Kreacher is the house-elf of Grimmauld Place. From the death of the last black inhabitants, he was left alone until Sirius moved in. Then his ownership passed on to Harry.  Harry had him work at Hogwarts until DH when he had him back to Grimmauld Place. Some trust started to build between them, but after the Ministry of Magic chaotic escape, the house became accessible to Death Eathers (or at least presumably). Harry even thought about Kreacher for some time, and how he would be saddened by their disappearance and the invasion of the Death Eaters.  Nevertheless, we never get to know what happens to him afterward, IIRC. Is there any information about him after Harry, Ron and Hermione broke into the Ministry of Magic?
In  Deathly Hallows, when Harry, Ron, and Hermione are at No. 12, Grimmauld Place, Harry summons Kreacher from the kitchens at Hogwarts where the elf has been working on Harry’s orders given a year previously.  When the trio’s heist in the Ministry of Magic ends with Hermione inadvertently bringing Yaxley inside the perimeter of the Fidelius charm that protects Grimmauld Place, the trio leg it and dare not return there for fear that it’s now crawling with Death Eaters.  Before they leave for the Ministry in the morning, Kreacher is talking about making a stew or pot roast or something for dinner. After the trio go on the run and camp out in the tent, Harry thinks of poor Kreacher preparing a dinner they’ll never return to eat, but they dare not summon him lest a Death Eater tag along (like how Dobby ‘tagged along’ with Kreacher when Harry called him to have him follow Draco in Half-Blood Prince).  The next time we see Kreacher is during the Battle of Hogwarts where he leads the house-elves from the kitchen with battle-cries of “For Master! For Harry Potter!” (or something like that), so he’s clearly lost none of his new-found loyalty to Harry.  Harry is still Kreacher’s master throughout the book, and since he does not communicate at all with Kreacher after their escape from the doorstep, he cannot have ordered Kreacher to return to Hogwarts to work there again.  We know that house-elves can go against their masters’ wishes, but this usually entails them punishing themselves rather brutally, and Kreacher does not seem to have been constantly injuring himself for the better part of a year at Hogwarts.  So when and how does Kreacher leave his master’s house without orders and return to Hogwarts?
At the moment I'm taking an algebraic topology course. In our current exercise we have to prove the following: Consider any topological space $X$ and any two maps $f, g : X \to S^n$, such that $f(x) \neq -g(x)$. It follows that $f$ is homotopic to $g$.  My idea is to use the mapping cylinders $M_f, M_g$ to construct a homotopy via $$M := M_f \amalg M_g/ (x, 0)_f \sim (x,0)_g$$ But this doesn't seem to work. In addition I don't know how $f(x) \neq -g(x)$ is going to help me (I can see how this could help if $n=1$, but I have no clue how this would work for $n&gt;1$).  I would appreciate any hints towards a solution.  Thanks in advance. Solving functional differential equation $f'(x)=2f(2x)-f(x)$ This question is essentially the same as one of my , except the only difference is that I am using "only" to change the text at the same location (i.e., ). In other words, I am destroying my earlier text as I go, and not adding new ones underneath the previous ones.   Since the text for the first "only" is shorter than the second "only", therefore the image will shift location. I would like for the image to remain at its location.   Here is an example     it may not be obvious but the second image is higher up than the first.  The code is here, can someone find a solution?  \documentclass{beamer}  \usepackage{graphicx}  \title[Title]{Presentation} \author{Sandro Botticelli} \institute{Italy} \date{1484}  \begin{document}   \begin{frame} \begin{figure} \includegraphics[scale = 0.8]{Venus}     \caption{Nascita di Venere} \end{figure}  \only&lt;1&gt;{In the centre the newly-born goddess Venus stands nude in a giant scallop shell. 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The Curves of Life : Being an Account of Spiral Formations and Their Application to Growth in Nature, to Science and to Art; with Special Reference to the Manuscripts of Leonardo da Vinci. — London : Constable and Company, 1914. — xxx + 479 p.   Here there are "30 pages + 478 pages" in total. Thanks! Possible Duplicates:             I heard a professor of mathematics explaining that the following conslusions are true.   $\textrm{False}\Rightarrow \textrm{True}$ $\textrm{False}\Rightarrow \textrm{False}$   Could someone please explain to me why both conclusions are true? I am especially interested in the first case. I'd like to hear two explanations if possible: One for mathematicians and one for non-mathematicians. :-) Triangle Inequality question with fractions Given $a, b \in \mathbb{R}$, prove that   $$\frac{|a + b|}{1 + |a + b|} \le \frac{|a|}{1 + |a|} + \frac{|b|}{1+|b|}$$  When does equality hold?  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In other words, why don't charges break apart?  About the possible duplicate: I want to know about charges in general, not just that of an electron. My response to :  Thank you very much for the extensive explanation. Unfortunately it's a bit above my level (I'm a first year electrical engineering bachelor student). This is how I understand it: If the charge is the sum of multiple separate charges, there has to be an external force that keeps this charges together. However we most often think of electrons and protons as point charges. In other words, we don't think of them as if they are made up from different (smaller) parts. But the idea of electrons and protons as point charges has its own problems. I didn't know that the electric field has a mass. From the formula, I understand that $m\propto \frac{1}{r}$. This would mean that $m\rightarrow \infty$ as $r \rightarrow 0$ (or $r = 0$), which of course is physically not possible. But there is a way to calculate the radius of an electron based on the mass, the speed of light and the constant of Planck. (Unfortunately I don't know why.) You can get around this by a technique called called renormalization, which causes the integral to converge. Unfortunately, I don't understand the rest of the answer due to my level in physics. Nevertheless I am grateful for you answer. It would be awesome if you could confirm or correct my understanding.