By using set identities, prove that for all sets A and B
Proofs involving $ (A\setminus B) \cup (A \cap B)$
Can trusted users be trusted to suggest tag synonyms without a score of 5 in a tag? (on smaller sites, not Stack Overflow) This has been asked before but I feel the original question was more focussed on Stack Overflow than the Stack Exchange network at large.   Related:   Since there are now a lot of sites in the network and many of them use the tag system for categorising questions based on the title of something (for example, Arqade (Gaming), Movies &amp; TV, Sci Fi &amp; Fantasy, etc) I feel that the required knowledge of the subject matter the tag is representing is less of an issue than it perhaps is on other Stack Exchange sites.   In the below screenshot, for example, I shouldn't really have needed a score of 5 in either of the tags to perform what is essentially a cleanup process. I could of course edit all of the posts to delete one tag, so why can't I suggest it as a synonym?    This could be implemented as a per site setting so that it can only be enabled on certain sites if need be.   Can trusted users be trusted to suggest tag synonyms without a score of 5 in a tag? Find where java class is loaded from Does anyone know how to programmaticly find out where the java classloader actually loads the class from?    I often work on large projects where the classpath gets very long and manual searching is not really an option.  I recently had a  where the classloader was loading an incorrect version of a class because it was on the classpath in two different places.  So how can I get the classloader to tell me where on disk the actual class file is coming from?  Edit: What about if the classloader actually fails to load the class due to a version mismatch (or something else),  is there anyway we could find out what file its trying to read before it reads it? How can it be that, if like charges repel, they don't repel themselves? In other words, why don't charges break apart?  About the possible duplicate: I want to know about charges in general, not just that of an electron. My response to :  Thank you very much for the extensive explanation. Unfortunately it's a bit above my level (I'm a first year electrical engineering bachelor student). This is how I understand it: If the charge is the sum of multiple separate charges, there has to be an external force that keeps this charges together. However we most often think of electrons and protons as point charges. In other words, we don't think of them as if they are made up from different (smaller) parts. But the idea of electrons and protons as point charges has its own problems. I didn't know that the electric field has a mass. From the formula, I understand that $m\propto \frac{1}{r}$. This would mean that $m\rightarrow \infty$ as $r \rightarrow 0$ (or $r = 0$), which of course is physically not possible. But there is a way to calculate the radius of an electron based on the mass, the speed of light and the constant of Planck. (Unfortunately I don't know why.) You can get around this by a technique called called renormalization, which causes the integral to converge. Unfortunately, I don't understand the rest of the answer due to my level in physics. Nevertheless I am grateful for you answer. It would be awesome if you could confirm or correct my understanding. Omitting 'that' in this sentence How to get Intel Dual Band Wireless-AC 3168 work on Ubuntu 16.04? I searched many solutions on this site but most of them were referring to Broadcom Wifi drivers while mine has Intel one.   Output of lspci -knn | grep Net -A2  arijit@HP-AU134TX:~$ lspci -knn | grep Net -A2  05:00.0 Network controller [0280]: Intel Corporation Device [8086:24fb] (rev 10) DeviceName: Intel Dual BandWireless-AC 3168 802.11 ac 1x1 WiFi + BT 4.0 Combo Adapter Subsystem: Intel Corporation Device [8086:2110]   Output of rfkill list  arijit@HP-AU134TX:~$ rfkill list  arijit@HP-AU134TX:~$    i.e No Output  Output of uname -a; dmesg | grep iwl  arijit@HP-AU134TX:~$ uname -a; dmesg | grep iwl  Linux HP-AU134TX 4.4.0-31-generic #50-Ubuntu SMP Wed Jul 13 00:07:12 UTC 2016 x86_64 x86_64 x86_64 GNU/Linux Are there any age limits (formal, informal, or guidelines) that schools use when deciding to accept someone into a PhD program? I'm most curious about the upper age limits. For example, will most schools accept someone in their 40's? How about their 50's? I've asked several questions about 4th Edition D&amp;D because I may be able to start a game. However, I'd need to purchase (some) materials. I am aware of the 4 book set (core) and the expected Essentials Red Box.  So now I need to know: presumably buying either the 4 book set or essentials will be enough to get my game on. Would there be something 'lacking' though? Would I be missing character types, for instance, as I've seen that the splatbook strategy is still in effect?  