I was curious to figure out what the purpose of using &quot;~&quot; operator in Latex is. I noticed the following piece of code which stated: Figure~\ref{fig:x} However, if you had to, for example, write the following segment: Figure \ref{fig:x} the exact same output is obtained.
As title, what is the difference between a   and ~?  I see no differences after the compilation.  MWE:  \documentclass[a4paper]{article}  \usepackage[english]{babel} \usepackage[utf8]{inputenc} \usepackage{amsmath} \usepackage{graphicx} \usepackage[colorinlistoftodos]{todonotes}  \title{Your Paper}  \author{You}  \date{\today}  \begin{document} \maketitle  I am cool.  I~am~cool.  \end{document}
How can I avoid "please insert CD/DVD" error when updating Ubuntu? I can't update 12.04 LTS. The Update Manager shows this:  CD/DVD 'Ubuntu 12.04 LTS _Precise Pangolin_ - Release i386 (20120423)' is required Please insert the above CD/DVD into the drive '/media/cdrom/' to install software packages from it.   The image I downloaded and burned to CD is from  the PC (Intel x86) desktop CD version. I burned many CDs but the result is the same. Who built the fortress of Dol Goldur, and when was it built? And was it always a place for evil, or Sauron just occupied it?  Despite the fact that Mordor is much better fortified; artificially by the old fortifications,  and geographically, as it is surrounded by the mountains. Also considering the presence of mount doom that seem to enhance Sauron's power, Sauron chose to start his campaign from Dol Goldur, despite Mordor being ready for quite some time.  Why did he choose this enigmatic fortress, and not go straight for Mordor or even go earlier? Is logical implication always determinable from just the given statements? I am reading introductory texts on logic and am having a hard time understanding intuitively logical implication.  Specifically, I am wondering if logical implication can always be determined by two statements, P and Q alone? I'll try to give an example to clarify my question.  For example if the statement P is $x = 2$, and the statement Q is $x^2 &lt; 6$, then determining the truth of P $\Rightarrow$ Q is fairly straightforward:   If $x$ does equal 2, then $4&lt;6$ and Q is true, so $P \Rightarrow Q$ is true. And if $x$ does not equal 2, $P \Rightarrow Q$ is vacuously true.  In this example, with two statements P and Q, I can evaluate the truth/falsity of the statement P implies Q. However, what if the statement Q is changed such that it is now the statement $y = 5$?  So now I am trying to prove from that $$x = 2 \Rightarrow y = 5$$  Here is where things get murky for me. Say $x$ is not equal to 2, then $P \Rightarrow Q$ is vacuously true. But what about when $x$ does equal 2, now the statement Q could be true, but it could also be false, depending on some 'context'.   For example, if I said at the start: "Here are two statements, P and Q. Know that x = 2 and y = 4." Given this 'context', $P \Rightarrow Q$ is false. But if I say "Here are two statements, P and Q. Know that x = 2 and y = 5", then now $P \Rightarrow Q$ is true.  My discomfort here is that I assumed with two statements P and Q, I can determine the truth/falsity of $P \Rightarrow Q$ with just this information. Is this the wrong assumption?  As a secondary question, how can the truthfulness of the statement $P \Rightarrow Q$ be variable, depending on context? I have read on some other questions that implication can be interpreted as a promise. So how could it be that depending on some context the promise is sometimes true but sometimes false? Wouldn't this actually mean the promise is not always consistent and is therefore always false?  Some further information in case it is relevant, I recognise that in the first example, the statement Q is 'dependent' in some way on the statement P, since the object under examination in both statements is $x$, whereas in the second example there are two different objects which I am trying to make a logical connection between, which I am thinking may not be valid for some reason. I also recognise that there is still 'context' in the first example, in that we need information on whether $x$ is equal to 2 or not, however in that case it doesn't matter, since $P \Rightarrow Q$ is always true anyway.  