the CSS class that I'm looking for.  I just need to be able to, in an if statement, do a comparison based on the existence of that class on the element. Installing PHP 5.6 on Xenial (16.04) I need to use php5 (5.6) because I'm working on a Drupal 7 project which doesn't support php7. The problem is I recently updated to ubuntu 16.04 Xenial which seems to .   I tried to install it from , however when It try to install it with apt I always get similar errors.  $ sudo apt-get install php5-cli Cependant les paquets suivants le remplacent :   php7.0-cli:i386 php7.0-cli`   Which means it's obsolete. How can I install PHP 5.6 on Xenial? How do I select text nodes with jQuery? I would like to get all descendant text nodes of an element, as a jQuery collection. What is the best way to do that? Suppose $n$ is a non negative integer $\geq 4$ and $\sigma\in S_n$ a permutation. Conjugacy classes in $S_n$ are completley determined by the cycle structure of $\sigma$. If we let the alternating group $A_n$.act on $S_n$ by conjugation, the orbits coincide with the conjugacy classes in $S_n$ except for those (even) permutations whose cycle structure is composed of even cycles (which corresponds to cycles of uneven length) of distinct order (fixed points are treated as cycles of length $1$.) See   For instance a $3$-cycle in $A_4$ has cycle structure $[3,1]$ which fits the bill, and a $5$-cycle in $A_5$ or $A_6$ works too, since the cycle structure is $[5]$ and $[5,1]$ respectively. On the other hand, a $3$-cycle in $A_5$ will does not satisfy the criteria as it has cycle structure $[3,1,1]$ with two fixed points.  In this case, the conjugacy class of $\sigma$ inside $S_n$ splits in $2$ orbits of the conjugation action of $A_n$, and if $\tau$ is any odd permutation, for instance $(12)$, then we have $$\mathrm{Conj}_{S_n}(\sigma)=\mathcal{O}(\sigma)\coprod\mathcal{O}(\tau\sigma\tau^{-1})$$  My question is How can we differentiate between the two orbits? Does it have geometric meaning?    I can see that this information can be useful when looking at representations of the alternating groups, since there are as many irreducible representations as conjugacy classes. As for some geometric interpretation, I ask because there is one for the three cycles in $A_4$. If $S_4$ is understood as the group of symmetries of a regular tetrahedron, then $A_4$ is the subgroup of direct symmetries, and a $3$-cycle corresponds to a rotation of angle $\frac{2\pi}{3}$ or $\frac{4\pi}{3}$ with axis passing through one of the vertices. Conjugacy classes inside $A_n$ preserve the angle, and that's how we can tell them apart. In this case the computations are so simple we don't really need this picture, but I think it is a nice way to understand the fact $3$-cycles in are not all conjugate inside $A_4$... Question Close Updates Why does the following behave unexpectedly in Python?  &gt;&gt;&gt; a = 256 &gt;&gt;&gt; b = 256 &gt;&gt;&gt; a is b True           # This is an expected result &gt;&gt;&gt; a = 257 &gt;&gt;&gt; b = 257 &gt;&gt;&gt; a is b False          # What happened here? Why is this False? &gt;&gt;&gt; 257 is 257 True           # Yet the literal numbers compare properly   I am using Python 2.5.2. Trying some different versions of Python, it appears that Python 2.3.3 shows the above behaviour between 99 and 100.  Based on the above, I can hypothesize that Python is internally implemented such that "small" integers are stored in a different way than larger integers and the is operator can tell the difference. Why the leaky abstraction? What is a better way of comparing two arbitrary objects to see whether they are the same when I don't know in advance whether they are numbers or not? cp - cannot create regular file : permission denied I want copy to /lib/udev but gives error  cp -f /tmp/ok_pcscd_hotplug.sh /lib/udev  cp:cannot create regular file /lib/udev/ok_pcscd_hotplug : Permission denied   I am root , but I can't understand . what can I do ? Appendix in LaTeX dpkg: error: dpkg status database is locked by another process