How to visualize spacetime curvature?
Better explanation of the common general relativity illustration (stretched sheet of fabric)
limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$ "Questions that may already have your answer" box overlaps onto the footer I'm currently rebuilding our entire intranet from scratch, mostly because the tech behind is out-dated and it has been proved that a lot of information is difficult to find.  Though that is beside the point, what I am wondering is what would be the optimum amount of users to use for qualitative and quantitative testing with a userbase of around 1000 users?  Is there a general rule of thumb for both based on the total number of users you have? or is it just say 5 for qualitative and 10 for quantitative?  What would be the best approach? Is there a word to describe someone who tends to disagree with others only to upset them? I have two Python dictionaries, and I want to write a single expression that returns these two dictionaries, merged (i.e. taking the union).  The update() method would be what I need, if it returned its result instead of modifying a dictionary in-place. &gt;&gt;&gt; x = {'a': 1, 'b': 2} &gt;&gt;&gt; y = {'b': 10, 'c': 11} &gt;&gt;&gt; z = x.update(y) &gt;&gt;&gt; print(z) None &gt;&gt;&gt; x {'a': 1, 'b': 10, 'c': 11}  How can I get that final merged dictionary in z, not x? (To be extra-clear, the last-one-wins conflict-handling of dict.update() is what I'm looking for as well.) I have a small server which I use for testing and programming. Currently it runs Debian 9.4 stretch with 4.14.0-0.bpo.3-amd64 kernel.  Today I tried to connect through SSH but I couldn't then I tried to ping it and it was unreachable. Therefore I had to hard-restart it by unplugging power cable. Then I went to  /var/log/syslog  and I found a strange line containing exactly 6140 characters like the following  ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@  then nothing else until new log entries of system restart. This is actually the first time it happens.  Does someone know what could it be? Automatically run commands over SSH on many servers There is a list of IP addresses in a .txt file, ex.:  1.1.1.1 2.2.2.2 3.3.3.3   Behind every IP address there is a server, and on every server there is an sshd running on port 22. Not every server is in the known_hosts list (on my PC, Ubuntu 10.04 LTS/bash).  How can I run commands on these servers, and collect the output?  Ideally, I'd like to run the commands in parallel on all the servers.  I'll be using public key authentication on all the servers.  Here are some potential pitfalls:   The ssh prompts me to put the given servers ssh key to my known_hosts file. The given commands might return a nonzero exit code, indicating that the output is potentially invalid. I need to recognize that. A connection might fail to be established to a given server, for example because of a network error. There should be a timeout, in case the command runs for longer than expected or the server goes down while running the command.   The servers are AIX/ksh (but I think that doesn't really matter. I have a couple of big text files and in the file UNIQS.txt I have a list of strings to grep from another file. The code I use is  grep -f UNIQS.txt EEP_VSL.uniqs.sam &gt; UNIQ_templates.sam   which does nothing - the file generated is empty. But when I do  grep -F -f UNIQS.txt EEP_VSL.uniqs.sam &gt; UNIQ_templates.sam   it works correctly. This confuses me because I didn't think grep would interpret the entries in UNIQS.txt as regexp patterns without quotes and slashes and so on being in the file (which there aren't). Is it the case in general that if you are getting the patterns from a file then it will automatically think that they are regexp patterns?  Edit: In the UNIQS.txt file, there are newline separated strings of the form  HWI-ST365:215:D0GH0ACXX:2:1101:10034:186783   (called template names) and the file EEP_VSL... tab separated columns, with about 14 columns and the first column is the template name, so basically I want to extract the line corresponding to each template in the file. Let's say we have a Python script my_script.py that does some data processing with Blender. This script accepts some arguments (e.g. arg1 arg2). How can I forward arguments to the script when using it with the CLI?  blender --background --python my_script.py   The only trick I could find was :  echo "arg1 arg2" &gt; my_input.txt &amp;&amp; blender --background --python my_script.py   Is this really the only way to do that?  Update  I found a  on Stack Overflow. Did Emperor Palpatine know about Luke and Leia from the beginning? Did Emperor Palpatine know that Padme had given birth to twins while it was happening?  Was it his long-term goal to have Vader only discover his offspring later, at some "correct" moment? PHPExcel Sumif and Skip how many linearly independed vectors combinations may I extract from a set of vectors Consider the set of $n$ vectors $\mathcal{S}=\{v_1,v_2,\dots,v_n\}$, with $v_i \in \mathbb{R}^n$. Let that set generate a subspace $\mathbb{V}$ of $\mathbb{R}^n$ of dimension $r\le n$. How many different unordered basis of  $\mathbb{V}$ may be constructed by picking up $r$ vectors from $\mathcal{S}$. There is a method to find them? Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$ I have some problems with this exercise. I don't know if it can be done. Consider the polynomial $ x^n - a \in \mathbb{Q} $ Can I compute the Galois group of this over $\mathbb{Q}$? Maybe having a nice "basis.  The splitting field is given by $\mathbb{Q}(\zeta_n,\alpha)$ , where $\zeta_n$ is a primitive root of unity , and $\alpha$ is some number such that $\alpha^n = a $. Well first of all, I want a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different (in this case the computation of the galois group is very simple).  