Voldemort clearly had a good thing going for him when Fudge was in denial about his return. This allowed him free reign to move, manipulate and organise without Ministry interference. In his pursuit of the prophecy, Voldemort purposefully crafted his plan in such a way as to protect that secret. By operating at a distance he could get what he wanted by forcing Harry to take it for him. The risk of disclosure here is next to nil. The idea that Voldemort might show up was so fanciful that Bellatrix laughed at the suggestion.     "Why couldn’t he come and get it himself?”   “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?”   (Order Of the Phoenix, Chapter 35, Beyond the Veil)   This makes it all the more surprising when Voldemort does show up in the Atrium a short while later. The Ministry find him and his return is revealed.  I'd understand his thinking if he wanted to bypass Harry altogether, sneak into the Ministry using stealth (who'd doubt that he'd be capable of this?) and take it himself in the middle of the night, causing the minimum of fuss.  I'd understand his thinking too if he'd stuck with his original plan - make Harry take the prophecy, entrust the Death Eaters to collect it and keep his distance.  What I don't understand is encouraging a fight with Harry and co followed by a personal appearance that blows his cover.  Why did he pursue this course of action?  (Related questions  and  in my view don't fully address Voldemort's stated desire to remain undercover).
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?
How can I approach implementing the following in Drupal 7?  What I need to do is create a module which defines a new fieldable entity, called 'Company'. I have a list of, let's say, 20 fields that need to be filled out by each Company instance. These questions are predefined and some may contain custom validation.  At the moment, I'm at the point when I can add new fields to the Company entity. This works fine at the moment. My issue is that I need all these fields to be there as soon the module is installed, so adding them via the interface isn't an option.  I was wondering how can I approach this? I'd assume it comes down being able to do what can be done using the 'Manage Fields' UI programmatically. Suppose $f(x)$ is linear i.e. $f(x + y) = f(x) +f(y) $ and monotone on $[-\infty, +\infty]$, then $f(x) = ax$, a real.  We know $f(0)=0$, and if one assumes differentiability it is quite easy.      But how does monotone make up for differentiability? Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = \gcd(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$. Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = hcf(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.   We know that if $a|c$ and $b|c$ then $a\cdot b\cdot s=c$ (for some positive integer $s$). $(ab|c)$  Then doesn't $ab|dc$ since $ab|c$?  I feel like I'm misunderstanding my givens.    Can we say $\operatorname{lcm}(a,b)=c$,  $\operatorname{hcf}(a,b)=d$, and $\operatorname{lcd}(a,b) \operatorname{hcf}(a*b)=a*b$? Thus, $ab|dc$  as $dc = ab$. Render shadows over a transparent background in cycles I'm trying to make a logo that will be on the header of a website. What I'd like to achieve, and the reason I'm using Blender, is the projected shadows on the background looking perfectly real (as opposed to doing a drop-shadow in Photoshop).  The only thing is, the background has to be transparent because the website might use different background colors / gradients that make it impossible to match with the exported 3D background.  I made this dummy file (not the real logo... duh!):    If I use the file as is, the grey borders of the image will be impossible to match with the website's background color.  Here what I want:    So obviously, I tried masking the plane, which made the shadows disappear. I tried having a plane set to glass, which didn't work either.  And I looked around for a similar question, but didn't find one.   I am not trying to render , or a . I've seen  which has a very difficult solution to this problem, and  that is very similar but in Blender Internal.  