Thanks very much in advance. How long does it take for a black hole to form for an external observer? The well-known fable of an astronaut sending signals out to an external observer while falling toward an event horizon states that the time lapse between such signals becomes greater even if in the astronaut sends them out periodically (as judged in his inertial frame). When viewed from earth and weighing time-dilation due to the gravitational field of the collapsing star, how is it possible that a black hole can form in finite time (for any external observer) if it takes an infinite amount of time to "see" events occurring at the event horizon? Ted Shifrin on this site posed an interesting problem to me: show that   $$ \lim_{(x,y) \to (0,0)} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \frac\alpha\gamma + \frac\beta\delta &gt; 1, \,\,\,\,\text{where } \alpha, \beta, \gamma, \delta &gt;0$$  I think I've got the $(\Leftarrow)$ direction as follows: assume WLOG that $\gamma \leq \delta$. Then switch to polar coordinates and get   $$\frac {r^ {\alpha + \beta} (*) } {r^\gamma (1(*) + r^{\delta - \gamma}(*))},$$  where $(*)$ represents some trig stuff that is bounded near zero.   Edit: I need to make sure that $(1(*) + r^{\delta - \gamma}(*))$ is bounded below here.  The other direction is giving me some trouble. It seems we need a lower bound for the fraction (something to force to zero), and I'm not sure where to find one. In particular, I'm not sure what to do with the denominator. Any ideas? Navigation page buttons are orange everywhere On a similar note to this question:  (which only covers two examples only on the profile page, this question mentions more places on and outside the profile page where this bug occurs).  I've noticed that whenever there are these page buttons now, for example, the profile page, /questions and tag pages they are orange.  Per the other post I'm assuming this is a bug but if it was intentional then this is a feature request to make them theme-able per site again. Dobble card game - mathematical background In  there is a deck of 55 cards. Each of them has 8 symbols on it. There is a total of 50 different symbols and each two cards have one and only one in common.  I am wondering, given a total number of symbols N and a number of symbols on each card K, how do you find the biggest number of cards such that:   on each of those cards there are exactly K of the N symbols, every two cards have one and only one symbol in common. Is it necessary to dispose System.Timers.Timer if you use one in your application? I am using System.Timers.Timer class in one of the classes in my application. I know that Timer class has Dispose method inherited from the parent Component class that implements IDisposable interface. Instances of the class below are created many times during my application lifecycle; each of them has an instance of Timer class that generates Elapsed events continuously during the class's lifecycle. Should I implement IDisposable interface in the class that uses Timer class to dispose the timer object? (I have seen code that doesn't do this at all). I am afraid that some unmanaged resources will not be freed if I use the class below like this:  SomeClass someClass = new SomeClass(); someClass.DoSomething(); someClass = null;   The class:  using System.Timers;  public class SomeClass {     private Timer m_timer;      public SomeClass()     {                    m_timer = new Timer();         m_timer.Interval = 1000;         m_timer.Elapsed += new ElapsedEventHandler(m_timer_Elapsed);         m_timer.AutoReset = false;         m_timer.Start();                            }      public void DoSomething()     {      }      private void m_timer_Elapsed(object sender, ElapsedEventArgs e)     {         try         {             //Do some task         }         catch (Exception ex)         {             //Ignore         }         finally         {             if (m_timer != null)             {                 //Restart the timer                 m_timer.Enabled = true;             }         }     } } Using right-hand Riemann sum to evaluate the limit of $ \frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$ I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, so I proceed:  \begin{align*} \lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)&amp;=\lim_{n \to \infty} \sum_{k=1}^{n}\frac{n}{n^2+k^2}\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{n^2}{n^2+k^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{1}{1+(k/n)^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}f(\frac{k}{n})(\frac{k-(k-1)}{n})\\ &amp;=\int_{0}^{1}\frac{1}{1+x^2}dx=\frac{\pi}{4} \end{align*}  where $f(x)=\frac{1}{1+x^2}$. Is this correct, are there any steps where I am not clear? Simple way to transpose columns and rows in SQL? How does JavaScript .prototype work? I'm not that into dynamic programming languages but I've written my fair share of JavaScript code. I never really got my head around this prototype-based programming, does any one know how this works? var obj = new Object(); obj.prototype.test = function() { alert('Hello?'); }; var obj2 = new obj(); obj2.test();  I remember a lot discussion I had with people a while back (I'm not exactly sure what I'm doing) but as I understand it, there's no concept of a class. It's just an object, and instances of those objects are clones of the original, right? But what is the exact purpose of this &quot;.prototype&quot; property in JavaScript? How does it relate to instantiating objects? Update: correct way var obj = new Object(); // not a functional object obj.prototype.test = function() { alert('Hello?'); }; // this is wrong!  