How to put arrow head in latex?
Tikz: Arrowheads in the center
Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.   The hint I was given is to evaluate: $$E[X-Y;X&gt;a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$  which I can write as: $$\int_A(X-Y)dP +\int_B(X-Y)dP$$ where $A=\{X&gt;a, Y\leq a\}$ and $B=\{X\leq a,Y\leq a\}$.  But I need some more hints. Proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain We're proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain, using the norm function $$\nu (a + b\sqrt{2} ) = |a^2 - 2b^2|$$ and the first part says that since $\nu (a + b\sqrt{2} ) = |(a + b\sqrt{2})(a - b\sqrt{2})|$ it's clear that $\nu (xy) = \nu(x) \nu(y)$? ... Can someone please explain to me how this is clear? Install Spyder3 without Anaconda on ubuntu 16.04 When can lands be tapped? How do I make my glasses look realistic? How to implement Enums in Ruby? What's the best way to implement the enum idiom in Ruby? I'm looking for something which I can use (almost) like the Java/C# enums. What is the name of this game involving racing to the other side of the board and putting down barriers? Does anyone recognize this game, where players take turns in a square board, each turn a player can move his pawn or place a wall and the objective is to reach the other side of the board before the other player.  Here is a picture of a near end game. Harry is a half-blood and Ginny is a pure-blood, so what does that make his children? I've been told it means they are pure-bloods but I can't be sure. What do you think? Some time ago, we announced that . This has gone well and we're extremely thankful to everyone for their support and feedback. Because it has been successful so far, we're going to be extending the program to the majority of our graduated sites. Well, we've pretty much already done that, but there are quite a few sites that have ads enabled, yet still display nothing. The thing is, we don't really have an abundance of clients offering products that a  would find relevant, but Amazon has stand mixers, a selection of super fine chinois, deliciously sharp cutlery and enough cast iron to fill the Gulf Of Mexico with . Beginning as early as this week (week of 5 February 2018), our ad team is going to be hand selecting affiliate ads to be displayed on select sites. The use of inventory will of course depend on the availability of relevant ads to display which means:  Some sites might see very few, if any ads. Some topics simply don't come remotely close to orbiting anything worth advertising. Furthermore, while graduated, some sites simply don't have enough activity for ads to make sense.  Since these are hand-selected, it could take a while for ads to appear initially.  These would of course yield to inventory from clients if clients had products that were relevant to any given topic.   Nothing changes when it comes to the . If you're used to occasionally seeing a community promotion ad, you'll just notice that you'll also occasionally see an ad for something that is likely to be of interest to someone interested in the topic that your site covers. In fact, to be clear, this is simply a continuation of the announcement we made in November of 2016. We just wanted folks to know that we're going to be continuing the program where there's good potential to show relevant advertisements, even if we don't have clients lined up with a demand for it. Where can I give feedback about a certain ad on [site]? Open a meta post on your per-site meta and let us know. Please remember, the bulk of advertisements are an effort to get something small back from folks that hit the sites through search engines and are extremely unlikely to ever become contributors. Their soufflé stopped falling, and we'll probably never see them again. Maybe we can get a few bucks once they realized that copper ions help egg whites stiffen properly and buy a bowl or a proper bloom whisk. However, you're the experts on these topics and we're very likely going to listen to you. What sites, specifically? Are any sites exempt? If you're a former Stack Exchange 1.0 site and / or subsequently joined the network under a special charter, nothing changes - don't panic :) For the rest, we're just extending what we're currently doing with ads on graduated sites to also include those where there's an opportunity to make sensible use of affiliate ads. If a site has less than 15k (average) daily visits, or fewer than roughly 50k