I'm an experienced player and GM and plenty creative, so many holes I'd be able to fill in myself. I just don't want to get sucked into too much financial impact. What is a good complex analysis textbook, barring Ahlfors's? I'm out of college, and trying to self-learn complex analysis. I'm finding  text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.) Prove that $\left | \frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a} \right | < \frac{1}{8}.$ Let $a,b,$ and $c$ be the lengths of the sides of a triangle. Prove that $$\left | \dfrac{a-b}{a+b}+\dfrac{b-c}{b+c}+\dfrac{c-a}{c+a} \right | &lt; \dfrac{1}{8}.$$   The best idea I had was to expand the fractions to get something nicer. So we get $\dfrac{a-b}{a+b}+\dfrac{b-c}{b+c}+\dfrac{c-a}{c+a} = \dfrac{a^2 b-a^2 c-a b^2+a c^2+b^2 c-b c^2}{(a+b) (a+c) (b+c)}.$ Then I would try to relate this to side lengths of a triangle to prove the desired inequality but that is the step where I am unsure. Is voltage the speed of electrons? Why is potential energy negative when orbiting in a gravitational field? I have to deal with a text classification problem. A web crawler crawls webpages of a certain domain and for each webpage I want to find out whether it belongs to only one specific class or not. That is, if I call this class Positive, each crawled webpage belongs either to class Positive or to class Non-Positive.  I already have a large training set of webpages for class Positive. But how to create a training set for class Non-Positive which is as representative as possible? I mean, I could basically use each and everything for that class. Can I just collect some arbitrary pages that definitely do not belong to class Positive? I'm sure the performance of a text classification algorithm (I prefer to make use of a Naive Bayes algorithm) highly depends on which webpages I choose for class Non-Positive.  So what shall I do? Can somebody please give me an advice? Thank you very much! Splash Screen waiting until thread finishes I still have a problem with the splash screen. I don't want to use the property SC.TopMost=true.  Now my application scenario is as follows:  in progeram.cs:  [STAThread] static void Main() {     new SplashScreen(_tempAL);// where _tempAL is an arrayList     Application.Run(new Form1(_tempAL)); }   in SplashScreen class:  public SplashScreen(ArrayList _Data) {     DisplaySplash() }  private void DisplaySplash() {     this.Show();     this.TopMost = true;     this.CenterToScreen();     this.SetTopLevel(true);      _allServerNarrators = new string[10];     for (int i = 0; i &lt; _allServerNarrators.Length; i++)         _allServerNarrators[i] = null;      GetFromServer();      this.Hide();     _serverData = new ArrayList();     _thisData.Add(_allServerNarrators);     _thisData.Add(_serverNarrators);  } private void GetFromServer() {     _serverNarrators = new ArrayList();     string _file = "Suras.serverNar";      if (!Directory.Exists("c:\\ASGAQuraan"))         Directory.CreateDirectory("c:\\ASGAQuraan");      while (counter &lt; 4 &amp;&amp; _serverFiles == null)     {         if (Download("c:\\ASGAQuraan", _ftpServerIP, _file))         {             StreamReader _strReader = new StreamReader                          ("c:\\ASGAQuraan\\"+_file,System.Text.Encoding.Default);             string _line = _strReader.ReadLine();             string _word;              while (true)             {                 while (_line != null)                 {                     _word = _line.Substring(0, _line.IndexOf("*"));                     int _narId = Convert.ToInt32(_word);                     _line = _line.Substring(2);                     int k = 0;                     _serverNarratorNode = new ArrayList();                     while (true)                     {                         int ind = _line.IndexOf("*");                         if (ind &gt; 0 &amp;&amp; ind &lt; _line.Length)                         {                             string str = _line.Substring(0, (ind));                             if (k == 0)                             {                                 _allServerNarrators[_narId] = str;                                 _serverNarratorNode.Add(str);                             }                             else                             {                                 _serverNarratorNode.Add(str);                             }                             _line = _line.Substring(ind + 1);                             k++;                         }                         else                         {                             _line = null;                             break;                         }                     }                     _serverNarrators.Add(_serverNarratorNode);                     _serverFiles = "added";                 }                 _line = _strReader.ReadLine();                 if (_line == null)                 {                     break;                 }             }         }         else             counter++;     } }   What I want is something in the splash screen class which waits until the thread finishes.  