Show that $\dfrac{\phi (n)}{n}=\sum\limits_{d|n} \dfrac{\mu (d)}{d}$
Prove that, for every natural n, $\sum_{d|n} \frac{\mu(d)}{d}=\prod_{p|n}(1-\frac{1}{p})$
single email limit exceeded salesforce very frequently..? Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$ Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element? Restoring a backup to an older version of SQL Server When trying to restore a backup to a SQL Server Express 2008 database I got the following error:  Restore failed for Server '...\SQLEXPRESS'.  (Microsoft.SqlServer.SmoExtended) System.Data.SqlClient.SqlError: The database was backed up on a server running version 10.50.1600.  That version is incompatible with this server, which is running version 10.00.2531.  Either restore the database on a server that supports the backup, or use a backup  that is compatible with this server. (Microsoft.SqlServer.Smo)   Is there a way to get a backup which is compatible with the older (in my case 10.00.2531) version from the newer (in my case 10.50.1600) version of SQL Server Express? Let $p,q$ be primes, $p≠q$, then I have to show that $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$  So far I've tried a lot of things with minimal polynomials and bases, but I'm far from any argument that I find sufficient.  This is an example from my algebra course, first chapter about field extension. So far we know about basic definitions, minimal polynomials and their properties, algebraic elements and algebraic fields.  Could someone maybe give me a hint about how to proceed? Thanks a lot!!! What's the difference between Google Chrome and/or Chromium? What are the advantages/disadvantages to each? I used Google Chrome in Windows, and am now using Chromium in Ubuntu.  Is there any difference to the two programs?  What are the advantages and disadvantages to each program?  Which one seems better? Gauss's mean value theorem problem Suppose that  $f(z)$ is analytic and $f(z)\neq0$  on the disc $|z|&lt;1$ .  Show that for $0&lt;r&lt;1$,  $$ \exp\left[\frac{1}{2\pi}\int_{0}^{2\pi}{\log}|f(re^{i\theta})|\,d\theta\right] = |f(0)| $$  Any other theorems needed to solve this problem? I don't have any idea just with "Gauss's MVT"  Thank you in advanced. How should we deal with duplicate answers? Duplicate questions are "solved" &#8212; it's clear what to do with them, there're tools to deal with them... but what happens when two users post the same answer? This happens a fair bit, particularly on questions with simple answers ("Click this button"-type ones).  Do we upvote both answers, since they are both correct? Do we upvote the first one (although if the answers were posted within a few minutes of each other, after an hour it's impossible to tell as they both show up as "answered 1 hour ago")? Do we upvote the currently highest voted one? Do we upvote one and downvote the other? Do we flag one (seems a little excessive)? Something else? "Happen to know" meaning "Do you happen to know is if she has a boyfriend?"   What does the above sentence means . I heard those line in the movie 'the social network' and I would like to know when do we use this phrase "happen to know" . Smooth option sometimes produces incorrect arrow tips in PGFplots Why is "improve this question / answer" shown to anonymous users instead of "edit"? If I browse a site as an anonymous user, instead of "edit" I see: "improve this question / answer".        I don't get it, are anonymous users considered to be kinda dumb and don't understand "edit"?  Why to reword this action? Representing network flows? For a project I'm working on, I need to represent network flows over 1,000 zones with directionality.    Any suggestions on displaying the information in a nice and non overwhelming way?  Any examples on displaying network flows in heavy urban areas? Color suggestions and thickness schemes? any "not to dos" when representing network flows? Is there a theorem that says that QFT reduces to QM in a suitable limit? Of course, it should be, as QFT is relativisitc quantum mechanics. But, is there a more manifest one? such as Ehrenfest's theorem which dictates the transition from QM to Classical Mechanics.  I am curious as the basic object in QM is wave function whereas the basic object in QFT is quantized operator, which looks very different, and wondering how the comparison works.  (Of course, I am aware that QFT can calculate corrections for Coulomb potential energy, which may seem to be a transition from QFT to QM, but I want one more manifest than that.) Working with IPv6 Addresses in PHP After searching around somewhat thoroughly, I noticed a slight lack of functions in PHP for handling . For my own personal satisfaction I created a few functions to help the transition.  The IPv6ToLong() function is a temporary solution to that brought up here: . It will split the IP in to two integers and return them in an array.  /**  * Convert an IPv4 address to IPv6  *  * @param string IP Address in dot notation (192.168.1.100)  * @return string IPv6 formatted address or false if invalid input  */ function IPv4To6($Ip) {     static $Mask = '::ffff:'; // This tells IPv6 it has an IPv4 address     $IPv6 = (strpos($Ip, '::') === 0);     $IPv4 = (strpos($Ip, '.') &gt; 0);      if (!$IPv4 &amp;&amp; !$IPv6) return false;     if ($IPv6 &amp;&amp; $IPv4) $Ip = substr($Ip, strrpos($Ip, ':')+1); // Strip IPv4 Compatibility notation     elseif (!$IPv4) return $Ip; // Seems to be IPv6 already?     $Ip = array_pad(explode('.', $Ip), 4, 0);     if (count($Ip) &gt; 4) return false;     for ($i = 0; $i &lt; 4; $i++) if ($Ip[$i] &gt; 255) return false;      $Part7 = base_convert(($Ip[0] * 256) + $Ip[1], 10, 16);     $Part8 = base_convert(($Ip[2] * 256) + $Ip[3], 10, 16);     return $Mask.$Part7.':'.$Part8; }  /**  * Replace '::' with appropriate number of ':0'  */ function ExpandIPv6Notation($Ip) {     if (strpos($Ip, '::') !== false)         $Ip = str_replace('::', str_repeat(':0', 8 - substr_count($Ip, ':')).':', $Ip);     if (strpos($Ip, ':') === 0) $Ip = '0'.$Ip;     return $Ip; }  /**  * Convert IPv6 address to an integer  *  * Optionally split in to two parts.  *  * @see https://stackoverflow.com/questions/420680/  */ function IPv6ToLong($Ip, $DatabaseParts= 2) {     $Ip = ExpandIPv6Notation($Ip);     $Parts = explode(':', $Ip);     $Ip = array('', '');     for ($i = 0; $i &lt; 4; $i++) $Ip[0] .= str_pad(base_convert($Parts[$i], 16, 2), 16, 0, STR_PAD_LEFT);     for ($i = 4; $i &lt; 8; $i++) $Ip[1] .= str_pad(base_convert($Parts[$i], 16, 2), 16, 0, STR_PAD_LEFT);      if ($DatabaseParts == 2)             return array(base_convert($Ip[0], 2, 10), base_convert($Ip[1], 2, 10));     else    return base_convert($Ip[0], 2, 10) + base_convert($Ip[1], 2, 10); }   For these functions I typically implement them by calling this function first:  /**  * Attempt to find the client's IP Address  *  * @param bool Should the IP be converted using ip2long?  * @return string|long The IP Address  */ function GetRealRemoteIp($ForDatabase= false, $DatabaseParts= 2) {     $Ip = '0.0.0.0';     if (isset($_SERVER['HTTP_CLIENT_IP']) &amp;&amp; $_SERVER['HTTP_CLIENT_IP'] != '')         $Ip = $_SERVER['HTTP_CLIENT_IP'];     elseif (isset($_SERVER['HTTP_X_FORWARDED_FOR']) &amp;&amp; $_SERVER['HTTP_X_FORWARDED_FOR'] != '')         $Ip = $_SERVER['HTTP_X_FORWARDED_FOR'];     elseif (isset($_SERVER['REMOTE_ADDR']) &amp;&amp; $_SERVER['REMOTE_ADDR'] != '')         $Ip = $_SERVER['REMOTE_ADDR'];     if (($CommaPos = strpos($Ip, ',')) &gt; 0)         $Ip = substr($Ip, 0, ($CommaPos - 1));      $Ip = IPv4To6($Ip);     return ($ForDatabase ? IPv6ToLong($Ip, $DatabaseParts) : $Ip); }   Someone please tell me if I'm reinventing the wheel here or I've done something wrong.  This implementation converts IPv4 to IPv6. Any IPv6 address it doesn't touch. possessive apostrophe for two things or persons What is the difference between the two sentences below? Are they the same or different in meaning?     Jack and Laura's new house is almost ready.      Jack's and Laura's new house is almost ready.   To my perception, the first says two of them have one house and the second says each one has a (different) house (which is almost ready). Is my perception correct?  EDIT: If I don't use "houses" instead of "house" (for my second interpretation), does it mean the sentence is wrong? "To your left" versus "on your left" There is a security camera on the building to your left      There is a security camera on the building on your left   Is there any real difference in meaning here? Which sounds more natural to native speakers? Creation of a matrix Can I transfer saved data from my old PS3 hard drive to my new PS3? A while ago, my PS3 completely broke. Every time I turn it on, it just completely shuts down a few seconds later. I decided to buy a new ps3 to play my old games. The problem is that I don't have my saved data, but I still have the original hard drive. Is it possible to somehow transfer the saved data from the old hard drive to my new PS3? I was thinking maybe I could clone the hard drive, but I don't have a spare hard drive to do it on. I do however have a 2 TB external hard drive. Would it be possible to clone it on the external? Are there any alternative methods? How to display Trash and Home folder icons? I've googled quite a bit but nothing I've found works. I just want to display the Trash, Home folder, Computer etc icons on my Desktop in Ubuntu.  The closest I've come is installing dconf-tools, running it, going to org -&gt; gnome -&gt; desktop -&gt; background and checking show-desktop-icons then org -&gt; gnome -&gt; nautilus -&gt; desktop and checking the icons I want to show.  Nothing happens.  I've also tried looking in compizconfig-settings-manager and gconf-editor but can't find anything there.  Ideas? One of the mysteries that my numerics class left me with are "stiff systems". Our prof didn't really manage to explain what they are. Now that the class (and the test) are over, I'm still curious.   The most concise definition about "stiff systems" I've found so far is: "There is no universally accepted definition of stiffness". So, if there is none, I don't expect one.   I'd be thankful about information on both of the following points:   From what kind of (real-world) problems do stiff systems arise? What's so "stiff" about them? What's the good thing - or the bad thing - that I'd worry about when I have to numerically solve such a real-world problem? Limit of $\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} }$ Direct way leads to nowhere.  $$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} } = \lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{n \sqrt {1 +\frac{i}{n^2}} } = $$  $$ = \lim_{n\rightarrow \infty} \frac{1}{n} \cdot \lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {1 +\frac{i}{n^2}} } = 0 \cdot \sum^{n}_{i=1} 1 = 0 \cdot n $$  So, I tried to apply squeeze theorem:  $$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} } \leq \lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{n}  = 1$$  Could you help me to figure out the lower bound?