I read in a textbook that the explanation to this rule lies in the fact that division is the reverse operation to multiplication. Unfortunately, the author did not elaborate on this point. Based on this, can someone help me to understand why   $\frac{a}{b}$ / $\frac{c}{d}$ = $\frac{a}{b}$ * $\frac{d}{c}$?
I have looked up various sites online. They only explain it in a very basic arithmetic way. Would someone explain to me why is $\displaystyle\frac{x}{\frac{1}{2}}$ is $\displaystyle\frac{2x}{1}$?
Migrating from Evolution to Thunderbird How to Toggle Selected Objects Visibility with H shortcut as in Maya I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. Thank you very much A random variable that induces a $\sigma$-algebra the same as the one in the sample space What happened to the Yin half of Kurama? Did Naruto get both halves of Kurama after fighting Sasuke, or did the other half of Kurama die when Minato was killed? Electron shells in atoms: What causes them to exist as they do? I have seen similar posts, but I haven't seen what seems to be a clear and direct answer.  Why do only a certain number of electrons occupy each shell? Why are the shells arranged in certain distances from the nucleus? Why don't electrons just collapse into the nucleus or fly away?  It seems there are lots of equations and theories that describe HOW electrons behave (pauli exclusion principle), predictions about WHERE they may be located (Schrödinger equation, uncertainty principle), etc. But hard to find the WHY and/or causality behind these descriptive properties.  What is it about the nucleus and the electrons that causes them to attract/repel in the form of these shells at regular intervals and numbers of electrons per shell?  Please be patient with me, new to this forum and just an amateur fan of physics. is the Turkish version of mancala, with its own set of rules.  I am planning on doing a project in it. Are there any game engines that either specifically play it, or universal Mancala players that plays it after supplying the options?  EDIT:  I am not looking for an app, but for a game engine. For example in chess,  is a playing app, while  is an open-source engine. Putting char into a java string for each N characters I have a java string, which has a variable length.   I need to put the piece "&lt;br&gt;" into the string, say each 10 characters.   For example this is my string:  `this is my string which I need to modify...I love stackoverlow:)`   How can I obtain this string?:  `this is my&lt;br&gt; string wh&lt;br&gt;ich I nee&lt;br&gt;d to modif&lt;br&gt;y...I love&lt;br&gt; stackover&lt;br&gt;flow:)`   Thanks Given $f(x) = \frac{1}{2}x^TAx + b^Tx + \alpha $   where A is an nxn symmetric matrix, b is an n-dimensional vector, and alpha a scalar. Show that  $\bigtriangledown _{x}f(x) = Ax + b$   and  $H = \bigtriangledown ^{2}_{x}f(x) = A$   Is this simply a matter of taking a derivative with respect to X, how would you attack this one? mkdir error: no such file or directory on a linux machine running CENTOS v 7.8.2003 I am in a directory Neutrinos I now do  mkdir /pionloop/ this works. I then go to into this new directory pionloop and do mkdir /E_0.3 and this works as well. I thus have as a result a directory Neutrinos/pionloop/E_0.3.  Now, starting in Neutrinos I want to do this in one command and therefore do mkdir /pionloop/E_0.3 and get: `mkdir: cannot create directory ‘/pionloop/E_3.0’: No such file or directory  What is going wrong here?? Remove blank attributes from an Object in Javascript How do I remove all attributes which are undefined or null in a JavaScript object?  (Question is similar to  for Arrays) How to install Tor? I tried to install Tor on my new version of Ubuntu, but it seems that Tor is not available in the repositories.  How else can I install and use Tor? I am using the below script to move two days back when script runs at starting two days of the year and also check first and second days of every month and move two days back. if [$month=&quot;01&quot;] &amp;&amp; [$day=&quot;01&quot;]; then         date=&quot;$last_month/$yes_day/$last_year&quot;         fulldate=&quot;$last_month/$yes_day/$last_year&quot; else         if [$month=&quot;01&quot;] &amp;&amp; [$day=&quot;02&quot;]; then                 date=&quot;$last_month/$yes_day/$last_year&quot;                 fulldate=&quot;$last_month/$yes_day/$last_year&quot;         else                 if [ $day = &quot;01&quot; ]; then                         date=&quot;$last_month/$yes_day/$year&quot;                         fulldate=&quot;$year$last_month$yes_day&quot;                 else                         if [ $day = &quot;02&quot; ]; then                                 date=&quot;$last_month/$yes_day/$year&quot;                                 fulldate=&quot;$year$last_month$yes_day&quot;                         else                                 date=&quot;$month/$yes_day/$year&quot;                                 fulldate=&quot;$year$month$yes_day&quot;                         fi                 fi         fi fi  But my bad am getting the below error message Etime_script.sh: line 19: [06=01]: command not found Etime_script.sh: line 24: [06=01]: command not found Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $ Let $f:\Bbb R\to (0,\infty)$ be a differentiable function such that for some constant $k$ we have, $$|f'(x)-f'(y)|\le k|x-y|$$   for all $x,y \in\Bbb R.$ Then prove that, $$(f'(x))^2&lt;2 kf(x).$$   Due to lack of arguments I could not proof this inequality. But rather I proved that is true for the particular function,   $$f(x) =\cos^2(x)+1\implies f'(x) = -\sin(2x).$$ And we readily have,$$|\sin(2x)-\sin(2y)| = \left|\int_{2x}^{2y}\cos t dt\right|\le 2|x-y|$$ as well   $$(f'(x))^2 = 4 \sin^2 x\cos^2x &lt; 4(\cos^2x +1) = 4f(x).$$  It is also true for the function $x\mapsto \sin^2x +1.$   From these example I don't see how to prove the general case. Any hint will be welcome. I have an equation contained inside \[...\], which automatically makes a \sum with sub- and superscripts turn big--so that the summation sign looks awkward inside parenthesis.  