For more details, please tell me what I need to tell you. counting triangles in a graph or its complement Given a simple graph $G$ [no loops or parallel edges] on six vertices,  let $G^c$ denote its complement. It is known that either $G$ or $G^c$ must contain a triangle  $T$ in it. [An example of a Ramsey number I think.]   My question came up when I tried to find such a graph on six vertices  which had only one $T$ in it or its complement.  I believe if I did things right there are none of these. This I tried to check by drawing a copy of each graph $G$  on six vertices which has a $T$ in it,  and then looking at the complements to see if any had  no $T$'s in them. I found none.  One way to pose my question for a general number $n$ of vertices  is then as follows: Consider the set of all graphs $G$ on $n$ vertices,  and for each one construct $G^c.$  Count the number of $T$ in $G$ and add that to the number of $T$ in $G^c.$  Finally find the minimal such sum for the given $n$  and call that say $M(n).$ So my above six vertex conjecture  is that $M(6)=2.$   By considering a cycle graph on five vertices we find $M(5)=0.$   So going further one could ask if anything is known  about this minimum function $M(n)$ for larger $n$ [such as bounds, etc.] And does it have a name?  I would also like to see a simpler way to handle the six vertex case, one not involving drawing all the graphs. Update SQL Server table with local data Covariance between sum of iid random variables and sum of indicator functions The question is-     Let $X_1,X_2,..,X_n$ be iid random variables from a continuous distribution whose density is symmetric about $0$. Suppose $\mathbb{E}(|X_1|)=2$ and define $Y=\sum_{i=1}^{n}X_i$ and $Z=\sum_{i=1}^{n}I(X_i&gt;0)$. Then calculate covariance between $Y$ and $Z$.   My attempt:  $E(X_i)=0$ for all $i=1(1)n$ because $X$ is symmetric about $0$ and $E(|X|) $ exists.   Now,  $Cov (Y,Z)=E(YZ)-E(Y)E(Z)$ $=E(YZ)-0$ $=E[(\sum_{i=1}^{n}X_i)(\sum_{i=1}^{n}I(X_i&gt;0)]$  $=(\sum_{i=1}^{n}E[(X_i.I(X_i&gt;0))]$ $+\sum\sum_{i \neq j}E(X_i)E(I(X_j&gt;0)$ as $X_i,X_j$ are independent.   $=\sum_{i=1}^{n}E[(X_i.I(X_i&gt;0)] +0 $ as $E(X_i)=0$  $ =\sum_{i=1}^{n}\{E[X_i.I(X_i&gt;0)|I(X_i&gt;0)=1]×1/2] + E[X_i.I(X_i&gt;0)|I(X_i&gt;0)=0]×1/2]\}$  $=\sum_{i=1}^{n}E[X_i.I(X_i&gt;0)|I(X_i&gt;0)=1]×1/2] +0$  $=\sum_{i=1}^{n}E[X_i|X_i&gt;0]×1/2]$  $=2n×(1/2)$ $=n$  Is my reasoning correct ? Thanks in advance! I get the following error:     Strict warning: Only variables should be passed by reference in include() (line 18 of /home/sites/dev/theparce/sites/all/themes/parce/block--block--3.tpl.php).   This is the block code which is causing that error.  if ($user_gallery) {   print render(node_show($user_gallery));  // Line 18   print drupal_render ($user_gallery_edit);  } else {   print drupal_render($user_gallery_new); }   Why do I get that error, even if I get all printed as expected? How to prove that $\sum_{r=0}^n\binom{n}{r}2^r=3^n$ Why is the (non-relativistic) stress tensor linear and symmetric? From :     "[...] the stress vector $T$ across a surface will always be a linear function of the surface's normal vector $n$, the unit-length vector that is perpendicular to it. [...] The linear relation between $T$ and $n$ follows from the fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy’s equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric."   What is the more detailed explanation for this, and/or where can I find it? Copying protected files between servers in one line? I'd like to copy squid.conf from one server to another.   The servers don't talk to each other. I'd like to go through my workstation. Both servers have the file, so it will be overwritten on the target. The files have 600 permission and are owned by root. root login via ssh is disabled (PermitRootLogin no). I'd like to do it in one line, if possible, since it will be a part of a setup guide.   I know to do  ssh source 'tar czpf - -C /etc/squid/ squid.conf' | \     ssh target 'tar xzpf - -C /etc/squid/'   to copy files between servers and preserve permissions. However, in this case I will get "Permission denied".  I also know I can do this:  ssh -t source 'sudo cat /etc/squid/squid.conf'   This way the -t allows sudo to ask for the admin password before outputing the content of the file.  The problem is, I don't know how to combine those techniques into something that will ask for the sudo password on each server, and transfer the file to its destination. Is this possible?  UPDATE: Here's the best I could come up with:  ssh -t source 'sudo tar czf /tmp/squid.tgz -C /etc/squid squid.conf' &amp;&amp; \ ssh source 'cat /tmp/squid.tgz' | \     ssh target 'cat &gt;/tmp/squid.tgz' &amp;&amp; \ ssh -t source 'sudo rm /tmp/squid.tgz' &amp;&amp; \ ssh -t target \     'sudo tar xzf /tmp/squid.tgz -C /etc/squid &amp;&amp; sudo rm /tmp/squid.tgz'   Calling this a one-liner seems like a stretch. I think I'll just break it down to separate steps in the setup guide.