For example one easy case, it's when $a&gt;0$, then $(a)^{\frac{1}{n}} \in \mathbb{R}$ , so clearly the minimal polynomial of $(a)^{\frac{1}{n}} , \zeta_n$ are distincs, and I'm done. If $n$ is odd then , it's also easy, since one root it's also real, for example $x^3-3 $, the real root is $ \root 3 \of { - 3}  =  - \root 3 \of 3  $ , so I can consider the splitting field as $\mathbb{Q}(-\root 3 \of 3 , \zeta_3 )=\mathbb{Q}(\root 3 \of 3 , \zeta_3 )$. The difficult case is when $n$ is even and $a&lt;0$ , for example  $x^8+20$ or $x^4+20$ in some cases as in the second, there are particular cases since there exist algorithms for the Galois group of quartics, but in general. It can be done?  $\underline{Remark}$ I'm searching a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different since in this case the computation of the galois group is very simple. Intuition/Real-life Examples: Pairwise Independence vs (Mutual) Independence Would someone please advance/discuss some real-life situations falsities $1, 2$? I'd like to intuit why these are false. As a neophyte, since I still need to compute the probabilities for the examples in the two answers to apprehend them, I haven't naturalised these concepts yet. Thus, I beg leave to ask about other real-life examples which require less or no computations.  I tried and would appreciate less numerical examples than  and , and  Examples 1.22 and 1.23 on P39 of Introduction to Pr by Bertsekas.  $1.$ Pairwise Independence $\require{cancel} \cancel{\implies}$ (Mutual) Independence.      $2.$ Pairwise Independence $\require{cancel} \cancel{\Longleftarrow}$ (Mutual) Independence.         P38 defines (Mutual) Independence :    For all $S \subseteq \{1, ..., n - 1, n\} $ and events $A_i$, $Pr(\cap_{i \in S} A_i) = \Pi_{i \in S} A_i.$ Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$ I'm trying prove $$\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$$  It is a part of solution of another problem I try a lot to prove it but I could not be succeed.  I tried using exponent manipulation in prime factorizations but it didn't help me. Problem  sudo notify-send Test "Hello World"   Displays a notification as expected.  notify-send Test "Hello World"   Does not display a notification.  Further information  Ubuntu version 16.04.  The notifications appear to use notify-osd instead of notification-daemon. Running notify-send appears to launch a notify-osd process under the user that ran notify-send. I'm not sure what dbus is.   No error messages  There are no errors in the syslog. When I run the following code no error messages occur.   #include &lt;libnotify/notify.h&gt; #include &lt;stdio.h&gt; int main() {     gboolean x = notify_init ("Hello world!");     printf( "notify_init: %d\n", x );     NotifyNotification * Hello = notify_notification_new ("Hello world", "This is an example notification.", "dialog-information");     GError *err = NULL;     x = notify_notification_show (Hello, &amp;err);     printf( "notify_notification_show: %d\n", x );     if(err != NULL) {         printf("Error detected!\n");         printf("Error message:%s\n", err-&gt;message);     }     else {         printf("No error detected.\n");     }     g_object_unref(G_OBJECT(Hello));     notify_uninit();     return 0; }   I do not know what I'm doing. Thanks for the help.  Python Notify  In the past, I've used a python program that has working notifications. I tested a hello world notification and it works without sudo!   #!/usr/bin/env python # -*- coding: utf-8 -*- import gi gi.require_version('Notify', '0.7') from gi.repository import Notify  Notify.init("test") n = Notify.Notification.new('test', 'test2') n.set_urgency(Notify.Urgency.CRITICAL)     n.show()   Code from   I am not sure why this works. The python Notify is just wrapping the library used in the C example. I'd prefer not to use python but will if I must.  Edit  The problem occurred again. Resolved by adding urgency critical. Did the date of Frodo's rescue out of the fire of Mordor have any particular meaning in The Lord of the Rings? Did the date of Frodo's rescue out of the fire of Mordor have any particular meaning in The Lord of the Rings?  We read in The Return of the King (Book 6 Chapter 4) that Gandalf states to Frodo and Sam:     The fourteenth of the New Year, or if you like, the eighth day of April in the Shire reckoning. But in Gondor the New Year will always now begin on the twenty-fifth of March when Sauron fell, and when you were brought out of the fire to the King.   Did Tolkien have any particular reason for choosing this date for Sauron's fall and Frodo's rescue out of the fire of Mordor, either allegorically or otherwise?  We are aware that Tolkien was not a fan of , but preferred the term applicability instead. I was wondering about generic installations of all applications in linux. And what that means? Well, when I was using windows I knew that if I want to install an application I am double-clicking the .exe file and then next,next,next.  In linux, I have understood that maybe there is a common (not generic) way to install any application. Installing from source maybe? Well is there any step by step method that can be used to install  application like in windows or not?  I am asking because I do not want to keep asking the google, how to? So, I have managed to install recently from source freecad from  and I think that it would be a very nice start as common method, right?  But the thing then is where to find the right source and when an application has a very unique method of installation! Difference between @interface definition in .h and .m file At the start of A New Hope are Yoda and Obi-Wan really the last remaining Jedi? I have never read any Star Wars books and have no knowledge outside of the Star Wars films, but after seeing Episode III I am left wondering if there are any other Jedi that escape death and remain in hiding like Yoda did. I mean with regard to the "official" events in the Star Wars universe.