Lower bound for cyclic expression in four variables
Prove that, $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\ge 4$ where we do not use AM-GM inequality on the given statement to prove it.
Please add a spell checker to the Markdown editor It would be nice to see a built-in spell checker into the main text-panel for posts. Every time I post here or Stack&nbsp;Overflow I quickly open up Gmail, paste my post there, and use its spell checker to make sure I didn't make any stupid mistakes. I know other services have these types of features, and it would be nice to see Stack&nbsp;Overflow have one as well.  Agree/disagree?  (BTW, I found two small mistakes for this short post.)  I know I made a comment about this on ChrisF's answer, but I thought that I should also mention it here in the question. I know there are many third-party tools to add spell checker functionality to Stack&nbsp;Overflow. But some (if not many) people out there do not have these tools. They are either at work (like me), or some other place where they using a machine that is not owned by them, or they aren't power users and just don't have these tools installed.  If Stack&nbsp;Exchange becomes used for other non-technical sites, then you will probably have to count on the fact that people are using  or something just as bad. That's why it would just be nice to have a built-in spelling tool. Again, it may not be worth it, but I think it would be interesting to discuss it at least. How could Thranduil have met a dragon? Show that computation of all such expressions will result in the same element of $X$ Creating a .desktop file for a new application How to add a small rounded rectangle around a word inline I want to add a box with a very light pink background and red border around a word but inline and  within a sentence.   i.e. something like this (drawn with inkscape)    I know the mdframed package can be used for placing boxes around paragraphs, but it does not seem to work inline and also give colored boxes.   Possibly TikZ is a solution? From the little I know of TikZ, it can be used to draw pictures inline inside a sentence. How can I define a mathematical function as a LaTeX macro? When would you call java's thread.run() instead of thread.start()? When would you call Java's thread.run() instead of thread.start()? unzip error "End-of-central-directory signature not found" I try to unzip a zip file, but got an error:  $ unzip COCR2_100.zip  Archive:  COCR2_100.zip   End-of-central-directory signature not found.  Either this file is not   a zipfile, or it constitutes one disk of a multi-part archive.  In the   latter case the central directory and zipfile comment will be found on   the last disk(s) of this archive. note:  COCR2_100.zip may be a plain executable, not an archive unzip:  cannot find zipfile directory in one of COCR2_100.zip or         COCR2_100.zip.zip, and cannot find COCR2_100.zip.ZIP, period.   I googled but didn't find a solution. I was wondering why it is and how I should fix it? Thanks!  The zip file can be downloaded from . It is an application, and here is its website .  My OS is 10.10. Searching for questions that have ONLY a given tag (i.e. add a range operator for tags) I would like to create an alias to rm command in order to have a confirmation message after executing this command. So I am creating an alias like this alias rm='rm -i'. But as far as I know this is a temporary alias and it lives until you close the terminal.  As it is explained  to save alias permanently I need to execute ~/.bash_aliases or ~/.bashrc commands in terminal and add my alias there. But when I execute ~/.bashrc I get following error message :  bash: /home/bakhtiyor/.bashrc: Permission denied   When I run ~/.bash_aliases I get another error message like this:  bash: /home/bakhtiyor/.bash_aliases: File or directory doesn't exist.   What is the actual problem and how can I solve it? Adding dynamic text to layout from attribute table with Data Driven Pages I was trying to figure out how to use dynamic text from an attribute table in a collection of Data Driven Pages in ArcGIS 10.   I've looked in the help files, but can't find anything on dynamic text from attribute tables. How does the size of insert affects the rate of Homologous Recombination in yeast? When performing genetic knockouts in yeast using homologous recombination to replace a target gene sequence via a vector DNA, does the region between the flanking regions in the vector have to be the same length as the site of recombination in the host cell's gene? Wine pollutes "Open With" application list The dialog box in question here is the one you get with the context menu option "open with other applications". Wine seems to have inserted more than a dozen or so entries for each application I install, which makes it a pain to find the correct application:     What can I do to remove the duplicates?     