function MyObject() {} // a first class functional object MyObject.prototype.test = function() { alert('OK'); } // OK  Also these  really helped a lot. (How) Can SE sites change their reputation payout? A recent question on  reminded me that there is a  about awarding 10 rep per question upvote. This is mainly due to the relative difficulty and usefulness of asking/answering being different on the site (or appearing to be, at least).  The highly voted answer on PPCG is "Yes, we should do that," but it's been well over three years since it was posted. The answer there said that (at least at that time) meta.so had this implemented already, so it wouldn't/shouldn't be hard to do.  Given that it's been quite some time, I don't know if that's (still) true. I know that meat has undergone changes since then, but as long as I've been actively participating, meta.se (and meta.so before the split) has given 5 points per question upvote, not 10.  My question is: What would the process to change it look like? It's been discussed on meta.ppcg and agreed upon, so what's the next step? Or is it not feasible at all? How can I use multiple accounts with Telegram? Is there a way to have two copies of  app in Ubuntu each with a different account?  It seems copying the app in another folder does not work. Is there a word for colloquial forms of address? For example, "dude," "man," "buddy," "pal," etc, when used to stand in for someone's name.  "Hey, pal, how's it going?"  Is there a word for terms like these?  Or is "colloquialism" as close as we can get? How do I show for an ideal transformer $M^2=L_1L_2$? I've been stuck on this problem for about an hour.     In an ideal transformer, the same flux passes through all turns of the primary and of the secondary . Show that in this case $M^2=L_1L_2$, where $M$ is the mutual inductance of the coils, and $L_1$, $L_2$ are their individual self-inductances.   My first instinct was the following: If we isolate the response of each coil to the other one, we get the equations $N_1\Phi_1=MI_2$ and $N_2\Phi_2=MI_1$, where $\Phi$ is magnetic flux going through a single turn of the coil and $I$ is current. Likewise, if we isolate the response of each coil to itself, we get the equations $N_1\Phi_1=L_1I_1$ and $N_2\Phi_2=L_2I_2$. I would now naturally want to say "Now just multiply $N_1\Phi_1$ and $N_2\Phi_2$ both ways, and get the desired result!" i.e.,  $$(N_1\Phi_1)(N_2\Phi_2)=(MI_2)(MI_1)=(L_1I_1)(L_2I_2)\implies M^2=L_1L_2$$  But that's incorrect, because the response of the coils to themselves and to the other coil are different quantities (i.e. not necessarily equal).  Realizing this, I write down the more general equations.  $$N_1\Phi_1=MI_2+L_1I_1$$ $$N_2\Phi_2=MI_1+L_2I_2$$  Now even with the hypothesis that $\Phi_1=\Phi_2=\Phi$, I still can't see how the identity $M^2=L_1L_2$ follows. Could you help clear up my confusion?  *Taken from D. Griffiths Introduction to Electrodynamics (4th ed.), Problem 7.58. I know people love to close duplicates so I am not asking for a reference to start learning statistics (as ).   I have a doctorate in mathematics but never learned statistics. What is the shortest route to the equivalent knowledge to a top notch BS statistics degree and how do I measure when I have achieved that.   If a list of books would suffice (assuming I do the exercises lets say), that's terrific. Yes, I expect working out problems to be an implicit part of learning it but I want to fast track as much as realistically possible. I am not looking for an insanely rigorous treatment unless that is part of what statistical majors generally learn. I'm stuck in a teleporting loop. What can I do? Activate physics by keyframe I'd like to have a group of debris fall from a ceiling onto the floor, but not have them drop all at the same time. Is there a way to set a keyframe that dictates when the object enters the simulation? For example, object 1 starts falling on frame 0, object 2 on frame 5, object 3 on frame 24, etc. When the site is in read-only mode, I get a message like this:     This site is currently in read-only mode.   Would it be possible to update that message to provide links to these resources on SE's status:   : Official SE tweets on SE's status. : Official blog posts on SE's status. Calculating large Bell number modulo a composite number