Book trilogy about twins in a walled city where people are sorted by intelligence
Looking for a book series about a girl and her friends captured and sold as slaves
Enlarged partition, filesystem still reports old size Definition of the Pauli group and the Clifford group There seem to be two definitions of the Pauli group. In Nielsen and Chuang, the Pauli group on 1 qubit is defined as \begin{align*} \mathcal{P}_1 = \{\pm I, \pm iI, \pm X, \pm iX, \pm Y, \pm iY, \pm Z, \pm iZ\} = \langle X, Y, Z\rangle. \end{align*} But in other references, such as ,  \begin{align*} \mathcal{P}_1 = \{\pm I, \pm X, \pm iY, \pm Z\} = \langle X, Z\rangle. \end{align*} The second definition makes more sense in the stabilizer formalism, since we usually only consider the action of Pauli strings of $X$ and $Z$. What bothers me is that by the second definition, the phase gate $S$, which is an element of the Clifford group, results in \begin{align*} SXS^\dagger = Y \notin \mathcal{P}_1. \end{align*} I know that the prefactor $\pm i$ somehow doesn't matter, but can someone justify it from a more mathematical/group-theoretical perspective? From my (honestly limited) knowledge of what the Heat Death of the Universe is, I understand that HDotU is the point when energy levels across the entire Universe reach equillibrium, thus energy doesn't flow anymore absolutely anywhere, and thus nothing will ever happen anymore.  If that's the case, would time still be? Even when there's not a single event happening inside it and nobody's there to observe the ticking, does time (or space, come to think of it) exist? Or would HDotU cause something that ends the entire spacetime? Should a user be forced to add a comment when they vote to close? Can I pass a System.Net.MailMessage to a WCF service? I'd like to setup a WCF service to send emails. The System.Net.MailMessage doesn't seem to be serializable, and cannot be passed in a [DataContract]  The error I receive is      Type 'System.Net.Mail.MailAddress'   cannot be serialized. Consider marking   it with the DataContractAttribute   Any suggestions? Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups Batch export Photoshop layers to individual PNG files Can I disable GNOME legacy tray in 17.04? Can a decision in a node of a decision tree be based on comparison between 2 columns of the dataset? Assume the features in the dataframe are columns - A,B,C and my target is Y  Can my decision tree have a decision node which looks for say, if A&gt;B then true else false? Re-mounting an un-mounted USB stick Is there a way to mount back a USB stick that is once ejected, without physically detaching it and then plugging it in?  It was a live-usb of Linux Mint 14 in it, if it helps. I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great. Is it possible to experimentally show that observable universe is much larger than ~93B ly? Thought experiment:   Entangle a pair of particles.  Package one half of each pair in a special package, and send it out in a nano-spaceship keep the other half of the pair of entangled particles here on earth accelerate nano-spaceship to 99.99999999999999% of the speed of light using a number of    Due to Lorentz ,  $$ L = L _\text{0} \sqrt {1 - \frac {v^{2}}{c^{2}}} $$  In one years time, the spaceship would have traveled 93B ly away from us (in the spaceships relative frame)  After one year of travel, the spaceship could be preconfigured to detect the CMB in the direction of travel, and then trigger their quantum entangled particle to send a '0' or a '1' depending on whether the CMB was detected.  Repeat this process using any number of spaceships in any number of directions, and soon you would experimentally show how big the universe really is.  Any problems? I want to know, is there a way to make a tungsten lightbulbs filament withstand a mass amount of thermal shock while still producing a vast amount of heat? How to remove Firefox's "inspect element" button? This is driving me crazy! When FF 12 came out, Firefox's own 'Inspect Element' button on the right-click menu took the place of there the Firebug 'Inspect Element' button used to be.  As I've been doing web development in Firefox forever, my muscle memory still clicks there when I want to use Firebug.    Is there any way to remove the Firefox internal 'Inspect Element' and replace the button on the menu with Firebug's, returning it to its original position (marked NO in the screenshot)? I found several very-close-to-duplicate questions/answers. However, I want to restate some questions and discuss the answers from these questions.  Here the questions which I found:         I find it quite unfair that super simple questions and their answers upvote give the same amount of reputation points per upvote as complex questions. I think reputation points should be much closer linked to contribution. And I think that contribution in answering/asking a simple question (both in time, value and time saved to other developers) is way less than for some more complex questions.   Why doesn't Stack&nbsp;Overflow change this?  I saw several common answers:     if you change the game, then the only result is that people will game it differently.   I agree only partly with this. Sure, people will always game the system. However, any improvement can be rejected based on this argument. I believe if there is an obvious problem, it makes sense to fix it. And it looks like this question was asked number of times (meaning that it's not only me thinking that it should be changed).     Reputation is NOTHING more than an INCENTIVE to contribute.   Again. I agree only partly. Reputation doesn't have any real value (it can't be converted to anything). However, let's be honest. People are competitive and people like to have some measurement stick and most of software developers want to have a fair game. Based on this, reputation isn't NOTHING, reputation is very important tool which makes Stack&nbsp;Overflow popular. Calculating reputation one way or another way may lure or discourage people from participating.     Esoteric questions are only of interest to people working with that item.   I believe one of the biggest values of Stack&nbsp;Overflow that it contains a lot of information which is hard to find somewhere else.  So, sure, there are some rare questions which aren't valuable to the whole community. But, I would say that a question which saves 25 people a day of work (complex question) is more valuable than a question which saves a 1000 people 3 minutes of googling (simple question).  My summary:  I believe it makes sense to change a system a little bit. The simplest way would be the ability to mark a question as trivial. If some number of people marked the question as trivial (let's say 10), all upvotes for this question/answers become less valuable (for example, 1 or 2 points, instead of 5). This will close this gap. SSL certificate for a public IP address? I just tried buying a Comodo Positive SSL but it was rejected due to not supporting a public IP address, but instead they only support a domain name.  Does anyone know any SSL certificate provider that supports public IP address instead of a domain name?  My company has a dedicated server hosted with a web hosting company, which is used for running a bugtracker for multiple projects (for multiple clients). Since it's only used for a bugtracker, we don't need a domain name (our clients access it by typing the public IP in their browser). Double SOCKS5 proxy using SSH I was searching for a way to connect to one of my servers who is behind a jumpbox via SSH. I was able to do so by running the following command:  $ ssh -tt ptitpou@jumpbox ssh -tt ptitpou@targetserver   Now, I want to be able to use that server to access webservers on targetserver's network. For that, I followed  and I was able to successfully access the websites. The command I used was:   $ ssh -L 127.0.0.1:9999:localhost:9696 -tt ptitpou@jumpbox ssh -D 9696 -tt ptitpou@targetserver   The question I want to ask is how to use the ssh -o ProxyCommand to command in a slightly cleaner way. I tried to put the second part of the command as the proxy command and I had a successful SSH connection, but I was unable to use the proxy after that.   I've also tried  but the command line hangs after connecting to targetserver.  How can I do that? Am I wrong to try to use ProxyCommand option for this task? Multi-geometry shapefiles Since a shapefile can only contain a single geometry type (e.g. point, polygon, etc), how could I persist both?   If I must create a shapefile for each geometry type, how can I pass them to a colleague to load?  In kml I can just pass the single kml to load.  Is there a shapefile manifest equivalent that I am missing? Where can I find a repository for older unsupported releases? Office suite on Ubuntu Phone?