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During Order of the Phoenix, Voldemort tries his best to lure Harry to the Ministry so that Harry retrieves the prophecy for him. In the aftermath, while Harry follows Bellatrix towards the exit, he raises a valid question:  ”Why couldn’t he (Voldemort) come and get it (the prophecy) himself?”  This is a valid question, the prophecy is labeled as referring to the Dark Lord and maybe Harry Potter. Only those mentioned can retrieve the prophecy, so Voldemort should have been able to get it, assuming he is the Dark Lord of the prophecy. Bellatrix gives a good answer:  “Get it himself?” shrieked Bellatrix on a cackle of mad laughter. “The Dark Lord, walk into the Ministry of Magic, when they are so sweetly ignoring his return? The Dark Lord, reveal himself to the Aurors, when at the moment they are wasting their time on my dear cousin?”  Despite the cackle of mad laughter, the reasoning is sound, Voldemort didn't want to reveal himself. Yet he does exactly that when he appears in the Ministry and is seen by the Minister himself, and he exposes himself for no gain. Bellatrix may or may not have been able to escape without his help, but he couldn't know that before he came, and his other followers that were captured were freed again a few months later, so it wouldn't have mattered much, he would have freed her, too. So why did he go to the Ministry and reveal himself? Every principal ideal domain satisfies ACCP. Sole minimal element: Why not also the minimum? A minimal element (any number thereof) of a partially ordered set $S$ is an element that is not greater than any other element in $S$.  The minimum (at most one) of a partially ordered set $S$ is an element that is less than or equal to any other element of $S$.    Let's consider the power set $\mathcal P (\{x,y,z\})$ together with the binary relation $\subseteq$.  The Hasse diagram shows what element(s) we're looking for:    It's easy to see that:   $\emptyset$ is a minimal element $\emptyset$ is the minimum   Now if we remove $\emptyset$ and consider $\mathcal P (\{x,y,z\})\setminus \emptyset$ instead, we get the following:   $\{x\}$, $\{y\}$ and $\{z\}$ are minimal elements there is no minimum   (1) We know that a minimum is unique and it is always the only minimal element.  (2) And from the example above, it seems that, if a sole minimal element exists, it is always the minimum.  But I read that (2) is false. Why? What to conclude from this lasso plot (glmnet) Following is the plot of glmnet with default alpha(1, hence lasso) using mtcars data set in R with mpg as the DV and others as the predictor variables.  glmnet(as.matrix(mtcars[-1]), mtcars[,1])     What can we conclude from this plot regarding different variables, especially am, cyl and wt (red, black and light blue lines)? How would we phrase the output in a report to be published?  I thought of following:   wt is the most important predictor of mpg. It is negatively affecting the mpg. cyl is a weak negative predictor of mpg. am may be a positive predictor of mpg. Other variables are not robust predictors of mpg.    Thanks for your thoughts on this.  (Note: cyl is the black line which does not reach 0 till very close to it.)  Edit: Following is plot(mod, xvar='lambda') which shows x-axis in order reverse of above plot: What intuitively is "bias"? I'm struggling to grasp the concept of bias in the context of linear regression analysis.    What is the mathematical definition of bias? What exactly is biased and why/how? Illustrative example? I have an Ubuntu 14.04 server that has no external monitor connected.  I use NoMachine to remote control the machine.  When I do so, the Unity/Gnome interface doesn't see any monitors connected, so I can only use 800x600 when connecting with NoMachine.  If I plug in a monitor, I can set the resolution to whatever size the NoMachine window is on the remote computer.  