When we're talking about spatial rotations is quantum mechanics, why do we need to resort to $SU(2)$? Why isn't $SO(3)$ enough?  I've read that $SO(3)$ isn't , and I've read about , and , but I'm falling down deeper and deeper into pure mathematics (esp. topology). Is there a "topology for physicists" explanation of what's going on here?
$SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence?  I heard that this fact is related to the description of bosons and fermions. But how does it follow from the fact that $SU(2)$ is the double cover of $SO(3)$?
TikZ: Named nodes in nested `pic`s I am using the new TikZ/PGF 3.0 pic feature.  I use them heavily, including using a pic inside another pic.  The problem is that Tikz sometimes cannot find the node names of nested pics.  I expect the following to work:  \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{positioning} \begin{document} \begin{tikzpicture} \tikzset{complex/.pic={   \node(re) {re};   \node[right=0mm of re](im) {im}; }} \tikzset{su3v/.pic={   \path pic(c0) {complex};   \path pic[right=1mm of c0im](c1) {complex};   \path pic[right=1mm of c1im](c2) {complex}; }} \tikzset{spinor/.pic={   \path pic(v0) {su3v};   \path pic[below=2mm of v0c2re](v1) {su3v};   \path pic[below=2mm of v1c2re](v2) {su3v};   \path pic[below=2mm of v2c2re](v3) {su3v}; }}  \path pic {su3v};       % Works \path pic(name) {su3v}; % ! Package pgf Error: No shape named namec0im is known. \path pic {spinor};     % ! Package pgf Error: No shape named v0c1im is known. \end{tikzpicture} \end{document}   What is wrong here? I have a query to insert a row into a table, which has a field called ID, which is populated using an AUTO_INCREMENT on the column. I need to get this value for the next bit of functionality, but when I run the following, it always returns 0 even though the actual value is not 0:  MySqlCommand comm = connect.CreateCommand(); comm.CommandText = insertInvoice; comm.CommandText += "\'" + invoiceDate.ToString("yyyy:MM:dd hh:mm:ss") + "\', " + bookFee + ", " + adminFee + ", " + totalFee + ", " + customerID +  ")"; int id = Convert.ToInt32(comm.ExecuteScalar());   According to my understanding, this should return the ID column, but it just returns 0 every time. Any ideas?  EDIT:  When I run:  "INSERT INTO INVOICE (INVOICE_DATE, BOOK_FEE, ADMIN_FEE, TOTAL_FEE, CUSTOMER_ID) VALUES ('2009:01:01 10:21:12', 50, 7, 57, 2134);last_insert_id();"   I get:  {"You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near 'last_insert_id()' at line 1"} Find all n such that $\phi(n) = n/2$ My idea for the solution is something like this:  Since $2 | n$, $n = 2^a p_1^{e1} p_2^{e2} \cdots p_t^{et}$ where $a \geq 1$.  Then, $n/2 = \phi(2^a) \phi(p_1^{e1}) \phi(p_2^{e2}) \cdots \phi(p_t^{et})$.  Looking at the case that $t = 0$,  $n/2 = \phi(2^a) = 2^{a-1}$, and therefore $n = 2^a$.  Otherwise, $n = 2^a \phi(p_1^{e1}) \phi(p_2^{e2}) \cdots \phi(p_t^{et}) = n = 2^a p_1^{e1} p_2^{e2} \cdots p_t^{et}$. Since $\phi(n) &lt; n$, this is a contradiction, and therefore, the only $n$'s that apply are the powers of 2.  Is this right? Also, we were asked to find $n$ such that $\phi(n) = n/3$. How would I go about that one? How to best encapsulate monetary values in a database Error -504 on appliation installation Yes. I've read all other questions regarding this problem. Like this one:  It's the same problem as this one but with more background information and an other app:   The details: My device is an Ulefone Be Touch 2 running Android 5.0.  My connection is perfect. To be sure I tried different networks, among them are 3G, 4G and wifi on ADSL, VDSL, cable and fiber. I've also used Tor and 3 different VPN providers in 3 countries on 2 continents. Let's say I ruled the network out.   A single app does not install: OpenKeychain.  Currently I have 126 apps installed. It does not install via desktop browser > play store > install button.    I have enough free space (40GB). The app is 7MB.   Re-adding Google account and clearing caches did not work.    Even a complete device wipe and re-install didn't work.     Downloaded the apk and tried manual installation, same error.   I also own a Nexus 7 (2013 wifi) and the app installs fine on that device. What are the requirements to get the "Weed Eater" hat? The hat's description says     answer a question qualifying for the Tumbleweed badge that gets accepted   During the winter bash, I  a question that is a , and it got accepted; yet I did not get the hat. However  answer  and did get the hat. (What was the difference?)  So what are the exact requirements to get the  hat? Cleaning Contaminated Disc Rotors I've had an eternal struggle with my current bike over brake contamination. The levers and calipers are shimano ultegra r8000, so the latest and (almost) greatest.  I first noticed it when I took it to a repair shop (i no longer use) for a bleed, it came back and the front brake would squeal like crazy and at low speeds had little to no braking force. You can stop from speed but then struggle to come to a halt without using your foot.  At around 2000 miles I tried to replace the brake pads, when I took them out they were black with grime on both sets (front &amp; back), not to mention worn down to the metal. I replaced them with the L04C sintered pads, gave the callipers a good cleaning with a towel (no chemicals) and fitted the new set.  