Finding that a two variable limit does not exist trouble: $\lim_{(x, y) \to (0, 0)} \frac{y^2 \sin^2 x}{x^4 + y^4}$
How to evaluate $ \lim_{(x,y)\to(0,0)}\frac{y^2\sin^2x}{x^4+y^4} $
Preserving order with LINQ I use LINQ to Objects instructions on an ordered array.  Which operations shouldn't I do to be sure the order of the array is not changed? Does a wing in a potential flow have lift? Does there exists an Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition.  Attempt In my opinion It does not exists.  If there exists such an isomorphism say $\phi :\mathbb{C}^* \to \mathbb{C}$ then, from the fact that Isomorphism takes identity to identity we have $$\phi \left( 1 \right)=0$$  From the fact      Given $\phi:G \to H$ be an isomorphism, then For a fixed integer $k$ and a fixed group element $b$ in $G$, the   equation $x^k=b$ has the same number of solutions in $G$ as does   the equation $x^k=\phi\left(b\right)$ in $H$.   We have $x^4=1$ and $x^4=0$ must have same number of solutions, but the solutions to these equations are 4 and 1 only. Why doesn't Spotlight show invisible files that I know exist? I am running Mac OS X 10.8 (Mountain Lion) and had to edit an invisible file as part of an Android project. While I eventually found it, Spotlight was of no help, since it could not find an invisible file when searching for the exact name (.classpath) and with the 'File Visibility' option set to 'Visible or Invisible'.  Why didn't Spotlight work in this instance? Why are most of the main characters White/English in Game Of Thrones? Okay, I was thinking about the series of Game of Thrones and all the battles and stuff, and I thought why aren't there any characters beside White/English characters that have main/important roles? Can you name any that are alive that are anything else beside White/English?   This is a short list of some of the main characters (all White/English):   Cersei - Queen and important political Lannister. Jaime - Guy who is father to the late king and is King slayer. Tyrion - Short guy in politics who is currently on the run(that's what the wikia told me). Daenerys - Queen who is freeing people and dragon owner.   And the list goes on and on, why aren't there any characters other than Caucasian characters that are in the series?  I've read a lot of articles on the internet about this after Googling around, and none of the articles I read had any scientific evidence or research - it was just conjecture.  So, are there any actors/characters from racially diverse backgrounds? I'm hoping for canon evidence (pictures, links, quotes) from the book and/or TV show. In the case of actors, can you also list their role?    This is not intended as a racist question. I just want to know why, since the story takes place over a large area, there doesn't seem to be any diversity. Multiple monitors, multiple workspaces? I connected my laptop to my LCD. Using the laptop display hotkey I can switch between different modes - only laptop monitor on, only LCD monitor on, both on and display the same view, both on and the desktop is stretched (divided).  But, I'm looking for something else - I would like to have two workspaces, one visible on the laptop monitor and the other on the LCD.  Is that possible? What are the TeX definitions for the hyphen and dashes -, --, and ---? I've tried latexdef which does not handle --- as the operand. And \texemdash is defined as a character. So that leaves me once again leaning on the collective wisdom of the group :) The square of an integer is congruent to 0 or 1 mod 4 This is a question from the free Harvard online abstract algebra .  I'm posting my solutions here to get some feedback on them.  For a fuller explanation, see  post.  This problem is from assignment 6.  The notes from this lecture can be found .     a) Prove that the square $a^2$ of an integer $a$ is congruent to 0 or 1 modulo 4.   b) What are the possible values of $a^2$ modulo 8?   a) Let $a$ be an integer. Then $a=4q+r, 0\leq r&lt;4$ with $\bar{a}=\bar{r}$.  Then we have $a^2=a\cdot a=(4q+r)^2=16q^2+8qr+r^2=4(4q^2+2qr)+r^2, 0\leq r^2&lt;4$ with $\bar{a^2}=\bar{r^2}$.  So then the possible values for $r$ with $r^2&lt;4$ are 0,1.  Then $\bar{a^2}=\bar{0}$ or $\bar{1}$.  b) Let $a$ be an integer. Then $a=8q+r, 0\leq r&lt;8$ with $\bar{a}=\bar{r}$.  Then we have $a^2=a\cdot a=(8q+r)^2=64q^2+16qr+r^2=8(8q^2+2qr)+r^2, 0\leq r^2&lt;8$ with $\bar{a^2}=\bar{r^2}$.  So then the possible values for $r$ with $r^2&lt;8$ are 0,1,and 2.  Then $\bar{a^2}=\bar{0}$, $\bar{1}$ or $\bar{4}$.  Again, I welcome any critique of my reasoning and/or my style as well as alternative solutions to the problem.  