Any idea how to make the parenthesis completely enclose the whole summation?  \documentclass{article} \begin{document} We have: \[ \sum_{i=1}^n i = (\sum_{i=1}^{n-1} i) + n = \frac{(n-1)(n)}{2} + n = \frac{n(n+1)}{2} \] \end{document} Why align and equation environment do not tolerate empty lines? How do you configure an OpenFileDialog to select folders? In VS .NET, when you are selecting a folder for a project, a dialog that looks like an OpenFileDialog or SaveFileDialog is displayed, but is set up to accept only folders.  Ever since I've seen this I've wanted to know how it's done.  I am aware of the FolderBrowserDialog, but I've never really liked that dialog.  It starts too small and doesn't let me take advantage of being able to type a path.  I'm almost certain by now there's not a way to do this from .NET, but I'm just as curious how you do it from unmanaged code as well.  Short of completely reimplementing the dialog from scratch, how do you modify the dialog to have this behavior?  I'd also like to restate that I am aware of the FolderBrowserDialog but sometimes I don't like to use it, in addition to being genuinely curious how to configure a dialog in this manner.  Telling me to just use the FolderBrowserDialog helps me maintain a consistent UI experience but doesn't satisfy my curiosity so it won't count as an answer.  It's not a Vista-specific thing either; I've been seeing this dialog since VS .NET 2003, so it is doable in Win2k and WinXP.  This is less of a "I want to know the proper way to do this" question and more of a "I have been curious about this since I first wanted to do it in VS 2003" question.  I understand that Vista's file dialog has an option to do this, but it's been working in XP so I know they did something to get it to work.  Vista-specific answers are not answers, because Vista doesn't exist in the question context.  Update: I'm accepting Scott Wisniewski's answer because it comes with a working sample, but I think Serge deserves credit for pointing to the dialog customization (which is admittedly nasty from .NET but it does work) and Mark Ransom for figuring out that MS probably rolled a custom dialog for this task. How to open new tab in browser in selenium 3.0? I have tried multiple actions but it is not working.  It always close the first tab and then it open link in the same page.  // Driver is launched package WebBrowserCommands;  import org.openqa.selenium.By; import org.openqa.selenium.Keys; import org.openqa.selenium.WebDriver; import org.openqa.selenium.chrome.ChromeDriver; import org.openqa.selenium.interactions.Actions;  public class OpenBrowser   {     public static void main(String[] args) throws Exception {         System.setProperty("webdriver.chrome.driver","/Users/ss108/Desktop/Util/SELENIUM SUPPORT/chromedriver");          WebDriver driver = new ChromeDriver();          driver.manage().window().maximize();         // Driver is launched          //We have two ways         boolean Indriectway = true;         //Boolean to understand both method          if(!Indriectway){             //Using String to take value.             String URL = "https://www.google.com";             driver.get(URL);              String PageSourceFirst = driver.getPageSource();             //We have took title in PageSourceFirst             System.out.println(PageSourceFirst);              //Get Title Command             String TitleFrist = driver.getTitle();             //We have took title in TitleFirst             System.out.println("Tile of WebSite is " + TitleFrist);              //Get Current URL             String CurrentURLFirst = driver.getCurrentUrl();             System.out.println("Current URL is " + CurrentURLFirst);              Actions act=new Actions(driver);             act.contextClick(driver.findElement(By.linkText("Gmail"))).perform();             Thread.sleep(10000);             act.contextClick(driver.findElement(By.linkText("Gmail"))).sendKeys(Keys.ARROW_DOWN).sendKeys(Keys.RETURN).build().perform();                    // Try to open page using right click and action arrow down **but it is not working**              driver.close();          }else {             //Using directly             driver.get("https://www.facebook.com");             String PageSourceSecond = driver.getPageSource();             System.out.println(PageSourceSecond);              //Get Title Command             System.out.println("Tile of WebSite is " + driver.getTitle());             //Get Current URL             System.out.println("Current URL is " + driver.getCurrentUrl());             driver.quit();          }     } }   Second method  1- Same result getting   System.setProperty("webdriver.chrome.driver","/Users/ss108/Desktop/Util/SELENIUM SUPPORT/chromedriver");     WebDriver driver = new ChromeDriver();     driver.get("https://www.facebook.com")     Robot robot = new Robot();     robot.keyPress(KeyEvent.VK_META);     robot.keyPress(KeyEvent.VK_T);     robot.keyRelease(KeyEvent.VK_T);     robot.keyRelease(KeyEvent.VK_META);       driver.get("https://www.google.com")   Third method  String selectLinkOpeninNewTab = Keys.chord(Keys.CONTROL,Keys.RETURN);  driver.findElement(By.linkText("urlLink")).sendKeys(selectLinkOpeninNewTab);   however none of them work for what I am trying. How to get Scilab 6.0.1 working on Ubuntu 18.04 LTS? I have installed clean Ubuntu MATE 18.04 LTS and installed Scilab from repository here with    sudo apt-get update sudo apt-get install scilab   After installation I tried to launch it from GUI (Applications → Other → Scilab) - it does not start. Then I tried to launch it from terminal with scilab command. Terminal output is as follows:  $ scilab /usr/bin/scilab-bin: error while loading shared libraries: libjava.so: cannot open shared object file: No such file or directory   so it can't find Java. I reported  about this problem.  I checked existence of libjava.so:  $ dpkg -S libjava.so   openjdk-11-jre-headless:amd64: /usr/lib/jvm/java-11-openjdk-amd64/lib/libjava.so   so it is in place.  What should I do to get Scilab 6.0.1 working on Ubuntu 18.04 LTS? How to reset Kali linux root password