What does singularity mean in the context of black holes?
Why must a singularity form inside a black hole?
If granted omniscience and we know that $\beta_1$ in a multiple linear regression model is truly 0, what does that mean in words (and math notation)?  The model is: $Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \epsilon$  Does $\beta_1 = 0$ mean:   Empirically: "If we make infinite, perfect measurements of our data, $\hat\beta_1$ approaches 0 at the limit?" [Math: ??] Formally: "In the true, finite population of our data, $X_1$ has 0 association with $Y$ and their vectors of values are perfectly linearly independent, uncorrelated, and orthogonal?" [Math: ??]   What does $\beta_1 = 0$ mean, precisely? Let $X, Y$ be an exponentially distributed random variables with parameters $a, b$. Then $X$ has pdf:    $$f_X(x) =\begin{cases} a e^{-a x},&amp; x\geq 0\\ 0,&amp; \text{otherwise}.\end{cases}$$ Suppose $X$ and $Y$ independent. Show that $$\mathbb P(X&gt;Y) = \frac{b}{a+b}.$$   Now I thought the following: $$f(x,y) = f_X(x)\ f_Y(y) = abe^{-ax -by},\qquad\text{for } x,y &gt; 0.$$ And then $$\mathbb P(X&gt;Y) = \int_0^\infty \int_0^x a b e^{-ax -by}\,dydx$$ However, if I solve this (manually or using Wolframalpha), I can't seem to end up with $\frac{b}{a+b}$. Any ideas? Associated elements in a ring Please help me to find elements $a,b$ in a ring $R$ such that $a\mid b$ and $b\mid a$, but there does not exist any unit $u$ in $R$ such that $a=ub$. Sometimes I see a comma after the proper name:     Hello Mr. Black,      In order to give you....   But my native language is not English and I think that the comma in this phrase should be placed before the proper name:      Hello, Mr. Black.      In order to give you....   What is the correct punctuation in English? I execute quite a lot of sudo commands.  I noticed that if I execute one and then another one within few minutes, the 2nd time I don't get the message:     [sudo] password for my_username:   but I do get it when there is more time between the execution of the two commands.  What's the period of time in which the message is not appearing? How can we check it / update it? I am new to Game Development, All I have developed is some 2d Games, using Game Maker by . There game development is much more easy, just as simple as Drag and Drop. But, Now I wish to evolve in Game Development, and wanna try some hands with the common programming languages like, Java, C++ or C. In order to achieve this, I came around the first topic, Game Engine. So, What is Game Engine.? That is a broad question, with various answers. I came out with some conclusion after reading various links,  Wikipedia: A game engine is a system designed for the creation and development of video games. The core functionality typically provided by a game engine includes a rendering engine for 2D or 3D graphics, a physics engine or collision detection (and collision response), sound, scripting, animation, artificial intelligence, networking, streaming, memory management, threading, localization support, and a scene graph.  It exists to abstract the (sometime platform-dependent) details of doing common game-related tasks, like rendering, physics, and input, so that developers (artists, designers, scripters and, yes, even other programmers) can focus on the details that make their games unique. Engines offer reusable components that can be manipulated to bring a game to life. Loading, displaying, and animating models, collision detection between objects, physics, input, graphical user interfaces, and even portions of a game's artificial intelligence can all be components that make up the engine.  Now what I understand is, Game Engine takes care of all the common work, like physics, loading etc... As far my question is concerned, what is a Game Engine (Programmatically)? Is it a Library? With pre-defined functions and classes, which can be inherited? Or what so ever, what is it? If a charge travels in a circle it must accelerate, thereby producing EM.  However, a wire in a circular loop is analogous to many charges moving in a circle. So, why don't circular currents produce EM? (I have not found any evidence that circular currents can produce EM).  A similar question is: . However, my question asks about circular wires and direct current. Thanks. 