Annoyingly all of my braking issues are back, now I probably should have cleaned the rotors, I simply used some muc off disc brake cleaner on them. I would like to know the best way to clean the rotors, I've inspected the pads and they are looking fairly black now, with plenty of tread left. I'm going to abandon these pads and fit a new set. For the rotors I plan to completely remove them from the wheels (they are center lock fitted) and try and treat them off the bike, what is the best way to fully clean them? They have plenty of wear left in them but you can see these black grooves in them, and plenty of muck on the back that I struggle to get off.  Below are images of the rotors after soaking them in disc brake cleaners, waiting and then wiping them down with a clean, microfiber towel.   Front Wheel Outside Rotor   Front Wheel Inside Rotor - note the increased dirt on this side  Looking at ways people clean their disc rotors, I've seen plenty of suggestions that I would need to burn the contaminants off the discs. Would the best approach be to remove them and use a blowtorch, should I soak them in some nasty chemicals overnight before I try to heat them? Do I try baking them in the oven?  And finally, what is the best way to prevent the brakes from becoming contaminated in the future? it's really off putting when they squeal and don't perform how I'd expect because it means I'm more reserved whilst out riding on the bike, leading to reduced time riding around outside. I commute on this bike (50 miles a week) and then ride about 50 on the weekends also, so it see's it's fair share of road use. I tend to wash it once after my big ride on saturdays and will rinse it off if it's been raining, any day that occurs (since I live in north east england, the acid rain..). I've owned disc brake bikes before and never had half the issues I've had with this, I had a cheap trek with tektro brakes and the brakes were sharper and never had any issues with contamination whatsoever (i did around 500 miles on that bike before a motorist happened).  Thanks for any and all advice :) Can i use the /testfor command to check for certain words in chat? What is an quick way to create a new blank file in Lion? In Windows, one can create a new file using Right Click -> Add New -> Type of File. Is there an analogous action in Lion?  I just want to create a new blank file on my Desktop. Why Faddeev-Popov ghost cannot exist in external line? How do the tens­es and as­pects in English cor­re­spond tem­po­ral­ly to one an­oth­er? Non-na­tive speak­ers of­ten get con­fused about what the var­i­ous tens­es  and as­pects mean in English. With in­put from some of the folk here I've put to­geth­er a di­a­gram that I hope will pro­vide some clar­i­ty on the mat­ter.  I of­fer it as the first an­swer to this ques­tion. Con­sid­er it a liv­ing doc­u­ment. In­put is wel­come, and good sug­ges­tions will be  in­cor­po­rat­ed in­to the di­a­gram.    No­ta bene: What this is not is a dis­cus­sion of whether there are  more than two tens­es in English. , to which this ques­tion is not in­tend­ed to sup­ply ar­gu­ments one way or the oth­er.  Here, the aim is to pro­vide an overview of what con­struc­tions English-speak­ing peo­ple use for con­vey­ing in­for­ma­tion about ac­tions re­fer­ring to past, present, and fu­ture, and to pro­vide it first and fore­most to pre­cise­ly the peo­ple who are like­ly to use "tense" as a catch-all term in their search, rather than to lin­guists who know bet­ter.  Break­ing News There is now an ex­cel­lent ELU blog ar­ti­cle ti­tled  . It is high­ly rec­om­mend­ed read­ing. Is there any point of using "return !0" in javascript? Using `M` instead of `k` when rep is >= 1M on https://chat.stackexchange.com/users I read on  ()    Shouldn't it be 1.03m or (if we really follow the conventions of the metric system) 1.03M instead? How can I visualize which folders and files are taking up all of the space on my hard drive?  I'm getting some conflicting reports on the size of hard drive contents. Namely what is and isn't there and what folders are actually using the space.   I need to know which of the files or folders the culprits behind all this hidden bloat. Also there should be a print option to get it on paper. Lie Algebra of $SL_n(\mathbb H)$ The Lie algebra of $SL_n(\mathbb C)$ are the matrices where the trace is $0$. But what is the Lie algebra of $SL_n(\mathbb H)$ where $\mathbb H$ is the quaternions? How do I recover my accidentally lost Windows partitions after installing Ubuntu? MySQL SELECT LAST_INSERT_ID() for compound key. Is it possible? Can I get the LAST INSERT ID() for a compound key in MySQL? Export each image from a collection in Google Earth Engine Isn't $HOME supposed to be the location of the user's home directory? I'm having trouble with a web application I'm writing: The web application, which runs as user nobody, needs to launch an inkscape process to manipulate some SVG files. Because this runs as user nobody, there is no home directory. What I am doing is creating a temporary directory under /tmp and trying to set that as the home directory via export HOME=/tmp/someUniqueId. I then set a few things in that "temporary" home directory, like a symlink .fonts to the folder for the font files to use.  I know this works for finding the ~/.fonts directory. But when I launch Inkscape, all I get is a message that Inkscape could not create the .inkscape directory:    Doesn't $HOME set the location of my home directory? Why is Inkscape still trying to access / as my home directory? How to get pretty HTML source code from PHP generated output?