Is there a way to create a "fake" monitor device on Ubuntu so I can set a desktop resolution in Unity/Gnome? $P[X=x]=0$ when $X$ is continuous variable My storage was not updated Recurrence relation for the integral, $ I_n=\int\frac{dx}{(1+x^2)^n} $ Express recurrence relation  of the integral  $$ I_n=\int\frac{dx}{(1+x^2)^n} $$  [My Answer]  $$ I_n = \int\frac{1+x^2}{(1+x^2)^n}dx-\int\frac{x^2}{(1+x^2)^n}dx $$  $$ I_n=I_{n-1}-\int x\cdot\frac{x}{(1+x^2)^n}dx $$  $$ I_n=I_{n-1}-\frac{x}{2(1-n)(x^2+1)^{n-1}}+\frac{1}{2(1-n)}I_{n-1} $$  $$ I_n=\frac{2n-3}{2(n-1)}I_{n-1}+\frac{x}{2(n-1)(x^2+1)^{n-1}} \ \ \ \ (n&gt;1) $$  $$ I_1=\arctan(x) $$  Is my answer correct? I need backup in pressing my case that the phrase “choose from one of four options” is grammatically incorrect. Is there some resource that can prove my case, that the incorrect phrase should be replaced with one of the two following ones?        Choose one of four options   Choose from four options I am trying to find a good rough guesstimate of how much of the maximum wattage usage of a graphics card is turned into hot air.  Say a graphics card is rated at using X watts at maximum usage. How much of that is released as heat into the air via the cooling setup, and how much is not (If not where does it go? To ground?)  Yes I know there are a lot of variables here, I am just looking for a guesstimate from someone who would understand better than me. I'm specifically using  (Radeon 6990 375 watts max) Is there a way to make Git mark a file as conflicted? I recall from complex analysis a while ago that the series $$\sum_{n=1}^\infty \frac{z^{n!}}{n}$$ has radius of convergence 1, diverges on roots of unity, and converges elsewhere on the circle. It's simple enough to see that it has to diverge on roots of unity: consider $z$ such that $z^k = 1$ for some $k \in \mathbb{N}.$ Plug this into the series and you see that past the $k$'th term, $z^{n!} = (z^k)^m = 1$ for some $m \in \mathbb{N}$; hence, the series becomes harmonic and must diverge.  But I don't recall how to show that the series converges for other values on the circle. If this is simple I'd like just a hint, if it is involved I would prefer a sparse outline of the proof. Thank you! Can I manually limit the %CPU used by a process? When can I use a forward declaration? Package Listings Warning: Text dropped after begin of listing when calling function I'm trying to this:  \newcommand{\code}[1] {\begin{lstlisting} {#1} \end{lstlisting} }   But I keep getting this error:  Package Listings Warning: Text dropped after begin of listing   I'm calling \usepackage and all so, do any of you knows how to solve this? How to work with "FIFO" in C# .NET? Does a fighter's Action Surge trigger Extra Attack? Context  A player in my game will play a fighter with a greatsword. At 5th level he has Extra Attack and the option to use Action Surge. PHB p.72 states:     Action Surge. Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On your turn, you can take one   additional action on top of your regular action and a possible bonus   action.      Once you use this feature, you must finish a short or long rest before   you can use it again. Starting at 17th level, you can use it twice   before a rest, but only once on the same turn.        Extra Attack. Beginning at 5th level, you can attack twice, instead of once,   whenever you take the Attack action on your turn.       The number of attacks increases to three when you reach 11th level in   this class and to four when you reach 20th level in this class.   Confusion  The questions and answers that I find on here deal with multi-classing or Two Weapon fighting specifically, which don't really give a clear yes or no on my question.  The way I read these two features is that when a fighter takes an Attack action, he gets the Extra Attack. After using Action Surge he gets the opportunity to take another Attack action, and with that another Extra Attack. This results in four attacks with the greatsword in one turn, which sounds amazing but also rather powerful. Is this correct, or am I missing something? I edited a question, on Stack overflow and when submitting preview, got red bar telling me my account isn't allowed to make suggested edits.  On the Sandbox android question it works fine, but that is editing an answer to my question.   The message I receive is:     Account is not allowed to suggest edits   Are the rules for suggested edits different for Android app? shade plane with a color function color(x,y)