Thanks. An Olympiad problem in prime numbers :can we proof that "n" is prime in this case " Let x,n ∈ $\mathbb{N}$  such that :  1+$x^1$+$x^2$+$x^3$+....+$x^{n-1}$ is prime .  Proof that n is prime .  I actually suppose that n isn't prime absurdly so n=p×k such as p is prime ,I also use the $a^n$-$b^n$ formula  and find that $x^k$-1/$x^1$-1 so that means that k=1 and then  n=p which is absurd. Does Anyone has another answer ? I would like to redirect a URL with uppercase characters http://localhost/A/B to a lowercase version http://localhost/a/b using the .htaccess file. Using regex I can capture A and B but is it possible to convert them to lowercase? I'm suppose to prove the following...  Let $A$ be a nonempty subset of $\mathbb{R}$. If $\alpha = \sup A $ is finite show that for each $ \epsilon &gt; 0 $ there is an $a \in A $ such that $\alpha - \epsilon &lt; a\leq \alpha $.  This is what I have so far...  Proof: We claim that $\alpha = \sup A $   Then by definition of a supremum $\alpha $ is the least upper bound of set A  Thus $\forall a \in A$,  $a \leq \alpha$   Suppose $\epsilon &gt; 0$   Then $\alpha - \epsilon &lt; \alpha $.   Thus $\alpha - \epsilon &lt; a\leq \alpha$.   I feel like I'm missing some step between my last and second to last step so what am I forgetting? Also is the rest of my proof right? How can I find out the variable name in python? I know how I can find out the python command by clicking a button in the Blender interface, and then look in the info window:   For example, as can be seen, I added a cube, UV sphere, plane, camera, lamp, pressed A twice.  But how can I find out an object variable name, for example the y-rotation of an object. Because I can't find the y-rotation, when I point my mouse over it, I can only find the rotation: How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$?  Why isn't there other possibility?  Thanks :) Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$? I wonder  whether there is a function $f\colon\Bbb R\to\Bbb R$ with the folowing characteristic?   for every two real numbers $\alpha,\beta,\alpha\lt\beta$,  $$\{f(x):x\in(\alpha,\beta)\}=\Bbb R$$   I can't  say such a function does not exist, neither can I construct a example  Thanks a lot! Why can I use a function before it's defined in JavaScript? We know that wizards in the Harry Potter universe are subject to regular physical harm (e.g. witness injuries from being hit by Bludgers or just colliding with things when playing Quidditch).  Therefore, it's almost a certainty that an average wizard would be vulnerable to, say, a bunch of bullets.   There may likely be spell(s) to protect from projectiles - witness Dumbledore's  deployed when Voldemort sent a bunch of glass at him during the  at the end of Harry Potter and the Order of the Phoenix (according to , possibly an unusually powerful Protego or the Silver shield).   But:   those spells obviously take effort/energy/time to cast (not everyone's Dumbledore) AND more likely than not most Death Eaters wouldn't recognize a Muggle gun for a threat (or realize what kind of threat) till too late.   So, the guns should at best allow an over-matched OotP member (e.g. Harry or Hermione) to greatly equalize the power between any regular Death Eaters and themselves, and even perhaps help against Voldemort (who can't be killed with a bullet, but, at best, would need to expend magical energy and time into conjuring a shield, and, at worst, be stopped/interrupted/driven off).  So I'm looking for an explanation - ideally in-universe, but possibly just some statement by JKR - of why the Muggle-born OotP - who are likely very much at ease AND familiar with Muggle technology and live in the late 20th century - do not use guns. Or, for that matter, ANY Muggle technology of the late 20th century? Cue Arthur C. Clarke's "indistinguishable from magic" meme.  Even if they are on some kind of idiotic moral crusade to never kill their opponents (even the Jedi aren't this dumb), given the time-frame, non-lethal weapons would have already existed, such as rubber bullets and tasers.  NOTE: Please don't offer the "if they do it, Death Eaters would start using guns as well" theory. First, Death Eaters would have used the guns if they knew how to, anyway. Second, the guns, like any "secret weapon", could be reserved for strategic battles, say the defense of Hogwarts in Deathly Hallows.    The origin of the question stems from two things.  