After WWII (Wizarding War II), who went to teach each one of the classes?   Minerva Mcgonagall became the Headmistress, and IIRC the DADA teacher is no longer present (having been replaced by Voldemort to Dark Magic classes) so who takes her and the other teachers' places when the whole ordeal is over?  Update: This asks about all of the teachers including DADA, and is more broad as a question than a single teacher. For instance Professor Trelawney is sacked by Umbridge, but does she resume upon the rehashing of the school staff?
Seeing that Defense Against the Dark Arts is a crucial subject, both in Harry's life as well as a part of the school-going wizard's curriculum, it must be in the hands of a good professor.   Is it known who Minerva McGonagall appointed as DADA prof. once she was made Headmistress. Was it someone we know or was it just a random powerful wizard. Is there any information about this available?
If integral is zero and function is continuous and non negative, then what about the function? Questions with mathjax don't fit in the box This bug is intermittent, only showing up on some answers, but not others.  Sometimes, posts that have mathjax equations expand the math, but don't resize the text box, cutting off some text on the bottom.  Note that this is on the IOS public beta, so that may partially have caused the problem.     App Version: 1.5.3.5 Device: iPhone 5s (GSM) OS Version: Version 10.0 (Build 14A5322e) How are percentiles calculated in Careers 2.0? I noticed that a percentile is placed at the top of my resume in Careers 2.0.    How are these percentiles calculated? Is there a certain threshold you need to be over in order for it to be displayed? Can a wizard choose where to place their soul piece when creating a horcrux? If $f^3=\rm id$ then it is identity function Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f^3(x)=f\circ f\circ f(x)=x$ for all $x$.  How can I prove $f$ is the identity function? When a permanent magnet attracts some object, lets say a steel ball, energy is converted into for instance kinetic energy and heat when attraction happens, and they eventually collide. Does this imply that energy is drawn from the magnetic field and the magnet is depleted, making it weaker and weaker for each magnetic attraction performed?  (if the answer requires Quantum Mechanical explanations, please elaborate :)) Map composer issue Show that $e^{-\beta} = \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \frac{e^{-u}}{\sqrt{u}} e^{-\beta^2 / 4u} du$. Show that $e^{-\beta} = \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \frac{e^{-u}}{\sqrt{u}} e^{-\beta^2 / 4u} du$.  I'm not really sure of how I should proceed to show this, and it's pretty un-intuitive as well.  I've managed to manipulate the RHS into  $$\frac{e^{4\beta}}{\sqrt{\pi}} \int_0^\infty \frac{1}{\sqrt{u}} e^{-(\beta+2u)^2/4u} du$$ which now vaguely resembles a Gaussian. However, substituting $v = \beta + 2u$ in in order to to get something like $e^{-x^2}$ doesn't particularly help. I got $$ \frac{e^{4\beta}}{\sqrt{2\pi}} \int_0^\infty \frac{1}{\sqrt{v - \beta}} e^{-v^2/2(v-\beta)} dv. $$  I'm really unsure of how I should proceed; any help would be appreciated. How to find out who called a method? How can I obtain an 'unbalanced' grid of ggplots? A week ago I  people on this site what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of Linear Algebra. Of course, I'm just 1 week in, but I have some questions.    How is this going to be applicable to quantum physics? I have learned about matrices (addition, subtraction, multiplication and inversion) and about how to solve multiple equations with 3 unknowns using matrices, and now I am starting to learn about vectors. I am just 1 week in, so this is probably not even the tip of the iceberg, but I want to know how this is going to help me.  Also, say I master Linear Algebra in general in half a year (I'm in high school but I'm extremely fast with maths), what other 'types' of math would I need to self-study before being able to understand rudimentary quantum physics mathematically? Out of Bag Error makes CV unnecessary in Random Forests? How to compute $\lim _{x\to 0}\frac{x\bigl(\sqrt{3e^x+e^{3x^2}}-2\bigr)}{4-(\cos x+1)^2}$? I have a problem with this limit, I don't know what method to use. I have no idea how to compute it. Is it possible to compute this limit with the McLaurin expansion? Can you explain the method and the steps used? Thanks. (I prefer to avoid to use L'Hospital's rule.)  $$\lim _{x\to 0}\frac{x\bigl(\sqrt{3e^x+e^{3x^2}}-2\bigr)}{4-(\cos x+1)^2}$$ Terminal returns not found for most commands Mac OSX Is the collection of all functions between two sets a set? How can I get a list of all packages in a repository section from command line? Is it safe to put hot food in the fridge? I heard that putting really warm food in the fridge right after cooking it is not the right thing to do. It might cause bacterial growth? Is this true? I would like AUCTeX (that is, Emacs with auctex loaded) to suggest more LaTeX macros when using completion (M-Tab or C-Ret) than it does by default. I would like to have macros from packages I have downloaded on CTAN and macros I have written. I would also like it if it would complete \section or other macros that can be accessed via C-c C-s, C-c C-e etc.  Solution: I could not make the code quoted by  work. i get an error on loading emacs but did not looked much further in because I ca bear the nuisance of appending a few lines to my files.  The link given by Ben ( contains a working solution, thus I accept his answer. I had come across this page in my researchs but somehow messed up things. What eventually worked for me was using TeX-auto-generate TEX AUTO to create a bunch of .el files in the ~/.emacs.d/auctex/auto direcoty. Note that this directory must be created beforhand. For some reasons TeX-macro-private did not achieve the wanted result. Pullback of a Volume Form Under a Diffeomorphism. I have an exercise here, which I have no idea how to do.     Problem: Let $ U $ and $ V $ be open sets in $ \mathbb{R}^{n} $ and $ f: U \to V $ an orientation-preserving diffeomorphism. Then show that   $$   {f^{*}}(\text{vol}_{V}) = \sqrt{\det \! \left( \left[   \left\langle {\partial_{i} f}(\bullet),{\partial_{j} f}(\bullet) \right\rangle   \right]_{i,j = 1}^{n} \right)} \cdot   \text{vol}_{U}. $$   Notation:         $ \text{vol}_{U} $ and $ \text{vol}_{V} $ denote the volume forms on $ U $ and $ V $ respectively.   $ f^{*}: {\Lambda^{n}}(T^{*} V) \to {\Lambda^{n}}(T^{*} U) $ denotes the pullback operation on differential $ n $-forms corresponding to $ f $. How do I sort a vector of pairs based on the second element of the pair? If I have a vector of pairs: std::vector&lt;std::pair&lt;int, int&gt; &gt; vec;  Is there and easy way to sort the list in increasing order based on the second element of the pair? I know I can write a little function object that will do the work, but is there a way to use existing parts of the STL and std::less to do the work directly? EDIT:  I understand that I can write a separate function or class to pass to the third argument to sort.  The question is whether or not I can build it out of standard stuff.  I'd really something that looks like: std::sort(vec.begin(), vec.end(), std::something_magic&lt;int, int, std::less&gt;());