combined total posts - it might not yet be time to invest in selecting ads to be shown there. But, part of this exercise is also seeing what works best. I don't really have feedback, just some suggestions Ads aren't just an opportunity to sell stuff, they're an opportunity to really show people that we understand a given topic. That, of course, makes suggestions from people that do that stuff every day quite valuable. Post on your per-site meta and our community / ad critters will take a look. Questions? Comments? Kvetches? Let us know in an answer or comment below. If anything about the plan changes, we'll update this post to let folks know. Use a rectangular contour to evaluate the integral $$\int_{-\infty}^{\infty} \frac{\cos(mx) dx}{e^{-x}+e^x} = \frac{\pi}{e^{m\pi /2}+e^{-m\pi /2}}$$  I need to evaluate the above integral specifically using a rectangluar contour and at some point applying the residue theorem. I should be ok with the residue part, but I'm confused at to how to get this into the example form shown in the book:  $$I =\int_{-\infty}^{\infty} \frac{e^{ax}dx}{1+e^x}=\frac{\pi}{\sin(a\pi)} $$  I tried using the identity $\cos (mx) = \frac{e^{ix}+e^{-ix}}{2}$ but I end up with $\frac{1}{2} \int_{-\infty}^{\infty}\frac{e^{ix}(e^{imx}+e^{-imx})dx}{1+e^{2ix}}$, which almost resembles the example but not quite... Evaluate the integral $\int_{-1}^1\frac{dx}{(e^x+1)(x^2+1)}$. Evaluate:      $$\int_{-1}^1\frac{dx}{(e^x+1)(x^2+1)}$$ In my game, XP needed to reach the next level is Current Level × Level Threshold.  Given that, how can I get my current level from the total XP ever earned?    For example:  Level Threshold = 50 Current Level = 1   Starting at level 1, I would need (1 × 50) = 50 XP to get to level 2, and so on.  Level 1: 50 XP needed to reach level 2 Level 2: 100 more XP needed to reach level 3 Level 3: 150 more XP needed to reach level 4   In other words, the progression table is as follows:  Level 1: 0 XP to 49 XP Level 2: 50 XP to 149 XP  Level 3: 150 XP to 299 XP Level 4: 300 XP to 499 XP   If I have 300 XP, I'll just reached level 4. How can I calculate that generally? Contraction mapping in the context of $f(x_n)=x_{n+1}$. Induced metric on a null hypersurface Consider a metric $g_{\mu\nu}(x)$ and a hyper surface ${\cal H}$ defined by $~f(x) = c$. One (or at least I) usually finds the "induced metric" on ${\cal H}$ by solving$~f(x) = c$ for one of the coordinates and plugging it back into the metric to give us $\gamma_{ij}$. For null hypersurfaces, I seem to be finding that the resultant induced metric is singular.         Is it true in general that the induced metric as defined above is singular on a null hypersurface?   How does one define the induced metric on a null hypersurface? How to load up CSS files using Javascript? Is it possible to import css stylesheets into a html page using Javascript? If so, how can it be done?  P.S the javascript will be hosted on my site, but I want users to be able to put in the &lt;head&gt; tag of their website, and it should be able to import a css file hosted on my server into the current web page. (both the css file and the javascript file will be hosted on my server). First off, here is some code:  int main()  {     int days[] = {1,2,3,4,5};     int *ptr = days;     printf("%u\n", sizeof(days));     printf("%u\n", sizeof(ptr));      return 0; }   Is there a way to find out the size of the array that ptr is pointing to (instead of just giving its size, which is four bytes on a 32-bit system)? Why does Captain America's Shield make noise? How do I earn SC2's "Hurry up, it's raid night" achievement? The "Hurry up, it's raid night" achievement in Starcraft 2 involves "beating Wings of Liberty, on normal, in 8 hours or less."  Is there a recommended mission path to take? Which mission should I take for the branching paths? Does it count time spent on the Hyperion in between missions, or is it exclusively time elapsed on missions. Does replaying a mission affect the time? General tips / hints from those who've unlocked the achievement?  Edit: Just got this achievement tonight, finally! The answer was perfect. Is there difference in force when the height increases? Consider a man with mass 50kg.   When he jumps from a 2 meter height, the total force is $F=ma$ ie (mass * gravity), but nothing happens to him. When he jumps from a 50 meter height, the total force is (mass * gravity ) same, but he dies.   Why is this happening? Can't kill a sleeping process