10 servers to administer and I'm a history major Through reasons that don't warrant exhaustive discussion, I find myself in charge of 10 servers:   A domain controller-~500 hosts/~350 users IIS web server-This is where we make our money SQL server-The crown jewels Exchange server Linux box for data entry AV server Backup server A few others tossed around   The company where I work believes everybody is replaceable and therefore believes they can pay a minimum wage for any position. The IT manager and Sysadmin quit recently and I think I was the only person who did not take a big step backward when the call went out for volunteers. This also explains why someone with my background is in this position. That is the reality, as much as I wish it otherwise.  What are the things I should be doing to keep those systems running? There is no written procedure left behind and I crammed the A+ and Network+ certs in the last two months but that leaves me with some theory and no practical experience.  I am in the process of teaching myself powershell but from here to there is a long way. I have no scripting or programming experience.  What tasks should I be performing? What practices should I implement?  I understand I am probably hosed but a lifeline to get me through would be helpful. What's that icon (Oneplus one)? does anybody know what's the meaning of that icon ?  I have a new oneplus one since several days and today this icon appeared in my status bar. Why is my iPhone screen stuck on the 'connect to iTunes' screen? I need help with my iPhone. It's stuck on the 'connect to iTunes' screen.   iTunes does recognize the iPhone, but when I select 'restore', it goes through a few progress bars, then errors again. I've already tried to hard reset the phone, but that doesn't work either. Let me say first that I am fairly new here (about 3 weeks) and absolutely love Stack Overflow. At first I let some of the negative attitudes get to me a little, left a few snarky comments, but that phase has passed. I see the tremendous value for this site in the long run and sincerely do want to help make it a better place.  I find myself going for  the downvote very frequently -- currently about 35% of my votes are downvotes. I even make an extra effort to upvote good content to offset this a bit, so the number should probably be more like 60%. I always try to leave constructive comments and try to help out the confused ones who are really trying.  I truly can't tell if it is just me, or if these folks deserve it. I don't know if you guys noticed, but there are a lot of really bad questions being posted, and a lot of people trying to pass off things like "use jQuery", "use Firebug", and "use Google" as good answers (and getting upvotes!). I even saw someone copy/paste my "What does Firebug say?" comment as an answer and get 3 votes within moments. Sometimes This answer is unclear or not useful just doesn't do it justice.  I feel like some of the people just drive me nuts, but then again I feel this way IRL :) Am I being too harsh, or is it really that bad on SO? Has there recently been a surge of new users, and the quality of content is degrading, or was it like this in the "good old days"? Have I perhaps misunderstood the expected behavior on Stack Overflow -- should I just ignore these crappy questions and answers rather than downvote them? Typesetting UML class diagrams What's the best way to draw UML class diagrams in LaTeX? Exercise from Atiyah-Macdonald, Chapter 1, 2.iv) Let $A$ be a ring and let $A[x]$ be the ring of polynomials in an indeterminate $x,$ with coefficients in $A.$ Let $f=a_0 + a_1x+\cdots+a_nx^n \in A[x].$ $f$ is said to be primitive if $(a_0,a_1,\ldots,a_n)=(1).$     Prove that if $f,g\in A[x],$ then $f$ and $g$ are primitive $\Rightarrow$ $fg$ is primitive.   I know Gauss lemma holds true for UFDs, but in this case it isn't required $A$ to be UFD: is it possible to drop the assumption, or is some hypothesis missing here? Clarification for non-answers One of my flags for "non-answer" was declined and I was wondering if I misunderstood what "non-answer" means. I saw someone post an answer that was just a block of code, with nothing in it that was even the slightest related to the question, it was just a random block of code with no explanation. Should I have not used the "not an answer" tag even though the description explicitly says that it should be used when answers "do not attempt to answer the question"? Let $B$ be a commutative domain and $A \subseteq B$ a subring. Suppose that the extension $A \to B$ is finite, i.e. $B$ is finitely generated as an $A$-module. If $I \cap A = \lbrace 0 \rbrace$ for some ideal $I$ of $B$, can we conclude that $I = \lbrace 0 \rbrace$?  