Tricky proof involving limit points
Multiples of an irrational number forming a dense subset
Why aren't these loops homotopic? Let $S^1 = \{z \in \mathbb{C} : |z| = 1\}$.  Take the loops $f,g : [0,1] \rightarrow S^1$, $f(t) = 1$, $g(t) = e^{2\pi it}$.  I know these represent different elements in $\pi_1(S^1, 1)$, but I don't see why $F(t,s) = e^{2\pi its}$ isn't a homotopy between $f$ and $g$. How to find and replace using sed on records starting with Specific number I have forgotten my Facebook account password and also I have no access to the number which I use to log-in to Facebook account.   On this page (see the image below) they are asking for a government photo-ID to verify my account. The problem is that I don't have a driving license, voter ID card, passport etc. All I have is a school ID card but that hasn't worked for me. Is there any other way to recover my password? Characterization convex function. Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function such that for all $a,b\in [0,1]$ with $a&lt;b$ $$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_a^b f(x)\,dx.$$ How to prove that $f$ is a convex function? Thank you for your help. What statistical test can I use to compare two ratios from two independent samples. The ratios are after to before results. I need to compare the after/before ratios for two independent models and show whether they are have significant difference or not. Please help! What permissions should my website files/folders have on a Linux webserver? This is a  about File Permissions on a Linux web server.   I have a Linux web server running Apache2 that hosts several websites. Each website has its own folder in /var/www/.  /var/www/contoso.com/ /var/www/contoso.net/ /var/www/fabrikam.com/   The base directory /var/www/ is owned by root:root. Apache is running as www-data:www-data. The Fabrikam website is maintained by two developers, Alice and Bob. Both Contoso websites are maintained by one developer, Eve. All websites allow users to upload images. If a website is compromised, the impact should be as limited as possible.  I want to know the best way to set up permissions so that Apache can serve the content, the website is secure from attacks, and the developers can still make changes. One of the websites is structured like this:  /var/www/fabrikam.com     /cache     /modules     /styles     /uploads     /index.php   How should the permissions be set on these directories and files? I read somewhere that you should never use 777 permissions on a website, but I don't understand what problems that could cause. During busy periods, the website automatically caches some pages and stores the results in the cache folder. All of the content submitted by website visitors is saved to the uploads folder. How to bond two different internet connections I have two internet connection, each with 4mbps bandwidth. I want to join/merge both of them to get 8mbps speed, both routers have WLAN and LAN capability.    How to Double my speed using two routers? Can I bond two connections in Windows and double my internet connection speed? Any download manager that accepts two internet connection to download? How to open a new Terminal with shell prompt and running multiple commands in Command Line? I want to Open a new Terminal in from within terminal.  It should have Shell prompt as well as already run few commands.  How to do it ? jQuery .ready in a dynamically inserted iframe We are using jQuery  to dynamically display an iframe when someone clicks on a picture.  In this iframe, we are using  a javascript library to display multiple pictures.  The problem seems to be that $(document).ready in the iframe seems to be fired too soon and the iframe content isn't even loaded yet, so galleria code is not applied properly on the DOM elements.  $(document).ready seems to use the iframe parent ready state to decide if the iframe is ready.  If we extract the function called by document ready in a separate function and call it after a timeout of 100&nbsp;ms. It works, but we can't take the chance in production with a slow computer.  $(document).ready(function() { setTimeout(ApplyGalleria, 100); });   My question: which jQuery event should we bind to to be able to execute our code when the dynamic iframe is ready and not just it's a parent? C++11 does not deduce type when std::function or lambda functions are involved I want to do something like:  MyObject myObj = GetMyObj(); // Create and fill a new object MyObject newObj = myObj.Clone();   And then make changes to the new object that are not reflected in the original object.  I don't often need this functionality, so when it's been necessary, I've resorted to creating a new object and then copying each property individually, but it always leaves me with the feeling that there is a better or more elegant way of handling the situation.  How can I clone or deep copy an object so that the cloned object can be modified without any changes being reflected in the original object? How to remember whether box is checked or unchecked (on click) with localstorage and also save a value? How can I do a Cache Refresh in Google Chrome? I don't know what exactly it's called, by cache refresh I mean, refresh the page after clearing its cache. I don't want to clear the entire browser cache.  I can't seem to cache refresh my pages. In Firefox, I know it to be Shift+Refresh.  In Chrome, I've tried Ctrl+R, Ctrl+Refresh, Alt+Refresh, Shift+Refresh but none of them work. I have an iPhone 4s with iOS7 and somehow I see iCloud music which I haven't downloaded yet on my iOS device. How can I get rid of these, I do not need to see them. Volumetric Lighting is not working I have a simple setup to try out Volumetric Lighting. I have a spot lamp and Volume Scatter set for the World. The cube has a window for the light of the spot lamp.     But no Volumetric Lighting. I have read most questions on this subject. But I have no idea where I am going wrong. How can I convert monsters from D&D 3 or 3.5 to 2ed monsters? How to rename multiple files by removing the extension? I have a number of tiff files named:    sw.001.tif sw.002.tif ...   and I want to remove the .tif at the end of each of the files.  How can I use the rename command to do this? 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? /testfor for item enchantment What I have now is:  /testfor @a {SelectedItemSlot:0,Inventory:[{Slot:0b,id:"diamond_pickaxe"}]}   How can I use the /testfor command to test a specific slot for a specific item that is enchanted? How to find P(2X>4)? This is a Binomial question. I am required to find P(2X>4), and given that X~B(10,0.2), how do I convert it to 2X~B(n,p)?  Have tried converting from P(2X>4) to P(X>2), but understand that it doesn't work that way.  Do i have to use integration or any formula that i should refer?