Update: Neither of the two solutions really work. The bug is interesting, but the symptoms does not match my problem (I'm not having problem with uninstalling applications, but rather the things that are inserted after installing them), and with the other one, all references to the Wine application are removed, which actually makes the problem worse (although it may be an acceptable solution if nothing else can be found). So this is still an open question; any takers? When is the limit of a sum equal to the sum of limits? I was trying to solve a problem and got stuck at the following step:  Suppose ${n \to \infty}$ .  $$\lim \limits_{n \to \infty} \frac{n^3}{n^3} = 1$$  Let us rewrite $n^3=n \cdot n^2$ as $n^2 + n^2 + n^2 + n^2 \dots +n^2$,$\space$ n times.  Now we have  $$\lim \limits_{n \to \infty} \frac{n^3}{n^3} = \frac {n^2 + n^2 + n^2 + n^2 + n^2 \dots +n^2}{n^3} $$  As far as I understand, we can always rewrite the limit of a sum as the sum of limits ...  $$\dots = \lim \limits_{n \to \infty} \left(\frac{n^2}{n^3} + \frac{n^2}{n^3} + \dots + \frac{n^2}{n^3}\right)$$  ...but we can only let ${n \to \infty}$ and calculate the limit if all of the individual limits are of defined form (is this correct?). That would be the case here, so we have:  $= \dots \lim \limits_{n \to \infty} \left(\frac{1}{n} + \frac{1}{n} + \dots + \frac{1}{n}\right) =$[ letting ${n \to \infty}]$ $= 0 + 0 + \dots + 0 = 0$  and the results we get are not the same.  Where did I go wrong? How to copy files that have too long of a filepath in Windows? What are the advantages of 10-bit monitors? To support 10-bit color the following are needed:   A monitor supporting it. A GPU supporting it (only AMD FirePro and NVIDIA Quadro support this?). Compatible software. Unless I am mistaken there are very few programs out there supporting 10-bit color. Photoshop is a notable example.   The questions are about how 10-bit monitors perform in comparison with 8-bit monitors:   In which situations would a 10-bit monitor give a noticeable advantage over an 8-bit monitor (say, for professional photography)? Have 10-bit monitors been compared against 8-bit monitors based on subjective or objective tests? What were the results? Human eyes can see only 10m colors, so would using a monitor with 1b colors make a difference? I have a table that I want to insert on a page, but at least one (perhaps both) of the following conditions are met:   The table is too wide to fit within the text block or page. That is, I'm exceeding some horizontal restriction. The table is too tall to fit within the text block or page. That is, I'm exceeding some vertical restriction.   What are my options to make this table fit? If it doesn't fit, regardless of my attempts, what other options exist? Moving linux install to a new computer I know that it can, in some circumstances, be difficult to move a Windows installation from one computer to another (physically move the hard drive), but how does that work on linux? Aren't most of the driver modules loaded at bootup? So theoretically would it be that much of a hassle?  Obviously xorg configs would change and proprietary ATI drivers and such would have to be recompiled (maybe?). Is there more to it than I'm thinking of?  Assume the 2 computers are from the same era, ie both i7s but slightly different hardware.  Update: Thanks for the answers. This is mostly for my own curiosity. I have my linux system up and running at work, but eventually I'd like to move to a computer that I can get dual video cards into so I can run more than 2 monitors. But not any time soon I just read  on EE.SE and that triggered a doubt in me.  If I were asked whether there is any relationship between the electrical resistivity and the density of a substance I'd answer negatively, but is it really so?   Is there really no (relevant) relationship between those two bulk properties of a material? Can anyone confirm or disprove this and give some explanation thereof? Sometime I see many application such as msn, windows media player etc that are single instance applications (when user executes while application is running a new application instance will not created).  In C#, I use Mutex class for this but I don't know how to do this in Java.