Does there exist such a module?
A ring without the Invariant Basis Number property
Are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic? Complex number isomorphic to certain $2\times 2$ matrices? I have been trying to prove this, but I am having trouble understanding how to prove the following mapping I found is injective and surjective. Just as a side note, I am trying to show the complex ring is isomorphic to special $2\times2$ matrices in regard to matrix multiplication and addition. Showing these hold is simple enough.  $$\phi:a+bi \rightarrow \begin{pmatrix} a &amp; -b \\ b &amp; a \end{pmatrix}$$  This is what I have so far:  Injective: I am also confused over the fact that there are two operations, and in turn two neutral elements (1 and 0). Showing that the kernel is trivial is usually the way I go about proving whether a mapping is injective, but I just can't grasp this.  $$ \phi(z_1) = \phi(z_2) \implies \phi(z_1)\phi(z_2)^{-1} = I = \phi(z_1)\phi(z_2^{-1}) = \phi(z_2)\phi(z_2^{-1}).$$  So if we can just show that the kernel of $\phi$ is trivial, then it also shows that $z_1 = z_2$. The only complex number that maps to the identity matrix is one where $a = 1$ and $b = 0$, $a + bi = 1 + 0i = 1$.  Using a similar argument for addition we can just say that the only complex number $z$ such that $\phi(z) = 0\text{-matrix}$, is one where $a=0$ and $b=0$, $a+bi=0+0i=0$.   Surjective:  I forgot to add this before I posted, but I honestly don't really understand how to prove this because it just seems so obvious. All possible $2\times2$ matrices of that form have a complex representation because the complex number can always be identified by its real parts and since the elements of the $2\times2$ matrix are real then the mapping is obviously onto.  I have always had trouble understanding when I can say that I have "rigorously" proved something, so any help would be appreciated! How can I get ssh-agent working over ssh and in tmux (on OS X)? I have a private key set up for my github account, the passphrase to which is, I believe, stored in OS X's keychain. I certainly don't have to type it in when I open a terminal window and enter ssh git@github.com.  However, when I'm running bash over an ssh session, or locally inside a tmux session, I have to type in the passphrase every single time I attempt to ssh to github.   suggests that a similar problem exists with screen, but I don't really understand the issue well enough to fix it in tmux. There's also  which includes a fairly complicated solution, but for zsh.  EDIT:  In response to , from a local terminal I get the following output:  [~] $ echo $SSH_AUTH_SOCK /tmp/launch-S4HBD6/Listeners [~]  $ ssh-add -l 2048 [my key fingerprint] /Users/richie/.ssh/id_rsa (RSA) [~] $ typeset -p SSH_AUTH_SOCK declare -x SSH_AUTH_SOCK="/tmp/launch-S4HBD6/Listeners"   Whereas over ssh or in tmux I get:  [~] $ echo $SSH_AUTH_SOCK  [~] $ ssh-add -l Could not open a connection to your authentication agent. [~] $ typeset -p SSH_AUTH_SOCK bash: typeset: SSH_AUTH_SOCK: not found   echo $SSH_AGENT_PID returns nothing whatever shell I run it from. How to activate the Gedit terminal plugin? I have the following packages installed:   gedit gedit-common gedit-plugins   In Ubuntu 12.04 with these packages installed all that was needed was to activate the terminal plugin and the press Ctrl+F9 to show the bottom panel. This isn't the case with Ubuntu 13.10 (Gedit 3.8.3) and I can't figure out how to do it. I am trying to run a buffer tool as part of script that I will both use as a standalone script and import into ArcToolbox. I'm just wondering, if I use GetParameterAsText for the buffer distance, how do I set the linear units? E.g. I want it preset so they only have to type 300 and the script takes that to be meters? Here is what I have so far:  # Buffer the Thames feature class to create the Flood Risk Area in_features = "Thames" out_feature_class = "Flood_R_A" # Run the Buffer Tool arcpy.Buffer_analysis (in_features, out_feature_class, arcpy.GetParameterAsText(5),    "FULL", "ROUND", "ALL", "") Which derivation of drift velocity is correct? In the derivation of drift velocity I have seen two variations and want to know which one's correct.   $s=ut+\frac{at^2}{2}$ Assume that the drift velocity of any electron in any conductor is : $$\vec{v_d}=\frac{\vec{l}}{t}$$ Due to the electric field the acceleration of electrons in any conductor is: $$a=\frac{-e\vec{E}}{m}$$ Now the distance travelled by an electron after a long time (initial thermal velocity = 0) $$l=\frac{at^2}{2}\implies \frac{-e\vec{E}t^2}{2m}$$ the time between the collisions is $\tau$ $\therefore$ $$l=\frac{-e\vec{E}\tau^2}{2m}$$ thus the velocity is $$\vec{v_d}=\frac{-e\vec{E}\tau}{2m}$$ In another proof I saw the author using $v=u+at \implies v_d=\frac{-e\vec{E}\tau}{m}$   My question is which of the the two equations of motion can be used in the proof? Can they be used at all. I am using different operating systems on different machines on different locations. I am using a USB flash disk to keep my files (source codes etc.).  I am new to Mac OS X and now I realize there are lots of files created with a dot underscore ._ prefix.  Questions  Why are ._ files created?   What are their purpose?   How can I avoid them? What happens to Bruce Banner when Hulk loses a tooth? In Age Of Ultron we witnessed Hulk lose a tooth.   My question is: Has Bruce Banner lost a tooth as well? Irreducibility check for polynomials not satisfying Eisenstein Criterion. How do I prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$? for $x, y$ positive?  This should be easy, but I'm not seeing how.  A hint would be appreciated. Case-insensitive string comparison in C++ Where to get QGIS toolbar icons? Does anybody know where to download the QGIS toolbar icons? Any nice, stable ways to keep a window 'Always on top' on the Mac? I am looking for a stable way to keep any window 'always on top' on OSX.  , an app that does that requires SIMBL/easySIMBL, which it seems is not very stable.  Are there any better solutions? It's actually surprising that there is no native way to implement this. It is often stated that the Lagrangian formalism and the Hamiltonian formalism are equivalent.   We often hear people talk about eigenvalues of Hamiltonians but I have never ever heard a word about eigenvalues of Lagrangians.   Why is this so? Is it not useful? is it not possible to do it? How do I connect to my Bluetooth headset? I have an ACER C7 Chromebook with Ubuntu 12.04 installed.  Also I have two apps installed Bluetooth Manager and Bluetooth.    I am trying to connect an  to Ubuntu. I have tried to set up the new device and pair, but the headset is never recognized.  I checked hard and soft setting for Bluetooth and they are on.  I am new to Ubuntu 12.04 and Linux and need specific instructions. running rmi server, classnotfound A while back I read about the super Kamiokande detector detected a large neutrino flux and then several hours later a supernova was seen. Anyone know of this with sources? I don't recall the source at the moment. Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular. Travelling within the US as a foreign citizen without a passport I'm travelling within the US and am a British citizen. Can I travel without my passport and just show my UK driver's license instead? Wave states with great confidence that Incursio is weaker than Grand Chariot, because Incursio was a prototype for Grand Chariot, etc.  The early clashes between Wave and Tatsumi can lead the viewer to such conclusion, but Incursio's has the ability to evolve (from the beast it was made of), while Grand Chariot doesn't.  If you watched to the finale, we can see that      Tatsumi causes Incursio to evolve so much during the fight against Shikoutazer that it even gains the ability to fly. While it is enough to defeat the emperor, unfortunately not enough to save Tatsumi's life.   Considering this, I don't think that Incursio was the weaker one. Is it so?