is similar too but has never been solved. I'm only a beginner at Compositor, and I'm trying to learn more any chance I get, so any solution in Compositor is welcome if detailed enough. Although if there's a simpler or faster way, it's welcome too!  .blend file: Where can I find other RPG players? Enable replace image feature in LibreOffice Error "Settings to access printer 'printername' are not valid" when printing from Windows service 10 hour layover in Moscow Russia, do I need a transit visa? On June 29th 2015, I will be traveling for the first time to Italy from USA. I will have a 10 hour layover in Moscow Russia. My question is, do I need a visa to visit the city? Proving AM-GM with induction I am trying to prove AM-GM with the following steps:    Prove that AM-GM holds for two variables Prove that if AM-GM holds for $k$ variables then it holds for $2k$ variables Prove that if AM-GM holds for $k$ variables then it holds for $k-1$ variables   If I prove all of these, I will have proved AM-GM. I have proved (1) and (2), but I am still struggling with (3).   Proof for (1)  By the Trivial Inequality, we know that the first statement is true. Therefore \begin{align} (a-b)^2 \ge 0 \implies\\ a^2-2ab+b^2 \ge 0 \implies \\ a^2+2ab+b^2 \ge 4ab  \implies\\ a+b \ge 2\sqrt{ab}  \implies\\ \frac{a+b}{2} \ge \sqrt{ab}\hspace{3mm}\blacksquare\\ \end{align}  Proof for (2)  Let our numbers be $a_1, a_2, a_3, \ldots, a_{2k}$. Denote $$x = \frac{\displaystyle\sum_{i=1}^{k}a_i}{k}, y = \frac{\displaystyle\sum_{i=k+1}^{2k}a_i}{k}$$  We know that by AM-GM for 2 variables, that $$\frac{x+y}{2}\ge \sqrt{xy}$$ By our definition of $x, y$ this means that $$\frac{x+y}{2} = \frac{\frac{\displaystyle\sum_{i=1}^{2k}a_i}{k}}{2}=\frac{a_1+a_2+a_3+\ldots+a_{2k}}{2k}$$ Again by definition, $$\sqrt{xy} = \sqrt{\left(\frac{x_1+x_2+\ldots+x_k}{k}\right)\left(\frac{x_{k+1}+x_{k+2}+\ldots+x_{2k}}{k}\right)}$$ By AM-GM for $k$ variables, $$\sqrt{\left(\frac{x_1+x_2+\ldots+x_k}{k}\right)\left(\frac{x_{k+1}+x_{k+2}+\ldots+x_{2k}}{k}\right)} \ge \sqrt{\sqrt[k]{x_1x_2\ldots x_k}\sqrt[k]{x_{k+1}x_{k+2}\ldots x_{2k}}}$$ This is simply equal to $$\sqrt[2k]{x_1x_2x_3\ldots x_{2k}}\hspace{3mm}\blacksquare$$  However, I have no idea how to proceed on (3). I tried using the definition of AM and GM but did not come up with anything useful. Feel free to point out any errors or flaws on my proofs for (1) and (2). On equations $m^2+1=5^n$ I am looking for integer solutions of Diophantine equation $m^2+1=5^n$. I found that $m=0,n=0$ and $m=2,n=1$. I could not find any other solutions. I try to prove this but I could not. Could anyone help me to solve this equation? I have a problem regarding alignment for my equations.  This dummy program shows the problem:  \documentclass{article} \usepackage{amsmath} \begin{document}  \begin{enumerate}     \item First do this     \begin{align}     c^2 &amp;= a^2 + b^2     \end{align}      \item Then do this      \begin{align}     variable &amp;= c^2     \end{align} \end{enumerate}  \end{document}     I would like the = to be aligned under each other, but still be able to list them inside the enumerate.  Any suggestions? Describing integral closure of quadratic number fields I'm facing the following problem.      Let $p$ be a prime and $ K=\mathbb{Q}(\sqrt{p}) $. I'm trying to find the integral closure of $ \mathbb{Z} $ in $ K $.   I don't really know where to start. I've managed to prove that $ \mathbb{Z}[\sqrt{p}] $ is in the integral closure and I'm almost sure $ \mathbb{Q} $ is not.   I can't think of a systematic way to approach this. I would appreciate any help. Fixed background image with ios7 bound on the cardinality of the continuum? I hope not Suppose we don't believe the continuum hypothesis. Using Von Neumann cardinal  assignment (so I guess we believe well-ordering?), is there any "familiar" ordinal number $\alpha$ such that, for non-tautological reasons, $\aleph_\alpha$ is provably larger than the cardinality of the continuum? I would hope not since it would seem pretty silly if something like $\alpha = \omega_0$ worked and we could say "well gee we can't prove that $c = \aleph_1$, but it's definitely one of $\aleph_1, \aleph_2, \ldots , \aleph_{73}, \ldots$". I (obviously) don't know jack squat about set theory, so this is really just idle curiosity. If a more precise question is desired I guess I would have to make it     For any countable ordinal $\alpha$ is the statement: $c &lt; \aleph_\alpha$ independent of ZFC in the same sense as the continuum hypothesis?   