In the question , it is said that Snape could fly. However, I don't remember reading it anywhere in the book.  Was it explicitly mentioned in the books (if so, where?) or is it not in the canon?
In the movies, the Death Eaters leave a smoky trail behind them as they fly. As the Order arrive in the Ministry of Magic in Order of the Phoenix, they leave a silvery trail behind them. In the books, there’s no mention of how they’re doing this. And J.K. Rowling said in the books that only Voldemort and Snape could fly like this; first seen in the Battle of the Seven Potters.  Please can somebody explain how the Death Eaters are able to fly like this?
Let $A$ and $B$ be two constant matrices and let $x$ and $ y$ be two random vectors, what is the general formula for $Var(Ax+By)$? I know the formula for when $x$ and $y$ are scalar random variables and $A$ and $B$ are constants, but what about the matrix case? How to unlock Sweet Tooth in Rocket League? How do you get Sweeth Tooth's ice cream van (from Twisted Metal) in Rocket League? Program does not wait before output How much swap should I take for 1GB to 8TB of RAM on 14.04 or higher? I'm going to install 64 bit 14.04 on a new machine with something like 1 to maybe 128GB RAM and even more RAM later. How much space should I allocate to the swap partition?   And especially: WHY those recommendations for these numbers?    (None of the existing answers  and  explain any of the why and are a bit dated if we look at 1GB or RAM and up) Are there infinitely-many numbers that are both square and triangular? I just started to read "Friendly Introduction to Number Theory" but I am getting stuck in the first exercises.     1.1. The first two numbers that are both squares and triangles are 1 and 36. Find the next one and, if possible, the one after that. Can   you figure out an efficient way to find triangular–square numbers? Do   you think that there are infinitely many?     I found how to find out the number which is both square and triangle. (don't know if this is effective way)    However how can I answer "Do you think that there are infinitely many?"? I think I need to find the formula but no idea yet. Can you please help me? How do I calculate areas of an area shapefile in square meters (m²) or in acres (ha)? I didn't find that functionality in the Vector Tools. Legend inconsistent in QGIS map print layout After I did styling my vector data set to be classified by graduated circle and categorized by type, it appears beautifully consistent in the map canvas as shown below:   But when I create a print map layout and after adding the map, I add the legend and it come out ugly inconsistent like this:    So, I would like to ask for help to make the legend in the map print layout consistent as the legend shown in layer panel. I use the latest version QGIS 3.10.2 64bit. On the hardness of data to learn Almost in all texts which are discussing theorems of statistical learning, they assume analyzing arbitrary unknown distribution (the worst case). But in practice different problems (different data) have different levels of hardness, for example linear separable data is easier to learn than the data that are less (or not) separable by hyperplanes.  Are there any works on formalizing the hardness of data (similar to the stuff done in complexity theory)  Note: I am not sure if the same as analyzing VC-dimension or not, but I think it is not. Sweave.sty causing error in verbatim environment? What does Geordi see on the holodeck? We've seen how he interprets people and things with his VISOR, but if the holodeck is all force-fields and holographic imaging, how does he interact in a similar way to other members?   What does the holodeck look like to Geordi LaForge using his VISOR? Convergence of $\sum \frac{a_n}{S_n ^{1 + \epsilon}}$ where $S_n = \sum_{i = 1} ^ n a_n$ Amplitude of an electromagnetic wave containing a single photon Given a light pulse in vacuum containing a single photon with an energy $E=h\nu$, what is the peak value of the electric / magnetic field? Why is my variable local in one 'while read' loop, but not in another seemingly similar loop? What is a subnet mask, and the difference between a subnet mask of 255.255.255.0 and 255.0.0.0? My DSL/WiFi router provided by my ISP has an internal IP address of 10.0.0.138, the addresses provided by DHCP are in the format of 10.0.0.xxx.  Now, I want to start using static IP addresses on my local network, and I don't know what to provide for the subnet mask.  What is a subnet mask?  Should I be using 255.255.255.0 or 255.0.0.0 as my subnet mask? What is the difference? How do I address unchecked cast warnings? Eclipse is giving me a warning of the following form:     Type safety: Unchecked cast from Object to HashMap   This is from a call to an API that I have no control over which returns Object:  HashMap&lt;String, String&gt; getItems(javax.servlet.http.HttpSession session) {   HashMap&lt;String, String&gt; theHash = (HashMap&lt;String, String&gt;)session.getAttribute("attributeKey");   return theHash; }   I'd like to avoid Eclipse warnings, if possible, since theoretically they indicate at least a potential code problem.  I haven't found a good way to eliminate this one yet, though.  I can extract the single line involved out to a method by itself and add @SuppressWarnings("unchecked") to that method, thus limiting the impact of having a block of code where I ignore warnings.  Any better options?  I don't want to turn these warnings off in Eclipse.  Before I came to the code, it was simpler, but still provoked warnings:  HashMap getItems(javax.servlet.http.HttpSession session) {   HashMap theHash = (HashMap)session.getAttribute("attributeKey");   return theHash; }   Problem was elsewhere when you tried to use the hash you'd get warnings:  HashMap items = getItems(session); items.put("this", "that");  Type safety: The method put(Object, Object) belongs to the raw type HashMap.  References to generic type HashMap&lt;K,V&gt; should be parameterized. Why doesn't the "zig-zag" comb deformation retract onto a point, even though it's contractible? I am starting to read Hatcher's book on Algebraic Topology, and I am a little stuck with exercise 6(c) in Chapter $0$. Unfortunately a picture is involved so it doesn't quite make sense for me to repeat it here - but I'll do it anyways (maybe it helps)     Let $Z$ be the zigzag subspace of $Y$ homeomorphic to $\mathbb{R}$ indicated by the heavier line in the picture:       Show there is a deformation retraction in the weak sense of $Y$ onto $Z$, but no true deformation retraction."   Now the definition for deformation retraction and weak deformation retraction as as follows:  We say that $f: X \to X$ is a deformation retraction of a space $X$ onto a subspace $A \subset X$ if there exists a family of maps $f_t : X \to X$ with $t \in [0,1]$ such that $f_0 = \mathbb{1}$ (the identity) and $f_1 (X) = A$ and also $f_t$ restricts to the identity on $A$ for each $t$.   A weak deformation retraction is almost the same, only that we now relax the conditions $f_1(X) = A$ to $f_1 \subset A$ and, for each $t \in [0,1]$ we require that $f_t(A) \subset A$.  Hence, as far as I understand the question and the concepts involved, I need to show that basically no map can be constructed such that the bold zigzag - line stays put while the thin lines retract to the bold line over a finite time interval $0 \leq t \leq 1$. However, I should be able to show that I can pull the thin lines continously onto the thick line, provided I am allowed to move points on the zigzag line .. is that correct ?  Now, the problem is I can't see why I shouldn't be able to do the former, given that I could find a weak deformation retract. Any help would be great! Add alias URL for posts timeline iPhone asks 'Trust This Computer' again (I am on macOS 10.14.6 and iOS 12.4)   I am certain that I have trusted my Mac before. Without the latter trust, I can also access the phone in iTunes.   I have tried reset "Location and Privacy" on my phone. Then   When connecting the iPhone with iTunes running, it asks for the trust and I trust it.  Reboot my Mac and the phone asks for another trust.   Is there some known malware mimics system behavior and steal data from an iPhone?  (I do have installed some pir*ted software recently. That's my fault.) Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$.  How? What's the proof? Or maybe it is self apparent just looking at the above?   PS: This problem is known as "The sum of the first $n$ positive integers". Can I download and play games from a friend's PSN account on a PS4?