What is uncertainty principle exactly?
Is the uncertainty principle a property of elementary particles or a result of our measurement tools?
I am a new user, who has just been told to run / execute a command. The answer has some text with a grey background like this text here or  sometimes this text here.   However, I don't know what "run" or "command" means in relation to Ubuntu.  How do I do this?    This is intended for use as a Q&amp;A you can link to when you tell someone to run a command. Details on how the Yearling badge works Jawa asked me this  and I didn't know the answer. So I present it to the meta SO community:     [To get the Yearling badge] Do you need to earn the 200 rep within that one year or is it enough to get 200 at some point after being a member for a year? And does 'a year' mean the very first 12 months or the last (sliding window of) 12 months?    I would assume that if you hit 200 reputation after you've already been around for a year, then you would get the Yearling badge. But I don't know. Weak solutions to the Neumann's problem (Evans PDE) As far as I can tell, it's something like this:  damage reduction = armor / (armor + 100)   Is this accurate? "exec: gnome-session: not found" when logging in using lightdm On login the password is accepted but I'm returned to the login screen. Tried multiple times even after reboot... same behavior.  I left my laptop (Lenovo Ideapad z510) in suspend mode (lid down) last night, but it did not resume this morning, as it usually does. It had shutdown or crashed, and on booting with Ubuntu the login failure mentioned above started.  cat ~/.xsession-errors /usr/sbin/lightdm-session: 1: /usr/sbin/lightdm-session: ls: not found /usr/sbin/lightdm-session: 104: exec: gnome-session: not found   Okay. I've tried a bunch of suggestions from    Changed id to /usr/sbin/id in /etc/profile removed .Xauthority removed /tmp removed,  purged and re-installed lightdm checked .profile for syntax errors (source .profile worked fine, so I guess no syntax errors)   Nothing worked. My guest login is working fine, so another user should also work. I'm now going to create another user and continue working, but a solution to this problem is still warranted... it's likely going to repeat in future.  Installing and reconfiguring gdm also did not work.  sudo apt-get install gdm Given six numbers $x,y,z,a,b,c$ which satisfy the following relations. Express $x+y+z$ in terms of $a,b,c$ Given six numbers $x,y,z,a,b,c$ which satisfy the following relations $y^2+yz+z^2=a^2$ $z^2+zx+x^2=b^2$ $x^2+xy+y^2=c^2$ Express $x+y+z$ in terms of $a,b,c$    My attempt: $\dfrac{(y^3-z^3)}{(y-z)}=a^2,\dfrac{(z^3-x^3)}{(z-x)}=b^2,\dfrac{(x^3-y^3)}{(x-y)}=c^2$ But this step didn't seem to help. So I tried adding and expressing $k(a+b+c)^2$ but some other unmanagable elements came along. I could not think of any other way to approach the problem. Please help. If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$ I was checking my old set of homework problems that I found this one:     If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only   if $a$ is a cube in $G$.   One direction is easy. If $a$ is a cube in $G$ then there exists $y \in G$ such that $a=y^3$. Now with direct manipulations it is shown that $x=y^{-2}$ is a solution to $x^2ax=a^{-1}$.  The other direction is harder though. I first attempted to isolate $x$ and then somehow show that it's necessary for $a$ to be a cube. But since I had no idea how to isolate $x$ I gave up. Then I decided to prove the existence of $y \in G: a=y^3$ by showing that a particular map is bijective so I can come up with a nice variable change to conclude that $a$ must be a cube, but this attempt failed as well. I'm tired now, so I think I'm giving up. I'm looking for a hint or a complete solution if a hint is not possible at this stage. Prove that $ m_{a} \geq m_{g} \geq m_{h} $ using strict inequalities unless $ a = b $. $ m_{a} = \frac{1}{2} (a + b) $   $ m_{g} =  \sqrt{ab} $   $ \frac{1}{m_{h}} = \frac{1}{2}(\frac{1}{a} + \frac{1}{b})$   Attempted Solution:  I believe I have shown the first step, which is $ m_{a} \geq m_{g} $ but i'm stuck on showing $ m_{g} \geq m_{h} $  $ \frac{a + b}{2} &gt; \sqrt{ab} $  $ (\frac{a + b}{2})^2 &gt; (\sqrt{ab})^2 $  $\frac{a^2 + 2ab + b^2}{4} &gt; ab$  $a^2 + 2ab + b^2 &gt; 4ab $  $a^2 -2ab + b^2 &gt; 0 $  $(a - b)^2 &gt; 0 $   so if $a = b$ we have $(a -b)^2 \geq 0$  Therefore, $m_{a} \geq m_{g} $  Now, i'm not sure if that is correct or not, if not can you please guide me to the complete solution or if I am correct on this part, provide a solution for $m_{g} \geq m_{h} $ Proving $\mathbb{Q}$ isn't homeomorphic to $\mathbb{Z}$ Prove that $\mathbb{Q}$ isn't homeomorphic to $\mathbb{Z}$  I know that a homeomorphism is a bijection function from two sets where both the function and its inverse are continuos.   So I figure that if I show that there is no continuos invective mapping from $\mathbb{Q}\rightarrow\mathbb{Z}$ then I have done the proof. But I do not know how to do this (if his is the right way). How can I start to show this? In physics sometimes we find energy that is negative. What does the negative sign indicate? Difference between correlation and covariance: is covariance only useful if the relation is linear? I'm trying to understand better the difference between  covariance and correlation, besides the fact that the correlation coefficient is a dimensional and has values between $-1$ and $1$.   One unclear point is the following: the correlation coefficient can highlight only a linear relation between two variables, but does  covariance only work for linear relations too? Supposing that the relation is not linear, is the covariance not zero? Is it bigger than in the case in which the relation is linear?   Cauchy inequality says $$|\sigma_{xy}|&lt;\sigma_{x} \sigma_{y}$$  Which seems to say that the covariance is maximum when there is a perfect linear relation.    So I do not get what does happen if the relation is not linear, and, if covariance is not a suitable parameter in that cas I don't get the reason for that, since simply looking at the definition:  $$\sigma_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})}{N}$$  it looks like that this should be non zero for any kind of relation (like parabolic).  