I know that if $I$ contains a nonzero prime ideal, the statement holds since $A \to B$ is an integral extension, whereby no two comparable prime ideals of $B$ can have the same intersection with $A$. But does the statement still hold if $I$ does not contain a nonzero prime ideal? And what if $A \to B$ is only assumed to be an integral extension? I have run into an issue with WPF and Commands that are bound to a Button inside the DataTemplate of an ItemsControl. The scenario is quite straight forward. The ItemsControl is bound to a list of objects, and I want to be able to remove each object in the list by clicking a Button. The Button executes a Command, and the Command takes care of the deletion. The CommandParameter is bound to the Object I want to delete. That way I know what the user clicked. A user should only be able to delete their "own" objects - so I need to do some checks in the "CanExecute" call of the Command to verify that the user has the right permissions.  The problem is that the parameter passed to CanExecute is NULL the first time it's called - so I can't run the logic to enable/disable the command. However, if I make it allways enabled, and then click the button to execute the command, the CommandParameter is passed in correctly. So that means that the binding against the CommandParameter is working.  The XAML for the ItemsControl and the DataTemplate looks like this:  &lt;ItemsControl      x:Name="commentsList"     ItemsSource="{Binding Path=SharedDataItemPM.Comments}"     Width="Auto" Height="Auto"&gt;     &lt;ItemsControl.ItemTemplate&gt;         &lt;DataTemplate&gt;             &lt;StackPanel Orientation="Horizontal"&gt;                 &lt;Button                                                  Content="Delete"                     FontSize="10"                     Command="{Binding Path=DataContext.DeleteCommentCommand, ElementName=commentsList}"                      CommandParameter="{Binding}" /&gt;             &lt;/StackPanel&gt;                                 &lt;/DataTemplate&gt;     &lt;/ItemsControl.ItemTemplate&gt; &lt;/ItemsControl&gt;   So as you can see I have a list of Comments objects. I want the CommandParameter of the DeleteCommentCommand to be bound to the Command object.  So I guess my question is: have anyone experienced this problem before? CanExecute gets called on my Command, but the parameter is always NULL the first time - why is that?  Update: I was able to narrow the problem down a little. I added an empty Debug ValueConverter so that I could output a message when the CommandParameter is data bound. Turns out the problem is that the CanExecute method is executed before the CommandParameter is bound to the button. I have tried to set the CommandParameter before the Command (like suggested) - but it still doesn't work. Any tips on how to control it.  Update2: Is there any way to detect when the binding is "done", so that I can force re-evaluation of the command? Also - is it a problem that I have multiple Buttons (one for each item in the ItemsControl) that bind to the same instance of a Command-object?  Update3: I have uploaded a reproduction of the bug to my SkyDrive: Let $f: \mathbb N \rightarrow \mathbb N$ are increasing function such that $f\left(f(n)\right)=3n$. Find $f(2017)$ Is a Schengen visa/transit visa required for a transit flight through two Schengen airports? I am travelling form Delhi (DEL) to San Francisco (SFO), I have to make a connection flight in Vienna that would take me to Frankfurt (FRA). My flight to SFO is from FRA, I have one ticket for entire journey and the flights are operated by Lufthansa or its subsidiaries.  Do I need a transit visa? My flight arrives at Vienna international airport and leaves from the same, and for second leg it arrives at Frankfurt international airport.  I asked this question as an addendum to my  but was advised to make a new post.  I hold Indian passport and valid US visa. How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$? This question came from the prelim exam I took last month. I have a proof that seems a bit unwieldy to me (posted as an answer), so I'm opening it up to ask if there are other ways of showing this.     Let $x$ be any positive real number, and define a sequence $\{a_n\}$ by   $$ a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} $$   where $[x]$ is the largest integer less than or equal to $x$. Prove that $\displaystyle{\lim_{n \to \infty} a_n = x/2}$. We want to have a plain empty sandbox - one without any configuration metadata or data from Production.  Is it possible to have such a sandbox?  Please advise.