assuming that even makes sense. Thanks! Equation's symbols or parameters description are shown below, however, the = and the details of parameter are not aligned.     I want to make a parameters list for a equation like this, in which symbols, = and detail information are aligned in terms of vertical position.  where    ...    P   =   notional permeability factor    N   =   number of waves    Sd  =   damage level    ...   A tabular environment will produce more spacing before and after text. So how to make the = vertical aligned?   The codes is provided:  \begin{equation} \frac{H_s}{\Delta D_{n50} } = 1.0~ P^{0.13}~ \left(\frac{S_d}{N} \right)^{0.2} \xi_m^P~ \sqrt{\cot \alpha} \end{equation} where:   $H_s$ = significant wave height, equal to the average of the highest 1/3 of the waves  $\Delta$  = relative buoyant density, equal to $\rho_r / \rho_w - 1$, where     $\rho_w$ is the water density  $D_{n50}$ = nominal  diameter defined in Equation (2)  $P$ = notional permeability factor  $S_d$ = damage level   $N$ = number of waves  $\xi_m$ = breaker parameter based on mean wave period $T_m$  $\alpha$ = slope angle How to adopt SD card as internal storage on Samsung devices? In class we are currently covering upper/lower bounds, big Oh and omega and the like. I am pretty good on the "typical" functions one would do, but at a complete loss at "general" statements. This ons function, for example:  $$\forall f\in\mathcal{F}: \lfloor \sqrt{\lfloor f(n)\rfloor }\rfloor \in O(\sqrt{f(n)}).$$  the domain F by the same is all natural numbers that return a real positive number when inputted. My first instinct with something like this is to remove the floors, the general strategy I though to solve this is remove the outer floor, then get rid of the square root, and then the inner floor. But I have no idea how to go about this or if this is even the right way to do it.  Help please? Thanks in advance. How do I get HTML5 h.264 Video working on Firefox? According the the , h.264 video should now be working in Linux.  It does also say the appropriate gstreamer plug-ins need to be installed.  Does anyone know what these are?    I used this  and the  to verify that h.264 support works in Google Chrome on my Ubuntu 12.04 64-bit, but does not work on Firefox 26. If $a_0+\frac{a_1}{2}+\dots + \frac{a_n}{n+1}=0$ then $a_nx^n+\dots + a_0$ has a root in $[0,1]$ Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots +a_0$ be a real polynomial such that $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\dots + \frac{a_n}{n+1}=0$.  Show that $P$ has a root in $[0,1]$. I would like proof verification and alternative proofs.  My solution:      notice that the antiderivative of $P(x)$ is $\frac{a_nx^{n+1}}{n+1}+\frac{a_{n-1}x^n}{n}+\dots + \frac{a_1x^2}{2}+a_0x+C$. From here $\int_{0}^1P(x)dx=a_0+\frac{a_1}{2}+\frac{a_2}{3}+\dots + \frac{a_n}{n+1}=0$. If $P$ had no roots then $P$ would be a strictly positive(or negative) continuous function, so its integral would be strictly positive ( or negative), a contradiction. No package php-imap available on CentOS 7 I'm running a dedicated server with CentOS 7 64bit, here is the output when trying to install IMAP:    yum install php-imap   Loaded plugins: fastestmirror   Loading mirror speeds from cached hostfile    * base: ftp.ciril.fr    * extras: ftp.ciril.fr    * updates: ftp.ciril.fr   No package php-imap available.   Error: Nothing to do       Also the same when disabling exclusions:    sudo yum install --disableexcludes=all php-imap Loaded plugins: fastestmirror Loading mirror speeds from cached hostfile  * base: ftp.ciril.fr  * extras: ftp.ciril.fr  * updates: ftp.ciril.fr No package php-imap available. Error: Nothing to do