So can this be a difference between the use of covariance and correlation coefficient? (The first is useful for any kind of relation, the second one only for linear ones). The email for getting notifications can be changed by clicking the "preferences" tab in your profile. After entering a new email address, the "Save email settings" button becomes clickable:    Clicking the button turns it grey:    So all looks well. But if you don't confirm the email by clicking on the link sent to your email inbox, the next time you go back to the preferences, you will see your old email address with no indication that it changed.  You can subscribe to the weekly newsletter from the "edit" tab of your profile. If you click the option and save, the next time you visit the edit page before confirming your email you get this message:    Once you confirm the email, the resend link disappears so you know the subscription is set up:    Ideally, the two systems would be unified so that all email subscriptions happen the same way. But short of that, it would be very helpful to indicate that an email address is unconfirmed for notification emails just like newsletters. Word for "willing to try new and unfamiliar things" Example:  A: Hey, do you like to do bungee jumping? B: Um, OK. Haven't done it before, but let's try it.  I'm not looking for adventurous or something similar. That word should also fit in less action-packed activities:  A: I have a new board game. Do you like to play? B: I don't know this game and, actually, I don't like these varieties of games but I'll play. Maybe it's fun.  First I thought about open-minded but I think this means more willing to listen to people, think about their ideas, arguments, ... and accept them (not necessarily agree). What word describes person who is willing to try activities he doesn't know or hasn't experienced before? Nobody owner (99 99) in FTP caused by php functions? Are threads implemented as processes on Linux? I'm going through , Advanced Linux Programming by Mark Mitchell, Jeffrey Oldham, and Alex Samuel. It's from 2001, so a bit old. But I find it quite good anyhow.  However, I got to a point when it diverges from what my Linux produces in the shell output. On page 92 (116 in the viewer), the chapter 4.5 GNU/Linux Thread Implementation begins with the paragraph containing this statement:     The implementation of POSIX threads on GNU/Linux differs from the   thread implementation on many other UNIX-like systems in an important   way: on GNU/Linux, threads are implemented as processes.   This seems like a key point and is later illustrated with a C code. The output in the book is:  main thread pid is 14608 child thread pid is 14610   And in my Ubuntu 16.04 it is:  main thread pid is 3615 child thread pid is 3615   ps output supports this.  I guess something must have changed between 2001 and now.  The next subchapter on the next page, 4.5.1 Signal Handling, builds up on the previous statement:     The behavior of the interaction between signals and threads varies   from one UNIX-like system to another. In GNU/Linux, the behavior is   dictated by the fact that threads are implemented as processes.   And it looks like this will be even more important later on in the book. Could someone explain what's going on here?  I've seen this one , but it doesn't help much. I'm confused.  This is the C code:  #include &lt;pthread.h&gt; #include &lt;stdio.h&gt; #include &lt;unistd.h&gt;  void* thread_function (void* arg) {     fprintf (stderr, "child thread pid is %d\n", (int) getpid ());     /* Spin forever. */     while (1);     return NULL; }  int main () {     pthread_t thread;     fprintf (stderr, "main thread pid is %d\n", (int) getpid ());     pthread_create (&amp;thread, NULL, &amp;thread_function, NULL);     /* Spin forever. */     while (1);     return 0; } I have a question about subsets $$ A \subseteq \mathbb R $$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize this kind of sets? In a topological,measurable or in some way? For example, does there exist a function continuous on $\mathbb Q$ and discontinuous on the irrationals? Storing Images in DB - Yea or Nay? Proof $p \vdash q \Rightarrow p$ Firstly, how do I read it? Is below right? With braces   $$p \vdash (q \Rightarrow p)$$  The given proof is:    ${p}$, premise ${q}$, assumption  $p$ by (1) // what???  ${q} \Rightarrow p$ by implication introduction with 2 and 2.1 QED...    I can't really get the link between $p$ and $q$ in 2.1 ... Over the time I've been able to vote for closure on SO, I've gotten into the habit of clicking on the reason itself in order to select the radio button corresponding to it. Since the reasons have changed, there are also now links in the middle of the text, which I also happen to accidentally keep clicking on. More annoyingly, these links don't open in a new tab so I have to click back and redo it. This happens most often with the "FAQ" link in the off-topic dialog, which is right there in the top left corner where I normally click.        Basically, those pieces of text now have two functions. One function of it is selecting the corresponding radio button and the other function is loading a new page. Obviously loading the new page trumps simple selection.  This has become even more painful now with the new off-topic messages, where a large portion of the message is a link:        A simple feature request: can the vote-to-close dialog itself not render the links? Will the silhouette formula change depending on the distance metric? I am using Silhouette width to compute the best value for k in k-means. As I am performing document clustering, I am calculating the values of a and b as follows in Python:  a = distance(data[index], centroids[clusters[index]], metric=metric, p=p) b = min([distance(data[index], c) for i,c in enumerate(centroids) if i != currentindex]) score = float(b - a) / max(a, b) if max(a, b) &gt; 0 else 0.0   The original formula from the  is the following:    I am using a cosine similarity measure to compute the distances and was wondering if this formula needs to be changed or it can be left as is. In the above snippet of code, the function distance computes the cosine similarity metric.