First, Harry's fascination with all things that are different in the magical world (e.g. moving photographs), based on his knowledge of the Muggle material world, as well as the somewhat amusing cluelessness of the "Muggle artifacts" expert, Mr. Weasley, regarding Muggle items and how they work.   Basically, from the books, it's clear that the Wizarding world wouldn't know details about guns or what they are for or how they work.  The second thing is it's clear, at least initially, that wizards are afraid of Muggles finding out about them. Hence all the efforts to cloak their society (basically, do you REALLY want every Muggle gunning for you, even with Magic)? The idea of how control over the Muggle world seems to be more indirect, by Death Eaters asserting mind-control over the government. This means there's an implicit understanding by Death Eaters that they don't stand a chance against armed Muggles even given the magical disparity.  Considering Harry and Hermione's overall adaptability and inventiveness (and brains in the latter case), it's mind-boggling that the idea of fighting Death Eaters with modern technology completely slipped their mind. I mean, it's a basic part of human myth (see Steel vs. Magic themes in Conan books, or Beowulf, or heck, A Yankee in King Arthur's Court). So it's not like the idea of using technology against magic would be impossible to come up with. The question begs itself - why not? Considering the fact that Rowling generally tried to be logical and consistent in building the HP universe, I feel like there must be some in-universe explanation for this that I just didn't notice.   (in-universe answers only unless there are some authoritative out-of-universe statements that are part of canon). Looking for a short story about a man who is visiting a planet where you get your position/job through playing a game I read this story 8-15 years ago in German. I'm not sure but I think the book had 2 short, totally different stories in it.  The story was about a man who was sent (I think kinda forced to go) to a planet where he should play a game everyone on that planet plays. I think the goal was to beat their leader and conquer the world like this?  The better you are at the game, the higher is your position in the society. I remember that at the end he was playing against the leader and they had a long back and forth game, but I'm not sure who won.  They played the game on a different planet that had only one stripe of forest around the whole planet and a fire that goes around it. But thats only the location they were playing.  The man was chosen because he was the best player in every game if I remember correctly.  The game was a board like game where you have troops or armies that you move around. It was not an easy game everyone just could play. Limit Decimal Places in Android EditText I want to put some symbols in green color with lstdefinelanguage in latex, but it seems that symbols are not recognized.  The symbols I need to be colored are =, > and &lt;. Is it possible? How can I do this?  The main file:  \documentclass{article}   \usepackage{graphicx,color} \usepackage{booktabs} \usepackage{listings}   \lstdefinelanguage{NeoIDL}{   sensitive = true,   keywords = [1]{module, resource, enum, annotation, for, import, entity, path,   @get, @post, @put, @delete, require, ensure, otherwise, call},   %ndkeywords={int, \char{}  },   %ndkeywordstyle=\color{blue}\bfseries,   morekeywords=[2]{&gt;, &lt;, ==},   keywordstyle=[2]\color{green},   numbers=left,   numberstyle=\scriptsize,   stepnumber=1,   numbersep=8pt,   showstringspaces=false,   breaklines=true,   frame=top,   comment=[l]{//},   morecomment=[s]{/*}{*/},   commentstyle=\color{purple}\ttfamily,   stringstyle=\color{red}\ttfamily,   morestring=[b]',   morestring=[b]"   }  \begin{document}  \begin{figure}[htb] \begin{small} \lstinputlisting[language=NeoIDL,firstnumber=1]{store_pos_servico.neo} \end{small} \caption{Service 1} \end{figure}   \end{document}   The file to be formatted:  module store {    (...)     resource order  {      path = "/order/{id}";       @post   int postOrder (Order order)     require (call store.getOrder(id) == "NotFound"),     otherwise "InvalidPrecondition"    }; } How to cite sources written in a different language than English? Assume one wants to cite a paper written in a different language than the article citing said paper (e.g., ones own article is written in English, but the source is French).   How to properly handle citation in such a case?  Do I cite the original (French) title or a translated title? If I use a translated title, people might have a hard time to find the source; if the original (French) title is cited, people might have a hard time understanding it.