How to represent a number with a repeating decimal as a ratio of integers
Expressing $1.24\overline{123}=1.24\;123\;123\;123\;\ldots$ as the ratio of two integers
We know that for a position variable $x$ and momentum $p$, the uncertainties of the two quantities are bounded by  $$\Delta x \Delta p \gtrsim \hbar$$  Now, this is usually first explained with $x$ being a simple linearly measured position and $p$ being linear momentum.  But it should apply to any good coordinate and its conjugate momentum.  It should, for instance, apply to angle $\phi$ about the $z$ axis, and angular momentum $L_z$:  $$\Delta \phi \Delta L_z \gtrsim \hbar$$  The thing is, $\Delta \phi$ can never be greater than $2\pi$.  I mean, you have to have some value of $\phi$ and $\phi$ only runs from 0 to $2\pi$.  Therefore  $$\Delta L_z \gtrsim \hbar/\Delta \phi \geq \hbar/2\pi$$  But, uh-oh!  This means it is impossible for $\Delta L_z$ to be zero, and we should never be able to have angular momentum states with definite $L_z$ values.  Of course, it doesn't mean that.  But I have never figured out how this is not in contradiction with the Schroedinger eqn. calculations that give us states with definite values of $L_z$.  Can anyone help me out?  One answer I anticipate is that $\phi$ is sort of "abstract" in that if you chose your origin at some other point you will get completely different values of $\phi$ and $L_z$, and ipso facto, usual considerations don't apply.  I don't think this will work, though.  Consider a "quantum bead" sliding around on a rigid circular ring and you get the exact same problem with no ambiguity in $\phi$ or $L_z$.  (Well, there will be some limited ambiguity in $\phi$, but still, there won't be in $L_z$.) How do I have no default web browser, so the system asks what browser to use when I click on a link? I use Internet Explorer for some things and Chrome for others. I'd like to have a choice when I click on a link of which browser to open the link in. Is this possible in Windows? I am currently running Windows 8, although there should be similarities between versions.  To the best of my knowledge this isn't possible. Where are the database username and password stored? I'd like to send my Drupal installation to a colleague, but I don't want him to see the username and the password of my database. Where are these informations stored?  I've seen that the username and the password are written in sitename/sites/default/settings.php, but is it the only file with these informations? Is it sufficient if I change the username and the password in this file and then send my colleague all my Drupal files and my database? Is -1 followed by a singular or plural noun? Why are objects at rest in motion through spacetime at the speed of light? I read that an object at rest has such a stupendous amount of energy, $E=mc^2$ because it's effectively in motion through space-time at the speed of light and it's traveling through the time dimension of space-time at 1 second per second as time goes forward.  What troubles me here, is the fact that it is traveling through space-time at the speed of light. Why is it at the speed of light? Any canon Shadowrun material related to NY/PA/NJ areas? I'm looking for specific, canon (i.e. official, published) references to material for the Shadowrun game-world setting. My game setting is based around the Philadelphia area; I've turned New Jersey into a radioactive swamp, then dropped a nuke in it; I've got some things established for the Philly area, and NYC.  What I want to know is whether anyone knows of specific references to published material dealing with this region. Any mention of the big apple in Shadowrun's history? Any famous names in the area, or megacorps that are mentioned? Anything at all that's part of the published SR work, that I can incorporate, or at least avoid contradicting.  If you have any home-brewed material for that area, I would love to see a link in comments, but please don't put it as an answer. Thanks! =) Let's say you want to save a bunch of files somewhere, for instance in BLOBs. Let's say you want to dish these files out via a web page and have the client automatically open the correct application/viewer.  Assumption: The browser figures out which application/viewer to use by the mime-type (content-type?) header in the HTTP response.  Based on that assumption, in addition to the bytes of the file, you also want to save the MIME type.  How would you find the MIME type of a file? I'm currently on a Mac, but this should also work on Windows.   Does the browser add this information when posting the file to the web page?  Is there a neat python library for finding this information? A WebService or (even better) a downloadable database? What should I do if two libraries provide a function with the same name generating a conflict? What should I do if I have two libraries that provide functions with equivalent names? Elegant way of obtaining the envelope of oscillating function Origin of electric charge Baryons have charges that are the result of a polynomial calculation of their building blocks (quarks)'s fractional charges. But what gives these quarks electric charges? What interactions do they have with photons? And what about leptons, like muons and electrons? How do they get their -1 charge? And then, how do positrons and anti-muons get +1 charge? Same goes for quarks: how does up-antiquarks get -2/3 charge? Or I may want to formulate my question this way:  How does any elementary particle interact with the EM-Force to get an electric charge? Inequality involving sides of a triangle How can one show that for triangles of sides $a,b,c$ that   $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} &lt; 2$$  My proof is long winded, which is why I am posting the problem here.   Step 1: let $a=x+y$, $b=y+z$, $c=x+z$, and let $x+y+z=1$ to get   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}&lt;2$  Step 2: consider the function $f(x)=\frac{1-x}{1+x}$, and note that it is convex on the interval (0,1), so the minimum of   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}$  is reached when the function takes the extreme points. i.e. $x=y=0, z=1$.    But going back to the fact that this is a triangle, we note that $x=y=0 \implies a=0$ which is not possible, so the inequality is strict. I can't get to the bottom of this error, because when the debugger is attached, it does not seem to occur.  Collection was modified; enumeration operation may not execute  Below is the code. This is a WCF server in a Windows service. The method NotifySubscribers() is called by the service whenever there is a data event (at random intervals, but not very often - about 800 times per day). When a Windows Forms client subscribes, the subscriber ID is added to the subscribers dictionary, and when the client unsubscribes, it is deleted from the dictionary. The error happens when (or after) a client unsubscribes. It appears that the next time the NotifySubscribers() method is called, the foreach() loop fails with the error in the subject line. The method writes the error into the application log as shown in the code below. When a debugger is attached and a client unsubscribes, the code executes fine. Do you see a problem with this code? Do I need to make the dictionary thread-safe? [ServiceBehavior(InstanceContextMode=InstanceContextMode.Single)] public class SubscriptionServer : ISubscriptionServer {     private static IDictionary&lt;Guid, Subscriber&gt; subscribers;      public SubscriptionServer()     {                     subscribers = new Dictionary&lt;Guid, Subscriber&gt;();     }      public void NotifySubscribers(DataRecord sr)     {         foreach(Subscriber s in subscribers.Values)         {             try             {                 s.Callback.SignalData(sr);             }             catch (Exception e)             {                 DCS.WriteToApplicationLog(e.Message,                    System.Diagnostics.EventLogEntryType.Error);                  UnsubscribeEvent(s.ClientId);             }         }     }          public Guid SubscribeEvent(string clientDescription)     {         Subscriber subscriber = new Subscriber();         subscriber.Callback = OperationContext.Current.                 GetCallbackChannel&lt;IDCSCallback&gt;();          subscribers.Add(subscriber.ClientId, subscriber);                  return subscriber.ClientId;     }      public void UnsubscribeEvent(Guid clientId)     {         try         {             subscribers.Remove(clientId);         }         catch(Exception e)         {             System.Diagnostics.Debug.WriteLine(&quot;Unsubscribe Error &quot; +                      e.Message);         }     } } What happens in a circuit, when the wire and the battery are superconducting. And shorted This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, including more unit test resources, navigate to the  tag.  This canonical question is intended to address several classes of common questions by providing a quick summary and links to comprehensive resources:   How is code coverage obtained and calculated? How do I increase my coverage?  Why can't I cover these lines? Why isn't the body of my loop or conditional statement covered? Cleaning wax and pesticide from apple skin remote ssh tcpdump command, through a gateway, to be opened, in realtime, in Wireshark, on local machine I (Host A - Windows based using plink) am trying to connect to a remote linux server (Host C), through a gateway linux server (Host B), and run this tcpdump on Host C to be piped all the way back to Host A (through Host B I assume), and opened, in real time, with Wireshark on Host A:  Below is what I saw as the command when not using a gateway ("direct remote fetch").  So, how would I make this work going through my gateway host B scenario above?  Host A$ plink.exe -ssh -pw abc123 root@Host C "tcpdump -s0 -U -n -w - -i eth0 not port 22" | "C:\Program Files\Wireshark\Wireshark.exe" -k -i - Extruding a circular shape from a non-circular object I need to extrude a circular shape from a surface that's basically just a grid of quads. What's the best practice to do this without screwing up the topology too much?  I basically want to have a circular face I can extrude, in order to have a cylinder like protuberance on the object, something like this: Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$ I am looking for the combinatorial argument for the identity:  \begin{equation} k\binom{n}{k} = n\binom{n-1}{k-1} \end{equation}  This is easy to show algebraically as:  \begin{equation} \binom{n}{k} = \dfrac{n(n-1)(n-2)(n-k+1)}{k(k-1)!} \end{equation}   What is the combinatorial argument? What are some general ideas to get started?   Here is a clarification of 2. From what I have seen so far, proving (combinatorially) an identity with an addition sign usually implies that we need to partition a set (this makes sense because of the addition rule and provides a nice visual). On the contrary, the previous observation leads me to believe that multiplication in identities can be resolved with the multiplication principle, but what is the "visual/interpretation" for this? Could someone provide such an interpretation for the example identity given above? Multiple AES Key Derivation from a master key How to filter by messages, not by conversation? I have been emailing my Gmail account error messages from a project I am working on.   I am trying to use Gmail filters/labels to sort out the class of errors.  But because the subject line is basically always the same even if I have a specific filter for 3 known error cases, the label is applied to the conversation, instead of the specific message.  Therefore, all the errors get all the filters.  I do not yet want to delete the errors, but would rather file them separately.   I have not yet used the NOT contains string yet, but that looks to be non-scalable as I get to the fourth or fifth error case I am done with handling.  Is there any option other than turning off Conversations for all my email?  (I guess I could use one of my spare accounts to turn off conversations and do the work there but would be nice to have such an option).